Package {cograph}

cograph: Modern Network Visualization for R

Description

A modern, extensible network visualization package that provides high-quality static network plots and ggplot2 conversions. cograph accepts adjacency matrices, edge lists, or igraph objects and offers customizable layouts, node shapes, edge styles, and themes.

Main Functions

• cograph: Main entry point for creating network visualizations

• sn_layout: Apply layout algorithms

• sn_nodes: Customize node aesthetics

• sn_edges: Customize edge aesthetics

• sn_theme: Apply visual themes

• sn_render: Render to device

• sn_ggplot: Convert to ggplot2 object

Layouts

cograph provides several built-in layouts:

• circle: Nodes arranged in a circle

• spring: Fruchterman-Reingold force-directed layout

• groups: Group-based circular layout

• custom: User-provided coordinates

Themes

Built-in themes include:

• classic: Traditional network visualization style

• colorblind: Accessible color scheme

• gray: Grayscale theme

• dark: Dark background theme

• minimal: Clean, minimal style

• viridis: Viridis-based color theme

• nature: Nature-inspired color theme

Weight conventions

cograph's analytic functions follow a single convention for edge weights:

• Semantics. A weight is a strength: higher weight means a stronger connection (larger transition probability, thicker correlation, stronger tie). This matches the qgraph / tna convention and the intuition of most user-facing inputs.

• Path-based measures (betweenness, closeness, harmonic, eccentricity, stress, load, radiality, etc.) invert weights to distances via 1 / weight ^ alpha. The alpha argument (default 1) tunes how strongly weight differences compress paths. Controlled by the invert_weights argument, which auto-detects to TRUE for tna objects and FALSE for matrices/igraph (matching native igraph / sna defaults).

• Non-path measures (degree, strength, eigenvector, PageRank, transitivity, modularity, ...) use the raw weights as-is without inversion.

• Unweighted override. Passing weights = NA to any analytic function forces unweighted behavior regardless of what is attached to the graph.

Individual functions may document exceptions in their own help pages. Any deviation from this convention is a bug — please report.

Author(s)

Maintainer: Sonsoles López-Pernas sonsoles.lopez@uef.fi [copyright holder]

Authors:

• Mohammed Saqr [copyright holder]

See Also

Useful links:

• https://sonsoles.me/cograph/

• https://github.com/sonsoleslp/cograph

• Report bugs at https://github.com/sonsoleslp/cograph/issues

CographLayout R6 Class

Description

Class for managing layout algorithms and computing node positions.

Value

A CographLayout R6 object.

Methods

Public methods

• CographLayout$new()

• CographLayout$compute()

• CographLayout$normalize_coords()

• CographLayout$get_type()

• CographLayout$get_params()

• CographLayout$print()

• CographLayout$clone()

Method new()

Create a new CographLayout object.

Usage

CographLayout$new(type = "circle", ...)

Arguments

Returns

A new CographLayout object.

Method compute()

Compute layout coordinates for a network.

Usage

CographLayout$compute(network, ...)

Arguments

Returns

Data frame with x, y coordinates.

Method normalize_coords()

Normalize coordinates to 0-1 range with padding.

Usage

CographLayout$normalize_coords(coords, padding = 0.1)

Arguments

Returns

Normalized coordinates.

Method get_type()

Get layout type.

Usage

CographLayout$get_type()

Returns

Character string.

Method get_params()

Get layout parameters.

Usage

CographLayout$get_params()

Returns

List of parameters.

Method print()

Print layout summary.

Usage

CographLayout$print()

Returns

The object itself, invisibly.

Method clone()

The objects of this class are cloneable with this method.

Usage

CographLayout$clone(deep = FALSE)

Arguments

Examples

# Create a circular layout
layout <- CographLayout$new("circle")

# Apply to network
adj <- matrix(c(0, 1, 1, 1, 0, 1, 1, 1, 0), nrow = 3)
net <- CographNetwork$new(adj)
coords <- layout$compute(net)

CographNetwork R6 Class

Description

Core class representing a network for visualization. Stores nodes, edges, layout coordinates, and aesthetic mappings.

Value

A CographNetwork R6 object.

Active bindings

Methods

Public methods

• CographNetwork$new()

• CographNetwork$clone_network()

• CographNetwork$set_nodes()

• CographNetwork$set_edges()

• CographNetwork$set_directed()

• CographNetwork$set_weights()

• CographNetwork$set_layout_coords()

• CographNetwork$set_node_aes()

• CographNetwork$set_edge_aes()

• CographNetwork$set_theme()

• CographNetwork$get_nodes()

• CographNetwork$get_edges()

• CographNetwork$get_layout()

• CographNetwork$get_node_aes()

• CographNetwork$get_edge_aes()

• CographNetwork$get_theme()

• CographNetwork$set_layout_info()

• CographNetwork$get_layout_info()

• CographNetwork$set_plot_params()

• CographNetwork$get_plot_params()

• CographNetwork$print()

• CographNetwork$clone()

Method new()

Create a new CographNetwork object.

Usage

CographNetwork$new(
  input = NULL,
  directed = NULL,
  nodes = NULL,
  simplify = FALSE
)

Arguments

Returns

A new CographNetwork object.

Method clone_network()

Clone the network with optional modifications.

Usage

CographNetwork$clone_network()

Returns

A new CographNetwork object.

Method set_nodes()

Set nodes data frame.

Usage

CographNetwork$set_nodes(nodes)

Arguments

Returns

The object itself, invisibly.

Method set_edges()

Set edges data frame.

Usage

CographNetwork$set_edges(edges)

Arguments

Returns

The object itself, invisibly.

Method set_directed()

Set directed flag.

Usage

CographNetwork$set_directed(directed)

Arguments

Returns

The object itself, invisibly.

Method set_weights()

Set edge weights.

Usage

CographNetwork$set_weights(weights)

Arguments

Returns

The object itself, invisibly.

Method set_layout_coords()

Set layout coordinates.

Usage

CographNetwork$set_layout_coords(coords)

Arguments

Returns

The object itself, invisibly.

Method set_node_aes()

Set node aesthetics.

Usage

CographNetwork$set_node_aes(aes)

Arguments

Returns

The object itself, invisibly.

Method set_edge_aes()

Set edge aesthetics.

Usage

CographNetwork$set_edge_aes(aes)

Arguments

Returns

The object itself, invisibly.

Method set_theme()

Set theme.

Usage

CographNetwork$set_theme(theme)

Arguments

Returns

The object itself, invisibly.

Method get_nodes()

Get nodes data frame.

Usage

CographNetwork$get_nodes()

Returns

Data frame with node information.

Method get_edges()

Get edges data frame.

Usage

CographNetwork$get_edges()

Returns

Data frame with edge information.

Method get_layout()

Get layout coordinates.

Usage

CographNetwork$get_layout()

Returns

Data frame with x, y coordinates.

Method get_node_aes()

Get node aesthetics.

Usage

CographNetwork$get_node_aes()

Returns

List of node aesthetic parameters.

Method get_edge_aes()

Get edge aesthetics.

Usage

CographNetwork$get_edge_aes()

Returns

List of edge aesthetic parameters.

Method get_theme()

Get theme.

Usage

CographNetwork$get_theme()

Returns

CographTheme object.

Method set_layout_info()

Set layout info.

Usage

CographNetwork$set_layout_info(info)

Arguments

Returns

The object itself, invisibly.

Method get_layout_info()

Get layout info.

Usage

CographNetwork$get_layout_info()

Returns

List with layout information.

Method set_plot_params()

Set plot parameters.

Usage

CographNetwork$set_plot_params(params)

Arguments

Returns

The object itself, invisibly.

Method get_plot_params()

Get plot parameters.

Usage

CographNetwork$get_plot_params()

Returns

List of plot parameters.

Method print()

Print network summary.

Usage

CographNetwork$print()

Returns

The object itself, invisibly.

Method clone()

The objects of this class are cloneable with this method.

Usage

CographNetwork$clone(deep = FALSE)

Arguments

Examples

# Create network from adjacency matrix
adj <- matrix(c(0, 1, 1, 1, 0, 1, 1, 1, 0), nrow = 3)
net <- CographNetwork$new(adj)

# Access properties
net$n_nodes
net$n_edges
net$is_directed

CographTheme R6 Class

Description

Class for managing visual themes for network plots.

Value

A CographTheme R6 object.

Active bindings

Methods

Public methods

• CographTheme$new()

• CographTheme$get()

• CographTheme$set()

• CographTheme$get_all()

• CographTheme$merge()

• CographTheme$clone_theme()

• CographTheme$print()

• CographTheme$clone()

Method new()

Create a new CographTheme object.

Usage

CographTheme$new(
  name = "custom",
  background = "white",
  node_fill = "#4A90D9",
  node_border = "#2C5AA0",
  node_border_width = 1,
  edge_color = "gray50",
  edge_positive_color = "#2E7D32",
  edge_negative_color = "#C62828",
  edge_width = 1,
  label_color = "black",
  label_size = 10,
  title_color = "black",
  title_size = 14,
  legend_background = "white"
)

Arguments

Returns

A new CographTheme object.

Method get()

Get a theme parameter.

Usage

CographTheme$get(name)

Arguments

Returns

Parameter value.

Method set()

Set a theme parameter.

Usage

CographTheme$set(name, value)

Arguments

Returns

The object itself, invisibly.

Method get_all()

Get all theme parameters.

Usage

CographTheme$get_all()

Returns

List of parameters.

Method merge()

Merge with another theme.

Usage

CographTheme$merge(other)

Arguments

Returns

A new merged CographTheme.

Method clone_theme()

Clone the theme.

Usage

CographTheme$clone_theme()

Returns

A new CographTheme.

Method print()

Print theme summary.

Usage

CographTheme$print()

Returns

The object itself, invisibly.

Method clone()

The objects of this class are cloneable with this method.

Usage

CographTheme$clone(deep = FALSE)

Arguments

Examples

# Create a custom theme
theme <- CographTheme$new(
  background = "white",
  node_fill = "steelblue",
  edge_color = "gray60"
)

Abbreviate Labels

Description

Abbreviates labels to a maximum length, adding ellipsis if truncated.

Usage

abbrev_label(label, abbrev = NULL, n_labels = NULL)

label_abbrev(label, abbrev = NULL, n_labels = NULL)

Arguments

Value

Character vector of (possibly abbreviated) labels.

Examples

labels <- c("VeryLongStateName", "Short", "AnotherLongName")

# No abbreviation
abbrev_label(labels, NULL)

# Fixed max length
abbrev_label(labels, 5)  # "Very…", "Short", "Anot…"

# Auto-adaptive
abbrev_label(labels, "auto")

Add Edges to a Network

Description

Add Edges to a Network

Usage

add_edges(x, from, to, weight = 1, ..., keep_format = FALSE, directed = NULL)

Arguments

Value

A cograph_network with the new edges, or the input format when keep_format = TRUE. An edge that already exists has its weight replaced, and a cograph_edges_replaced warning says how many.

Note

When the igraph package is attached it masks this function with igraph::add_edges(), which takes an igraph object. Use cograph::add_edges() to be explicit.

See Also

remove_edges, add_nodes, bind_networks

Examples

adj <- matrix(0, 3, 3, dimnames = list(LETTERS[1:3], LETTERS[1:3]))
adj["A", "B"] <- adj["B", "A"] <- 1

add_edges(adj, from = "B", to = "C", weight = 0.5)

Add Nodes to a Network

Description

Add Nodes to a Network

Usage

add_nodes(x, labels, ..., keep_format = FALSE, directed = NULL)

Arguments

Value

A cograph_network with the new nodes appended (isolated until edges are added), or the input format when keep_format = TRUE.

See Also

remove_nodes, add_edges, mutate_nodes

Examples

adj <- matrix(c(0, 1, 1, 0), 2, 2)
rownames(adj) <- colnames(adj) <- c("A", "B")

add_nodes(adj, labels = c("C", "D"))
add_nodes(adj, labels = "C", group = "new")

Edge Aesthetics

Description

Functions for setting edge aesthetic properties.

Node Aesthetics

Description

Functions for setting node aesthetic properties.

Aggregate Layers

Description

Combines multiple network layers into a single network.

Usage

aggregate_layers(
  layers,
  method = c("sum", "mean", "max", "min", "union", "intersection"),
  weights = NULL
)

lagg(
  layers,
  method = c("sum", "mean", "max", "min", "union", "intersection"),
  weights = NULL
)

Arguments

Value

Aggregated adjacency matrix

Examples

nodes <- c("A", "B", "C")
l1 <- matrix(c(0, 1, 0, 1, 0, 1, 0, 1, 0), 3, 3, dimnames = list(nodes, nodes))
l2 <- matrix(c(0, 1, 1, 1, 0, 0, 1, 0, 0), 3, 3, dimnames = list(nodes, nodes))
layers <- list(L1 = l1, L2 = l2)

aggregate_layers(layers, "sum")           # total edge weight
aggregate_layers(layers, "mean")          # average edge weight
aggregate_layers(layers, "union")         # edge present in any layer
aggregate_layers(layers, "intersection")  # edge present in every layer

Aggregate Edge Weights

Description

Aggregates a vector of edge weights using various methods. Compatible with igraph's edge.attr.comb parameter.

Usage

aggregate_weights(w, method = "sum", n_possible = NULL)

wagg(w, method = "sum", n_possible = NULL)

Arguments

Value

A single numeric value, or 0 when no non-zero, non-NA weight remains.

Examples

w <- c(0.5, 0.8, 0.3, 0.9)
aggregate_weights(w, "sum")   # 2.5
aggregate_weights(w, "mean")  # 0.625
aggregate_weights(w, "max")   # 0.9

Motif Results as a Data Frame

Description

Returns the tables held by a motif result from motifs or subgraphs as tidy data frames.

Usage

## S3 method for class 'cograph_motif_result'
as.data.frame(
  x,
  row.names = NULL,
  optional = FALSE,
  ...,
  what = c("results", "types")
)

Arguments

Value

A data.frame. For what = "results" in a census, the columns are type and count, plus expected, z, p and sig when significance was tested. For subgraphs(), the columns are triad, node1, node2, node3, type and observed, plus the significance columns when tested. For what = "types", the columns are type and count.

See Also

motifs, subgraphs

Examples

census <- motifs(regulation_net, significance = FALSE)
as.data.frame(census)
as.data.frame(census, what = "types")

Cograph Network as a Data Frame

Description

The tidy accessor for a cograph_network: one row per edge (or per node), with endpoints given as labels rather than internal indices, so no caller has to reach into the object with $ or translate integer ids by hand.

Usage

## S3 method for class 'cograph_network'
as.data.frame(
  x,
  row.names = NULL,
  optional = FALSE,
  ...,
  what = c("edges", "nodes")
)

Arguments

Value

A base data frame. For what = "edges", one row per edge with columns from and to (node labels), weight, and any extra edge columns the network carries (for example session). For what = "nodes", one row per node with the node metadata columns (id, label, layout coordinates, and any custom columns).

This is the accessor, so it hands back everything the object holds, including columns mutate_edges computed. to_df is the narrower conversion verb: it returns from, to and weight only.

See Also

to_df, get_edges, get_nodes

Examples

adj <- matrix(c(0, .5, .8, 0,
                .5, 0, .3, .6,
                .8, .3, 0, .4,
                 0, .6, .4, 0), 4, 4, byrow = TRUE)
rownames(adj) <- colnames(adj) <- c("A", "B", "C", "D")
net <- as_cograph(adj)

as.data.frame(net)
as.data.frame(net, what = "nodes")

Convert to Cograph Network

Description

Creates a lightweight cograph_network object from various network inputs. The resulting object is a named list with all data accessible via $.

Usage

as_cograph(x, directed = NULL, simplify = FALSE, ...)

to_cograph(x, directed = NULL, ...)

Arguments

Details

The cograph_network format is designed to be:

• Lean: Only essential data stored, computed values derived on demand

• Modern: Uses named list elements instead of attributes for clean $ access

• Compatible: Works seamlessly with splot() and other cograph functions

Producer packages may attach optional plotting hints under meta$splot. The recognized fields are renderer (which cograph renderer to use), weight (the edge column or matrix to render as weight), and defaults (a named list of renderer arguments). Entries in defaults are defaults only — user-supplied arguments to splot always override them. renderer and weight define which view is rendered and are not overridden by plot arguments.

Use getter functions for programmatic access: get_nodes, get_edges, get_labels, n_nodes, n_edges

Use setter functions to modify: set_nodes, set_edges, set_layout

Value

A cograph_network object: a named list with components:

nodes

Data frame with id, label, and optional layout or metadata columns

edges

Data frame with from, to, weight columns

directed

Logical indicating if network is directed

weights

Full n×n weight matrix when available for matrix/TNA round-trips, or NULL

data

Original estimation data (sequence matrix, edge list, etc.), or NULL

meta

Consolidated metadata list with sub-fields: source (input type string), layout (layout info list or NULL), tna (TNA metadata or NULL), and optionally splot (producer-supplied rendering hints read by splot)

node_groups

Optional node groupings data frame

A cograph_network object. See as_cograph.

See Also

get_nodes to extract the nodes data frame, get_edges to extract edges as a data frame, n_nodes and n_edges for counts, is_directed to check directedness, splot for plotting

Examples

mat <- matrix(c(0, 1, 1, 1, 0, 1, 1, 1, 0), nrow = 3)
net <- as_cograph(mat)
get_nodes(net)
get_edges(net)
splot(net)
mat <- matrix(c(0, 1, 1, 1, 0, 1, 1, 1, 0), nrow = 3)
net <- to_cograph(mat)

Convert to mcml

Description

Convert various objects to the mcml class – a clean, tna-independent representation of a multilayer cluster network.

Usage

as_mcml(x, ...)

## S3 method for class 'cluster_summary'
as_mcml(x, ...)

## S3 method for class 'group_tna'
as_mcml(x, clusters = NULL, method = "sum", type = "tna", directed = TRUE, ...)

## S3 method for class 'mcml'
as_mcml(x, ...)

## Default S3 method:
as_mcml(x, ...)

Arguments

Value

An mcml object with components macro, clusters, cluster_members, and meta.

An mcml object.

An mcml object. When clusters is provided, macro$data contains the cluster assignments and macro$weights is NULL (the macro is the sequence of clusters, not a summary).

The input mcml object unchanged.

See Also

summarize_clusters, as_tna

Examples

# From cluster_summary
mat <- matrix(c(0.5, 0.2, 0.3,
                0.1, 0.6, 0.3,
                0.4, 0.1, 0.5), 3, 3, byrow = TRUE,
              dimnames = list(c("A", "B", "C"), c("A", "B", "C")))
clusters <- list(G1 = c("A", "B"), G2 = c("C"))
cs <- csum(mat, clusters, type = "tna")
m <- as_mcml(cs)
m$macro$weights

Convert cluster_summary to tna Objects

Description

Converts a cluster_summary object to proper tna objects that can be used with all functions from the tna package. Creates a macro (cluster-level) tna model and per-cluster tna models (internal transitions within each cluster), returned as a flat group_tna object.

Usage

as_tna(x)

## S3 method for class 'cluster_summary'
as_tna(x)

## S3 method for class 'mcml'
as_tna(x)

## Default S3 method:
as_tna(x)

Arguments

Details

This is the final step in the MCML workflow, enabling full integration with the tna package for centrality analysis, bootstrap validation, permutation tests, and visualization.

Requirements

The tna package must be installed. If not available, the function throws an error with installation instructions.

Workflow

# Full MCML workflow
net <- cograph(edges, nodes = nodes)
net$nodes$clusters <- group_assignments
cs <- csum(net, type = "tna")
tna_models <- as_tna(cs)

# Now use tna package functions
plot(tna_models$macro)
tna::centralities(tna_models$macro)
tna::bootstrap(tna_models$macro, iter = 1000)

# Analyze per-cluster patterns
plot(tna_models$ClusterA)
tna::centralities(tna_models$ClusterA)

Excluded Clusters

A per-cluster tna cannot be created when:

• The cluster has only 1 node (no internal transitions possible)

• Some nodes in the cluster have no outgoing edges (row sums to 0)

These clusters are left out of the result with a warning of class cograph_cluster_dropped, which names each cluster and the nodes that have no transition within it. The macro (cluster-level) model still includes all clusters.

Value

A group_tna object (S3 class) – a flat named list of tna objects. The first element is named "macro" and represents the cluster-level transitions. Subsequent elements are named by cluster name and represent internal transitions within each cluster.

macro

A tna object representing cluster-level transitions. Contains $weights (k x k transition matrix), $inits (initial distribution), and $labels (cluster names). Use this for analyzing how learners/entities move between high-level groups or phases.

<cluster_name>

Per-cluster tna objects, one per cluster. Each tna object represents internal transitions within that cluster. Contains $weights (n_i x n_i matrix), $inits (initial distribution), and $labels (node labels). A cluster that cannot become a tna model is left out with a warning (see Excluded Clusters).

A group_tna object (flat list of tna objects: macro + per-cluster).

A group_tna object (flat list of tna objects: macro + per-cluster).

A tna object constructed from the input.

See Also

csum to create the input object, plot_mcml for visualization without conversion, tna::tna for the underlying tna constructor

Examples

clusters <- list(C1 = c("Explore", "Reflect", "Discuss"),
                 C2 = c("Plan", "Create", "Share"),
                 C3 = c("Monitor", "Adapt", "Synthesize", "Evaluate"))
cs <- csum(regulation_net, clusters, type = "tna")
tna_models <- as_tna(cs)
names(tna_models)          # "macro", "C1", "C2", "C3"
splot(tna_models$macro)    # cograph renderer avoids tna's plot deps

Degree Assortativity Coefficient

Description

Computes the degree assortativity coefficient, measuring the tendency of nodes to connect to other nodes with similar degree. Positive values indicate assortative mixing (high-degree nodes connect to high-degree nodes), negative values indicate disassortative mixing.

Usage

assortativity(x, directed = NULL, type = NULL, digits = NULL, ...)

Arguments

Details

The degree assortativity coefficient is defined as the Pearson correlation coefficient between the degrees of nodes at either end of each edge (Newman 2002):

r = \frac{\sum_{jk} jk(e_{jk} - q_j q_k)}{\sigma_q^2}

where e_{jk} is the fraction of edges connecting degree-j to degree-k vertices, q_k is the excess degree distribution, and \sigma_q^2 its variance.

Because the Pearson correlation is invariant to subtracting a constant, the implementation computes the correlation of the raw (rather than excess) degrees at the two ends of each edge, counting every undirected edge in both orientations; this is numerically identical to the formula above.

For directed networks, the coefficient is the Pearson correlation between the source-end and target-end degrees over each edge in its stored orientation, with the degree mode at each end chosen by type (Foster et al. 2010).

The coefficient is NA when the network has no edges or when either degree vector has zero variance.

Value

An object of class "cograph_assortativity" with components:

coefficient

Numeric scalar: the assortativity coefficient in [-1, 1].

type

Character: the degree type used.

directed

Logical: whether the network was treated as directed.

n_nodes

Integer: number of nodes.

n_edges

Integer: number of edges.

network

The original input network.

References

Newman, M.E.J. (2002). Assortative mixing in networks. Physical Review Letters, 89(20), 208701. doi:10.1103/PhysRevLett.89.208701

Foster, J.G., Foster, D.V., Grassberger, P., & Paczuski, M. (2010). Edge direction and the structure of networks. PNAS, 107(24), 10815-10820. doi:10.1073/pnas.0912671107

See Also

assortativity_attribute, centrality, network_summary

Examples

# Assortative network (high-degree connect to high-degree)
adj <- matrix(c(
  0, 1, 1, 1, 0,
  1, 0, 1, 1, 0,
  1, 1, 0, 0, 1,
  1, 1, 0, 0, 1,
  0, 0, 1, 1, 0
), 5, 5)
rownames(adj) <- colnames(adj) <- LETTERS[1:5]
cograph::assortativity(adj)

Attribute Assortativity (Homophily)

Description

Computes assortativity with respect to a node attribute, measuring the tendency of nodes to connect to others with similar attribute values. For categorical attributes, this computes the modularity-based nominal assortativity. For numeric attributes, this computes the Pearson correlation between attribute values at edge endpoints.

Usage

assortativity_attribute(x, values, directed = NULL, digits = NULL, ...)

homophily(x, values, directed = NULL, digits = NULL, ...)

Arguments

Details

For categorical (nominal) attributes, the coefficient is:

r = \frac{\text{tr}(\mathbf{e}) - \|\mathbf{e}^2\|}{1 - \|\mathbf{e}^2\|}

where \mathbf{e} is the mixing matrix with e_{ij} = fraction of edges connecting type i to type j.

For numeric (scalar) attributes, the coefficient is the Pearson correlation between attribute values at edge endpoints (computed over both orientations of every edge when the network is undirected). Any non-numeric values vector (character or factor) is treated as nominal.

The coefficient is NA when the network has no edges, when a nominal attribute has a single category, or when either value vector has zero variance.

Value

An object of class "cograph_assortativity" with components:

coefficient

Numeric scalar: assortativity coefficient.

type

Character: "nominal" or "scalar".

directed

Logical.

n_nodes

Integer.

n_edges

Integer.

attribute_values

The attribute values used.

network

Original input.

References

Newman, M.E.J. (2003). Mixing patterns in networks. Physical Review E, 67(2), 026126. doi:10.1103/PhysRevE.67.026126

See Also

assortativity, detect_communities

Examples

adj <- matrix(c(0,1,1,0, 1,0,0,0, 1,0,0,1, 0,0,1,0), 4, 4)
rownames(adj) <- colnames(adj) <- c("A", "B", "C", "D")
groups <- c(A = "x", B = "x", C = "y", D = "y")
cograph::assortativity_attribute(adj, groups)

Binarize Edge Weights

Description

Replaces every surviving weight with 1, dropping edges at or below the threshold. The network equivalent of sna::event2dichot().

Usage

binarize(
  x,
  threshold = 0,
  absolute = TRUE,
  signed = FALSE,
  keep_isolates = TRUE,
  keep_format = FALSE,
  directed = NULL
)

Arguments

Value

A cograph_network whose weights are all 1 (or, when signed = TRUE, 1 for a positive edge and -1 for a negative one), or the input format when keep_format = TRUE. Nodes left without edges are kept and reported in a cograph_isolates_created warning, unless keep_isolates = FALSE.

References

Butts, C. T. (2008). Social network analysis with sna. Journal of Statistical Software, 24(6), 1–51.

See Also

threshold_edges, normalize_weights

Examples

adj <- matrix(c(0, .5, .8, 0,
                .5, 0, .3, .6,
                .8, .3, 0, .4,
                 0, .6, .4, 0), 4, 4, byrow = TRUE)
rownames(adj) <- colnames(adj) <- c("A", "B", "C", "D")

binarize(adj, threshold = 0.45)

Combine Two Networks

Description

Aligns two networks on node labels and combines their edges.

Usage

bind_networks(
  x,
  y,
  method = c("union", "intersection", "difference"),
  weight = c("sum", "mean", "max", "min", "first"),
  keep_format = FALSE,
  directed = NULL
)

Arguments

Value

A cograph_network over the combined node set, or x's format when keep_format = TRUE. Nodes are ordered with x's first, then any node only y has.

See Also

add_edges, plot_difference

Examples

a <- matrix(0, 3, 3, dimnames = list(c("A", "B", "C"), c("A", "B", "C")))
a["A", "B"] <- a["B", "A"] <- 1
b <- matrix(0, 3, 3, dimnames = list(c("B", "C", "D"), c("B", "C", "D")))
b["B", "C"] <- b["C", "B"] <- 2

bind_networks(a, b)
bind_networks(a, b, method = "difference")

Calculate Network Centrality Measures

Description

Computes centrality measures for nodes in a network and returns a tidy data frame. Accepts matrices, edge-list data frames, igraph objects, cograph_network, or tna objects.

Usage

centrality(
  x,
  type = c("basic", "extended", "all"),
  measures = NULL,
  include = NULL,
  mode = "all",
  normalized = FALSE,
  weighted = TRUE,
  directed = NULL,
  loops = TRUE,
  simplify = "sum",
  digits = NULL,
  sort_by = NULL,
  cutoff = -1,
  invert_weights = NULL,
  alpha = 1,
  damping = 0.85,
  personalized = NULL,
  transitivity_type = "local",
  isolates = "nan",
  lambda = 1,
  diffusion_method = NULL,
  k = 3,
  states = NULL,
  decay_parameter = 0.5,
  dmnc_epsilon = 1.7,
  membership = NULL,
  katz_alpha = 0.1,
  hubbell_weight = 0.5,
  shapley_k = 2,
  shapley_cutoff = 2,
  s_shell_a = 0.5,
  discount_p = 0.01,
  ncvote_theta = 0.5,
  comm_r = "max_intra",
  ld_radius = 2,
  enrenew_depth = 2,
  voterank_lambda = 0.1,
  contraction_rho = 5,
  wks_alpha = 1,
  wks_beta = 1,
  renewed_threshold = 2,
  kpath_k = 3,
  kpath_len = 3,
  epc_threshold = 0.5,
  epc_runs = 1000,
  epc_seed = NULL,
  betweenness_delta = 1,
  closeness_delta = 1,
  gravity_mass = "kshell",
  gravity_radius = 3,
  mdd_lambda = 0.7,
  volume_radius = 2,
  diffusion_q = 1,
  diffusion_steps = 3,
  ds_beta = 0.1,
  ds_mu = 1,
  ds_steps = 5,
  cda_alpha = 0.5,
  icc_alpha = 0.2,
  exogenous_base = "reverse_closeness",
  wlr_alpha = 1,
  alr_h_mode = "all",
  grc_gamma = 1,
  rwd_decay = 0.5,
  rwd_node_weights = NULL,
  linerank_aggregation = "probability",
  bridging_steps = 2,
  bridging_values = NULL,
  proximal_variant = "source",
  exf_alpha = 2,
  beta_direction = "positive",
  ninl_order = 3,
  ninl_radius = NULL,
  map_flow = "unrecorded",
  map_convention = "paper",
  sr_prior = 0,
  mcgm_radius = 2,
  mcgm_alpha = NULL,
  dkgm_radius = 2,
  nd_order = 2,
  nd_decay = 0.2,
  nd_mass = "degree",
  ira_mass = "coreness",
  ira_alpha = 1,
  ira_tol = 1e-06,
  ira_max_iter = 1000,
  iira_beta = 0.2,
  iira_steps = 50,
  hcc_delta = 0.5,
  lhc_radius = 2,
  tpr_alpha = 0.85,
  tpr_k = 0.85,
  tpr_decay = 1,
  tpr_tol = 1e-14,
  tpr_max_iter = 1000,
  rsp_beta = 0.01,
  rsp_cost = c("inverse", "weight"),
  re_indexes = c("degree", "closeness", "betweenness", "constraint"),
  re_negative = NULL,
  tna_network = NULL,
  psych_network = NULL,
  ...
)

Arguments

Details

The following centrality measures are available:

degree

Count of edges (supports mode: in/out/all)

strength

Weighted degree (supports mode: in/out/all)

betweenness

Shortest path centrality

closeness

Inverse distance centrality (supports mode: in/out/all)

eigenvector

Influence-based centrality

pagerank

Random walk centrality (supports damping and personalization)

authority

HITS authority score

hub

HITS hub score

eccentricity

Maximum distance to other nodes (supports mode)

coreness

K-core membership (supports mode: in/out/all)

constraint

Burt's constraint (structural holes)

transitivity

Local clustering coefficient (supports multiple types)

harmonic

Harmonic centrality - handles disconnected graphs better than closeness (supports mode: in/out/all)

diffusion

Diffusion degree centrality - sum of scaled degrees of node and its neighbors (supports mode: in/out/all, lambda scaling)

leverage

Leverage centrality - measures influence over neighbors based on relative degree differences (supports mode: in/out/all)

kreach

Geodesic k-path centrality - count of nodes reachable within distance k (supports mode: in/out/all, k parameter)

alpha

Alpha/Katz centrality - influence via paths, penalized by distance. Similar to eigenvector but includes exogenous contribution

power

Bonacich power centrality - measures influence based on connections to other influential nodes

subgraph

Subgraph centrality - participation in closed loops/walks, weighting shorter loops more heavily

laplacian

Laplacian centrality using Qi et al. (2012) local formula. Matches NetworkX and centiserve::laplacian()

load

Load centrality - fraction of all shortest paths through node, similar to betweenness but weights paths by 1/count

current_flow_closeness

Information centrality - closeness based on electrical current flow (requires connected graph)

current_flow_betweenness

Random walk betweenness - betweenness based on current flow rather than shortest paths (requires connected graph)

voterank

VoteRank - identifies influential spreaders via iterative voting mechanism. Returns normalized rank (1 = most influential)

percolation

Percolation centrality - importance for spreading processes. Uses node states (0-1) to weight paths. When all states equal, equivalent to betweenness. Useful for epidemic/information spreading analysis.

radiality

Radiality centrality (centiserve). Sum of (diam + 1 - d) normalized by n-1.

lin

Lin's centrality. Reachable nodes squared divided by sum of distances.

decay

Decay centrality. Sum of delta^d for parameter delta.

residual_closeness

Residual closeness. Sum of 1/2^d.

dangalchev

Dangalchev closeness (alias for residual closeness).

generalized_closeness

Generalized closeness. Sum of alpha^d.

harary

Harary centrality. Sum of 1/d^2 for all reachable pairs.

average_distance

Average distance (centiserve). Sum of distances / (n+1).

barycenter

Barycenter centrality. 1 / sum of distances.

wiener

Wiener index. Total sum of shortest path distances from node.

closeness_vitality

Closeness vitality. Drop in Wiener index when node removed.

communicability

Total communicability. Row sums of matrix exponential.

communicability_betweenness

Communicability betweenness. Fraction of communicability through each node.

random_walk

Random walk centrality. Inverse sum of random walk distances (requires connected graph).

stress

Stress centrality. Number of shortest paths through node.

flow_betweenness

Flow betweenness. Max-flow based betweenness.

lobby

Lobby index (h-index of neighborhood).

entropy

Graph entropy centrality. Entropy change on node removal.

semilocal

Semi-local centrality. Triple-nested neighborhood sum.

clusterrank

ClusterRank. Clustering coefficient times neighbor degree sum.

bottleneck

Bottleneck centrality. Count of shortest path trees where node is critical.

centroid

Centroid value. Minimum f(v,i) across all nodes.

mnc

Maximum Neighborhood Component size.

dmnc

Density of Maximum Neighborhood Component.

topological_coefficient

Topological coefficient. Shared neighbor ratio.

bridging

Bridging centrality. Betweenness times bridging coefficient.

local_bridging

Local bridging. (1/degree) times bridging coefficient.

effective_size

Burt's effective size. Degree minus redundancy.

diversity

Diversity centrality. Shannon entropy of edge weight distribution.

cross_clique

Cross-clique connectivity. Count of cliques containing node.

markov

Markov centrality. Inverse mean first passage time (requires connected graph).

integration

Integration centrality. Distance-based influence.

expected

Expected centrality. Sum of neighbor degrees.

gilschmidt

Gil-Schmidt power index. Sum of 1/d normalized by n-1.

salsa

SALSA authority scores (directed graphs only).

leaderrank

LeaderRank. PageRank with ground node (directed graphs only).

participation

Participation coefficient. Diversity of inter-community connections (requires membership).

within_module_z

Within-module degree z-score. Intra-community connectivity (requires membership).

gateway

Gateway coefficient. Inter-community brokerage weighted by centrality (requires membership).

distance_entropy

Normalized Shannon entropy of a node's hop-distance profile; 1 = distances spread evenly, 0 = all at one distance.

local_dimension

Growth exponent of the ball around a node (slope of \ln B_i(r) on \ln r); lower = more influential.

local_information_dimension

Entropy-weighted local dimension over boxes up to half the node's eccentricity; higher = more influential.

neighborhood_connectivity

Mean degree of a node's neighbors (average neighbor degree); isolates score 0.

modularity_vitality

Drop in modularity when the node is removed under a fixed partition; positive = community hub, negative = bridge (requires membership).

shapley_game1, shapley_game2, shapley_game3

Shapley value of the node in the coverage games of Michalak et al. (2013): one-hop coverage, shapley_k-neighbor coverage, and coverage within shapley_cutoff hops. Values sum to the node count.

access_information

Mean bits needed to reach every other node along shortest paths without a map; low = well connected.

hide_information

Mean bits others need to find the node; high = hidden.

rumor

Log rumor centrality on the node's BFS tree: log of the number of spreading orders that could start there.

community_hub_bridge

Community size times intra-community degree plus number of other communities touched times inter-community degree (requires membership).

entropy_variation_degree, entropy_variation_betweenness

Drop in the Shannon entropy of the degree (by mode) or betweenness distribution when the node is deleted; signed, nats.

s_shell

Shell index of the strength-based peeling with asymmetric topological link weights, exponent s_shell_a.

degree_discount, single_discount

Greedy seed-selection order under degree discounting (discount_p) or unit discounting, scored 1 for the first selected down to 1/n.

ncvoterank

VoteRank with voters weighted by normalized neighborhood coreness (ncvote_theta); election order scored like voterank.

community_based, comm_centrality, community_mediator

Links weighted by the size of the community they reach; Gupta's scaled intra/inter-degree combination (comm_r); base-2 entropy of the link distribution over communities times degree share (all require membership).

local_dimension_fixed, fuzzy_local_dimension, local_volume_dimension

Silva-Costa estimator at ld_radius; slope of the fuzzy ball (higher = more influential); slope of the degree volume (lower = more important).

wvoterank, enrenew, voterank_plus

Election orders of the weighted, entropy-based (enrenew_depth) and degree-weighted (voterank_lambda) VoteRank variants, scored like voterank.

node_contraction, node_contraction_improved

One minus the agglomeration ratio after contracting the node with its neighbors; the improved form adds the same score of its edges on the line graph (contraction_rho).

two_way_rw

Number of node pairs whose most likely two-way random-walk route passes through the node.

heatmap

Farness minus mean neighbor farness; lower = more central.

flow_coefficient

Share of neighbor pairs linked through the node but not directly.

local_entropy

-\sum_{j \in N(i)} k_j \ln k_j; lower = more central.

weighted_h_index

h-index over topological link weights k_i k_j repeated k_j times.

redundancy

Mean degree of the neighbors inside the ego network; degree minus effective size.

weighted_kshell

k-shell on (k^\alpha s^\beta)^{1/(\alpha + \beta)} after Garas' weight normalization (wks_alpha, wks_beta).

renewed_coreness

k-core of the graph after removing links whose diffusion importance is below renewed_threshold.

geodesic_kpath

Number of shortest paths of length at most kpath_k starting at the node.

local_efficiency

Global efficiency of the subgraph induced on the node's neighbors, the node itself removed. Note that igraph::local_efficiency() instead measures the distances between those neighbors through the rest of the network.

s_core

Largest strength threshold whose s-core still contains the node; the k-core number when weights are absent.

fragmentation

Distance-weighted fragmentation of the network after deleting the node. Higher means a more disruptive removal.

kpath

Number of simple paths of length at most kpath_len that the node lies on, endpoints included.

epc

Edge percolated component: mean size of the node's component over epc_runs bond-percolation realizations, as a share of the network. A Monte Carlo estimate.

length_scaled_betweenness

Betweenness with each separated pair weighted by 1 / d(s,t).

delta_betweenness

Betweenness with the pair weight (d(s,t) - 1)^{-\delta} (betweenness_delta).

ego_betweenness

Betweenness inside the node's own ego network.

delta_closeness

\sum_j d_{ij}^{-\delta} / (n-1) (closeness_delta).

truss, mdd

Node truss number (k-2 triangles convention) and mixed-degree shell threshold (mdd_lambda). Both use the simple undirected skeleton; see centrality_truss.

bridging_coefficient, godfather, support

Reciprocal-degree ratio, count of unconnected neighbor pairs, and count of triangle-supported relationships on the simple undirected skeleton.

volume

Sum of degrees in the closed volume_radius-hop neighborhood on the simple undirected skeleton.

mcc

Maximal clique centrality: sum of (|C|-1)! over incident maximal cliques of size at least two. Costly; see centrality_mcc for isolate and precision conventions.

diffusion_centrality

Finite-horizon weighted outgoing walks: \sum_{t=1}^{T}(qA)^t\mathbf{1}, with diffusion_q and diffusion_steps. Distinct from diffusion degree.

dynamical_importance

Relative spectral-radius loss on vertex deletion, evaluated by repeated eigendecomposition. Costly; see centrality_dynamical_importance for zero-radius graphs.

dynamics_sensitive

Finite-time spreading score including ds_beta, ds_mu and ds_steps; uses the simple undirected skeleton.

malatya

Sum of focal-to-neighbor degree ratios on the simple undirected skeleton; the reciprocal of the bridging coefficient on nonisolated vertices.

resistance_curvature

One minus half the incident conductance times effective-resistance sum. Weighted, componentwise and costly; see centrality_resistance_curvature.

extended_coreness

Sum of neighbors' neighborhood coreness; equivalently the squared simple adjacency times core numbers.

dkgm

Gravity with the degree k-shell index as the mass at both ends, default radius two; see centrality_dkgm.

neighbor_distance

Benchmark centrality plus its decayed sums over non-backtracking walks of up to nd_order steps; the Zoo's neighbor distance centrality at the defaults. See centrality_neighbor_distance.

ira

Steady state of a unit resource repeatedly reallocated to neighbors in proportion to their ira_mass; conserved, so the scores of a component sum to its size. Warns cograph_no_converge where no steady state exists. See centrality_ira.

iira

The same recursion with each share scaled by 1-(1-\beta)^{k_i} for the iira_beta spreading rate, run iira_steps times. Decays geometrically, so only the order is meaningful. See centrality_iira.

lnc

Local neighbor contribution: the cubed degree times the binomial own-contribution factor (1-1/d_i)^{d_i-1} times the neighbors' degree sum over n-1. Parameter-free; raw scores depend on the whole graph's order. See centrality_lnc.

ked

KED method: the degree times one plus the normalized entropy of the neighbors' degrees times \exp(K_i/N) for the neighbor-degree sum K_i and the whole graph's order N. Parameter-free. See centrality_ked.

hcc

Hybrid characteristic centrality: the extended degree \delta k_i+(1-\delta)\sum_{j\in N(i)}k_j over its maximum, plus the E-shell peeling round in which the node leaves over the number of rounds. Raw scores lie in [0,2] and are not component-local. See centrality_hcc.

ehcc

Extended hybrid characteristic centrality: the closed-neighborhood sum of hcc, the focal node counted once. See centrality_ehcc.

lhc

Lhc index: the degree-and-triangle-share influence C(v)=\sum_{u\in\Phi(v)}k_u(1+TP(u))/d^2(uv) over the ball of radius lhc_radius, summed over the open neighborhood. The triangle share is normalized by TNTS=\sum_u NTS(u), three times the number of distinct triangles, and is written as zero on a triangle-free graph. Raw scores are not component-local. See centrality_lhc.

iec

Immediate effects centrality: the reciprocal mean length of the influence sequences that end at a node, (n-1)/\sum_{i\neq j}m_{ij} for the mean first passage times M=(I-Z+EZ_{dg})\mathrm{diag}(1/c) of the influence chain W=A/\mathrm{rowSums}(A) built with a_{ii}=1. Direction-sensitive and costly (one eigenproblem and two dense solves). NA at every node when the chain is reducible or the graph has one node. Not the same measure as markov. See centrality_iec.

dil

Degree and importance of lines: the degree plus the share of each incident line's importance I_e=(k_m-p-1)(k_n-p-1)/ (p/2+1) that the node's own degree claims, k_i+\sum_{j\in\Gamma_i}I_{e_{ij}}(k_i-1)/(k_i+k_j-2), with p the number of triangles on the line. Two-hop local and component-local; never below the node's degree. See centrality_dil.

trust_pagerank

Trust-PageRank: a damped PageRank whose split of a node's score among its neighbors is the column-stochastic trust-value T(i,j)=(1-k)s(i,j)/\sum_{l\in N_j}s(j,l)+ k\,d_i/\sum_{l\in N_j}d_l, with s the fixed point of SimRank restricted to the lines of the graph. Scores sum to one when no node is isolated. NA at every node of a component that has lines but no triangle, where the similarity vanishes and the ratio is undefined. Costly (two fixed-point recursions over dense matrices). See centrality_trust_pagerank.

rsp_betweenness

Simple randomized shortest paths betweenness: the expected number of visits a node receives over the Boltzmann distribution on absorbing walks, summed over every ordered source-target pair. rsp_beta interpolates between the random-walk and shortest-path readings. Direction-sensitive, component-local, and costly (one dense inverse). See centrality_rsp_betweenness.

relative_entropy

Normalized geometric mean of several index distributions, the minimum-relative-entropy integration of re_indexes; sums to one. See centrality_relative_entropy.

mixed_gravity

Gravity with focal core-number and partner-degree masses, default radius three.

extended_mixed_gravity

Sum of immediate neighbors' raw mixed gravitational centralities.

extended_gravity

Sum of neighbors' raw k-shell gravity scores, with gravity_radius applied around each neighbor.

cda

Weighted degree and strength, adjusted by Barrat clustering, plus weighted neighbor contributions; uses cda_alpha.

improved_closeness

Closeness using distances divided by the number of shortest paths raised to icc_alpha.

exogenous

Contribution to all other nodes' base centrality, measured by deletion. Selects a base using exogenous_base.

global_structure

Exponential focal coreness times distance-discounted partner coreness (GSM).

hybrid_global_structure

Exponential degree-coreness influences with an adaptive distance exponent (H-GSM).

improved_global_structure

Exponential focal degree with partner degrees discounted by a global mean-degree distance exponent (IGSM).

weighted_leaderrank

Stationary scores with ground-node outgoing weights determined by original in-degree and wlr_alpha.

linerank

PageRank on the line graph, aggregated at endpoints; uses damping and linerank_aggregation.

expected_force

Entropy of onward boundary degrees over all two-event transmission sequences.

mcgm

Multi-characteristics gravity with degree, coreness and eigenvector masses; default radius two.

spectralrank

Outgoing Perron eigenvector with a unit-linked ground node; sr_prior supplies optional diagonal information.

controlrank

Smallest eigenvalue of each grounded symmetric row-Laplacian; see centrality_controlrank.

map_equation

Codelength saving on silencing a node, conditional on the supplied partition, flow model and coding convention.

ninl

Finite neighbor propagation of closed-neighborhood degree volume; uses ninl_order and ninl_radius.

beta_measure

BG power shared by successors among predecessors; beta_direction selects positive or negative orientation.

localized_bridging, extended_local_bridging

Betweenness in one-hop or two-hop ego networks times the original bridging coefficient.

modified_expected_force

Expected Force multiplied by log degree with the scaling parameter exf_alpha.

proximal_betweenness

First/last shortest-path intermediaries; uses proximal_variant on the directed unweighted skeleton.

x_degree

Counts four-edge nonbacktracking walks with each node at the middle, using original neighbor excess degrees.

coleman_theil

Concentration of dyadic Burt constraints across contacts; isolates zero and single-contact nodes one.

bridging_capital

Information-walk loss under single-entry deletion; uses bridging_steps and bridging_values.

random_walk_decay

Weighted sum of discounted first arrivals from random walks; uses rwd_decay and rwd_node_weights.

graph_regularization

Reciprocal diagonal of the inverse regularized weighted Laplacian, using grc_gamma.

adaptive_leaderrank

Stationary scores with destination weights determined by original H-indices using alr_h_mode.

Value

A base data.frame with one row per node, in the input's node order unless sort_by is given, and the columns:

• node: character, the node labels (the index as a string when the input carried no names)

• One numeric column per requested measure, with a mode suffix for the mode-aware measures (e.g., degree_in, closeness_all); see list_centralities for which measures carry a suffix. A measure that a tier supplied but that has no value on this input is an all-NA column.

Measures without a value on a given input

A few measures are undefined on some graphs – the community-partition measures without membership, or "relative_entropy" when one of its constituent indexes is zero at every node. Naming such a measure in measures or include raises a classed condition, because you asked for that measure. When a tier (type = "basic", "extended" or "all") supplied it, the condition becomes a cograph_undefined_measure warning and the column is NA, so one undefined measure does not take the rest of the tier with it.

Examples

# Built-in edge-list data
data(student_interactions)
centrality(student_interactions)

# Matrix input also works
adj <- matrix(c(0, 1, 1, 1, 0, 1, 1, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality(adj)

# Specific measures
centrality(adj, measures = c("degree", "betweenness"))

# Directed network with normalization
centrality(adj, mode = "in", normalized = TRUE)

# Sort by pagerank
centrality(adj, sort_by = "pagerank", digits = 3)

# PageRank with custom damping
centrality(adj, measures = "pagerank", damping = 0.9)

# Harmonic centrality (better for disconnected graphs)
centrality(adj, measures = "harmonic")

# Global transitivity
centrality(adj, measures = "transitivity", transitivity_type = "global")

Access and Hide Information

Description

Search-information centralities of Rosvall, Trusina, Minnhagen and Sneppen (2005) and Sneppen, Trusina and Rosvall (2005). A walker who knows only the shortest paths from i to j but has no map must be told which link to take at each step; the number of bits needed is

S(i \to j) = -\log_2 \sum_{p \in \{p(i, j)\}} \frac{1}{k_i} \prod_{l \in p,\, l \ne i, j} \frac{1}{k_l - 1},

summed over all shortest paths, with k_i the degree of the source and k_l - 1 the choices left at each intermediate node (the link the walker arrived on is excluded). Then

A_i = \frac{1}{N} \sum_j S(i \to j), \qquad H_i = \frac{1}{N} \sum_j S(j \to i),

with S(i \to i) = 0.

Usage

centrality_access_information(x, ...)

centrality_hide_information(x, ...)

Arguments

Details

Access information A_i: how many bits it costs, on average, to reach the rest of the network from i. A low value means the node reaches others with few decisions. Hubs score high: a walker leaving a hub has many links to choose from (on a star with five leaves the hub scores 1.93 bits, a leaf 1.33). Hide information H_i: how many bits it costs the rest of the network to find i. High values mark hidden, peripheral nodes; hubs score low (the star hub scores 0). The encyclopedia's prose states the star case the other way round; the formulas and the source papers give the values above.

On a directed graph every step uses the out-degree, 1 / k^{out}. On a disconnected graph the average runs over the nodes a walker can actually reach (or be reached from), so values stay finite; on a connected graph this is exactly the paper's 1 / N. Distances are hop counts; edge weights are ignored. Cost is O(N (N + M)) with an N \times N matrix in memory.

Validated against an independent enumeration of all shortest paths and against the worked values in both papers (star and complete bipartite graphs).

Value

Named numeric vector, one value per node, in bits.

References

Rosvall, M., Trusina, A., Minnhagen, P., & Sneppen, K. (2005). Networks and cities: An information perspective. Physical Review Letters, 94, 028701.

Sneppen, K., Trusina, A., & Rosvall, M. (2005). Hide-and-seek on complex networks. Europhysics Letters, 69(5), 853-859.

See Also

centrality for computing multiple measures at once.

Examples

star5 <- matrix(0, 5, 5)
star5[1, 2:5] <- 1; star5[2:5, 1] <- 1
rownames(star5) <- colnames(star5) <- LETTERS[1:5]
centrality_access_information(star5)
centrality_hide_information(star5)

Adaptive LeaderRank centrality

Description

Xu and Wang's adaptive LeaderRank computes original node H-indices, then adds a ground node with H-index one, joined bidirectionally to every original node. Each augmented arc from j to i has weight a_{ji}h_i. Row-normalized weights define the resource transition matrix. Raw stationary scores retain total augmented mass N, following initial score one on ordinary nodes and zero on ground. The ground score is omitted without redistribution. H-indices are not recomputed after ground edges are added.

Usage

centrality_adaptive_leaderrank(x, alr_h_mode = "all", ...)

Arguments

Details

The H-index is the largest integer h for which at least h original neighbors have degree at least h. The focal node is excluded from that neighbor list. This differs from cograph's existing closed-neighborhood centrality_lobby convention.

The paper evaluates directed and undirected networks but does not pin a directed H-index convention. alr_h_mode makes that choice explicit. Default "all" computes H-indices on the simple undirected skeleton, merging reciprocal arcs. "out" uses outgoing neighbors' out-degrees; "in" uses incoming neighbors' in-degrees. These directed H-index choices are explicit cograph conventions, not claims of the authors' directed-software behavior. In every case, resource flow retains the original directed arcs. Undirected edges become opposite arcs, and all H-index modes then coincide.

Input weights are ignored; the algorithm generates its own destination weights. Loops are removed and parallel arcs count once. The generic mode, inversion and cutoff arguments are ignored. Original nodes with H-index zero receive zero stationary score. If every H-index is zero, the ground transition row is undefined and all scores are NaN. This can occur on edgeless inputs or some directed inputs in in/out H-index modes. Empty input returns an empty vector. No H-index pseudocount is added.

A native ground-elimination solve obtains the unique stationary solution in O(N^3) time and O(N^2) memory, including periodic chains for which ordinary iteration need not converge. Optional final max normalization acts on the returned ordinary-node scores. Numerical definition agreement does not establish superior spreading predictions or author-code parity.

Value

Named numeric vector in input node order.

References

Xu, S., & Wang, P. (2017). Identifying important nodes by adaptive LeaderRank. Physica A, 469, 654-664. doi:10.1016/j.physa.2016.11.034.

Examples

centrality_adaptive_leaderrank(igraph::make_ring(4))

Alpha (Katz) Centrality

Description

Influence via all paths penalized by distance. Similar to eigenvector centrality but includes an exogenous contribution, making it well-defined even for directed acyclic graphs.

Usage

centrality_alpha(x, mode = "all", ...)

Arguments

Value

Named numeric vector of alpha centrality values.

See Also

centrality for computing multiple measures at once, centrality_eigenvector for a related measure.

Examples

adj <- matrix(c(0, 1, 1, 1, 0, 1, 1, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality_alpha(adj)

HITS Authority and Hub Scores

Description

Kleinberg's HITS algorithm. centrality_authority scores nodes pointed to by good hubs. centrality_hub scores nodes that point to good authorities.

Usage

centrality_authority(x, ...)

centrality_hub(x, ...)

Arguments

Value

Named numeric vector of authority or hub scores.

See Also

centrality for computing multiple measures at once.

Examples

adj <- matrix(c(0, 1, 0, 0, 0, 1, 1, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality_authority(adj)
centrality_hub(adj)

Average Distance Centrality

Description

Sum of shortest path distances divided by (n + 1). Lower values indicate more central nodes.

Usage

centrality_average_distance(x, mode = "all", ...)

Arguments

Value

Named numeric vector of average distance values.

See Also

centrality for computing multiple measures at once.

Examples

adj <- matrix(c(0, 1, 0, 1, 0, 1, 0, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality_average_distance(adj)

Barycenter Centrality

Description

Inverse of the total distance to all reachable nodes.

Usage

centrality_barycenter(x, mode = "all", ...)

Arguments

Value

Named numeric vector of barycenter centrality values.

See Also

centrality for computing multiple measures at once.

Examples

adj <- matrix(c(0, 1, 0, 1, 0, 1, 0, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality_barycenter(adj)

BG-index or beta power measure

Description

The positive beta-measure of node i is the sum, over its successors j, of one divided by the in-degree of j. Each node with predecessors shares one unit of domination power equally among those predecessors. This is van den Brink and Gilles' BG-measure (1992, definition 2.1), subsequently called the beta-measure (2000, definition 2.1). The negative variant applies the positive measure to the reversed graph (Boldi and Vigna 2014). It sums reciprocal source out-degrees over incoming neighbors.

Usage

centrality_beta_measure(x, beta_direction = "positive", ...)

Arguments

Details

Uses the simple unweighted graph, retaining direction. Loops and duplicate arcs are removed; weights, mode, inversion and cutoff are ignored. Undirected edges represent reciprocal arcs, so both variants coincide with the sum of reciprocal neighbor degrees. This does not implement the separately defined weighted extension of the original paper.

Nodes without successors have positive score zero; nodes without predecessors have negative score zero. Isolates score zero, and empty graphs return no scores. There is no division by a zero degree: every contributing successor has at least one predecessor. Raw positive scores sum to the number of nodes with nonzero in-degree; raw negative scores sum to the number with nonzero out-degree. In disconnected graphs this accounting applies independently to each component.

Dense matrix preparation and evaluation take O(n squared) time and memory. The score is an expected number of predecessor selections, not a probability distribution or a stationary random-walk centrality.

Value

Named numeric vector in input node order.

References

van den Brink, R. and Gilles, R. P. (2000). Measuring domination in directed networks. Social Networks, 22, 141-157. doi:10.1016/S0378-8733(00)00019-8.

Boldi, P. and Vigna, S. (2014). Axioms for centrality. Internet Mathematics, 10, 222-262. doi:10.1080/15427951.2013.865686.

Examples

centrality_beta_measure(igraph::make_graph("Zachary"))
centrality_beta_measure(igraph::make_star(5, mode = "out"),
                        beta_direction = "negative")

Betweenness Centrality

Description

Fraction of shortest paths passing through each node. Nodes with high betweenness act as bridges connecting different parts of the network.

Usage

centrality_betweenness(x, ...)

Arguments

Value

Named numeric vector of betweenness values.

See Also

centrality for computing multiple measures at once, centrality_load for a related measure.

Examples

adj <- matrix(c(0, 1, 1, 1, 0, 1, 1, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality_betweenness(adj)

Bottleneck Centrality

Description

Number of shortest path trees where the node appears in more than n/4 paths.

Usage

centrality_bottleneck(x, mode = "all", ...)

Arguments

Value

Named integer vector of bottleneck centrality values.

See Also

centrality for computing multiple measures at once.

Examples

adj <- matrix(c(0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0), 4, 4)
rownames(adj) <- colnames(adj) <- c("A", "B", "C", "D")
centrality_bottleneck(adj)

Bridging Centrality

Description

Product of betweenness and bridging coefficient. Identifies nodes that bridge communities.

Usage

centrality_bridging(x, ...)

Arguments

Value

Named numeric vector of bridging centrality values.

See Also

centrality for computing multiple measures at once, centrality_localized_bridging for the ego-network variant.

Examples

adj <- matrix(c(0, 1, 1, 1, 0, 1, 1, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality_bridging(adj)

Bridging capital from lost information walks

Description

Implements Jackson's section 3.3 definition:

Brid_i=\sum_j\sum_{s,t}v_{st}\sum_{h=1}^T [P^h-(P-P_{ij}E_{ij})^h]_{st}.

P contains per-contact transmission probabilities between zero and one. Rows need not sum to one: this is broadcast information flow, not a Markov chain. bridging_steps is the finite horizon T, default two, with zero giving an empty sum. Input edge weights supply P; unweighted edges use probability one. Finite nonnegative pair values v_st default to one, including diagonal entries. Named value matrices are reordered by labels.

Usage

centrality_bridging_capital(x, bridging_steps = 2, bridging_values = NULL, ...)

Arguments

Details

The source explicitly deletes one matrix entry P_ij and credits its criticality to i. On undirected input, opposite entries are therefore tested separately; deleting one leaves the reverse entry present. This is not simultaneous deletion of an undirected edge or of a whole node. Walks can repeat nodes and edges. A walk using the selected entry several times contributes once to that entry's deletion loss, not once per use.

Direction and loops are retained, as allowed by the source's formal definitions. Generic loops/simplify apply first. Remaining parallel weights sum into one matrix entry and must still be at most one; removal deletes that aggregate entry. Zero weights are absent. Mode, inversion and shortest-path cutoff do not affect results. Isolates score zero, empty inputs return no scores, and all-zero values or zero horizon give zeros. No renormalization follows entry removal.

The native implementation tracks walks that have and have not used the selected entry, avoiding cancellation in matrix-power subtraction. Dense cost is O(m T n cubed) time and O(n squared) memory, where m is the number of positive directed matrix entries. Request this costly measure explicitly. Nonrepresentable intermediate walk masses raise errors, even if a final rescaled result might exist. Raw valued-score overflow may be avoided by normalized=TRUE, which scales values first then divides final node scores by their maximum. This implements expected walk counts EInf, not the source's alternative probability-of-ever-hearing measure PInf.

Value

Named numeric vector in input node order.

References

Jackson, M. O. (2020). A typology of social capital and associated network measures. Social Choice and Welfare, 54, 311-336. doi:10.1007/s00355-019-01189-3.

Examples

centrality_bridging_capital(igraph::make_ring(4), bridging_steps = 2)

Gould-Fernandez Brokerage — Coordinator Role

Description

Coordinator brokerage (w_I): count of open directed 2-paths A \to V \to A passing through node V, where all three nodes belong to V's group. The broker mediates contact between two in-group members.

Usage

centrality_brokerage_coordinator(x, membership = NULL, ...)

Arguments

Details

Bit-exact match against sna::brokerage$raw.nli[, "w_I"]. Counts OPEN 2-paths only — those where no direct edge from a to c exists. Directed-only; returns NA with a warning on undirected input.

Value

Named integer vector of coordinator role counts.

References

Gould, R. V., & Fernandez, R. M. (1989). Structures of mediation: A formal approach to brokerage in transaction networks. Sociological Methodology, 19, 89-126.

See Also

centrality, centrality_brokerage_itinerant, centrality_brokerage_representative, centrality_brokerage_gatekeeper, centrality_brokerage_liaison.

Examples

adj <- matrix(c(0,1,1,0, 0,0,1,1, 0,0,0,1, 1,0,0,0), 4, 4, byrow = TRUE)
rownames(adj) <- colnames(adj) <- c("A", "B", "C", "D")
centrality_brokerage_coordinator(adj, membership = c(1, 1, 2, 2))

Gould-Fernandez Brokerage — Gatekeeper Role

Description

Gatekeeper brokerage (b_OI): count of open directed 2-paths A \to V \to B where V and B are in the same group and A is in a different group. The broker acts as a gate letting in-group members receive contact from outside.

Usage

centrality_brokerage_gatekeeper(x, membership = NULL, ...)

Arguments

Details

Bit-exact match against sna::brokerage$raw.nli[, "b_OI"]. Directed-only.

Value

Named integer vector of gatekeeper role counts.

References

Gould, R. V., & Fernandez, R. M. (1989). Structures of mediation: A formal approach to brokerage in transaction networks. Sociological Methodology, 19, 89-126. doi:10.2307/270949.

See Also

centrality_brokerage_coordinator.

Examples

adj <- matrix(c(0,1,1,0, 0,0,1,1, 0,0,0,1, 1,0,0,0), 4, 4, byrow = TRUE)
rownames(adj) <- colnames(adj) <- c("A", "B", "C", "D")
centrality_brokerage_gatekeeper(adj, membership = c(1, 1, 2, 2))

Gould-Fernandez Brokerage — Itinerant (Consultant) Role

Description

Itinerant brokerage (w_O): count of open directed 2-paths A \to V \to A where the two endpoints are in the same group but the broker V is in a different group. The broker mediates within another group as an outsider.

Usage

centrality_brokerage_itinerant(x, membership = NULL, ...)

Arguments

Details

Bit-exact match against sna::brokerage$raw.nli[, "w_O"]. Directed-only.

Value

Named integer vector of itinerant role counts.

References

Gould, R. V., & Fernandez, R. M. (1989). Structures of mediation: A formal approach to brokerage in transaction networks. Sociological Methodology, 19, 89-126. doi:10.2307/270949.

See Also

centrality_brokerage_coordinator.

Examples

adj <- matrix(c(0,1,1,0, 0,0,1,1, 0,0,0,1, 1,0,0,0), 4, 4, byrow = TRUE)
rownames(adj) <- colnames(adj) <- c("A", "B", "C", "D")
centrality_brokerage_itinerant(adj, membership = c(1, 1, 2, 2))

Gould-Fernandez Brokerage — Liaison Role

Description

Liaison brokerage (b_O): count of open directed 2-paths A \to V \to B where all three nodes belong to different groups. The broker mediates between two groups to neither of which they belong.

Usage

centrality_brokerage_liaison(x, membership = NULL, ...)

Arguments

Details

Bit-exact match against sna::brokerage$raw.nli[, "b_O"]. Directed-only.

Value

Named integer vector of liaison role counts.

References

Gould, R. V., & Fernandez, R. M. (1989). Structures of mediation: A formal approach to brokerage in transaction networks. Sociological Methodology, 19, 89-126. doi:10.2307/270949.

See Also

centrality_brokerage_coordinator.

Examples

adj <- matrix(c(0,1,1,0, 0,0,1,1, 0,0,0,1, 1,0,0,0), 4, 4, byrow = TRUE)
rownames(adj) <- colnames(adj) <- c("A", "B", "C", "D")
centrality_brokerage_liaison(adj, membership = c(1, 1, 2, 2))

Gould-Fernandez Brokerage — Representative Role

Description

Representative brokerage (b_IO): count of open directed 2-paths A \to V \to B where A and V are in the same group and B is in a different group. The broker represents their group outward.

Usage

centrality_brokerage_representative(x, membership = NULL, ...)

Arguments

Details

Bit-exact match against sna::brokerage$raw.nli[, "b_IO"]. Directed-only.

Value

Named integer vector of representative role counts.

References

Gould, R. V., & Fernandez, R. M. (1989). Structures of mediation: A formal approach to brokerage in transaction networks. Sociological Methodology, 19, 89-126. doi:10.2307/270949.

See Also

centrality_brokerage_coordinator.

Examples

adj <- matrix(c(0,1,1,0, 0,0,1,1, 0,0,0,1, 1,0,0,0), 4, 4, byrow = TRUE)
rownames(adj) <- colnames(adj) <- c("A", "B", "C", "D")
centrality_brokerage_representative(adj, membership = c(1, 1, 2, 2))

Clustering degree algorithm centrality

Description

Wang et al.'s CDA returns the propagation-capability score PC_i=CD_i+\sum_j(w_{ij}/w_{\max})CD_j, where CD_i=[\alpha d_i+(1-\alpha)s_i]/[1+\exp(-C_i^w)]. Here d is degree, s is strength, and Cw is Barrat's weighted local clustering coefficient. The maximum edge weight is taken over the whole graph, including other connected components. Inner CD scores remain raw until the final optional normalization of PC.

Usage

centrality_cda(x, cda_alpha = 0.5, ...)

Arguments

Details

Uses finite nonnegative weights on an undirected graph. Zero-weight edges are absent connections. Clustering is set to zero for nodes with fewer than two positive-weight neighbors; this convention agrees with the source's leaf example. Isolates and edgeless graphs score zero. Weights retain their original units: scaling all weights can change scores and rankings because degree and strength are combined. At alpha zero, uniform weight scaling scales scores proportionally; at alpha one, scores are invariant to that scaling. Binary inputs are independent of alpha because their degree and strength coincide.

Self-loops are removed. For weighted directed inputs, opposite arcs are added into undirected edge weights. Parallel weights are combined by simplify first, with any remaining parallel edges added. Without weights, the simple undirected skeleton is used. These are explicit cograph projections to the source's undirected domain. mode and shortest-path weight inversion do not affect CDA. Nonfinite intermediate strengths or scores raise an error, including when normalization is requested.

Value

Named numeric vector in input node order.

References

Wang, Q., Ren, J., Wang, Y., Zhang, B., Cheng, Y., & Zhao, X. (2018). CDA: A Clustering Degree Based Influential Spreader Identification Algorithm in Weighted Complex Network. IEEE Access, 6, 19550-19559. doi:10.1109/ACCESS.2018.2822844.

Examples

centrality_cda(igraph::make_ring(5), cda_alpha = 0.5)

Centroid Value

Description

Minimum difference between own and competitor's closer-node count. Measures how much a node is at the center of the graph.

Usage

centrality_centroid(x, mode = "all", ...)

Arguments

Value

Named numeric vector of centroid values.

See Also

centrality for computing multiple measures at once.

Examples

adj <- matrix(c(0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0), 4, 4)
rownames(adj) <- colnames(adj) <- c("A", "B", "C", "D")
centrality_centroid(adj)

Closeness Centrality

Description

Inverse of the average shortest path distance from a node to all others. For directed networks, centrality_incloseness and centrality_outcloseness measure incoming and outgoing closeness.

Usage

centrality_closeness(x, mode = "all", ...)

centrality_incloseness(x, ...)

centrality_outcloseness(x, ...)

Arguments

Value

Named numeric vector of closeness values.

See Also

centrality for computing multiple measures at once, centrality_harmonic for a variant that handles disconnected graphs.

Examples

adj <- matrix(c(0, 1, 1, 1, 0, 1, 1, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality_closeness(adj)

Closeness Vitality

Description

Drop in the Wiener index when a node is removed. Higher values indicate more critical nodes for overall connectivity.

Usage

centrality_closeness_vitality(x, mode = "all", ...)

Arguments

Value

Named numeric vector of closeness vitality values.

See Also

centrality for computing multiple measures at once.

Examples

adj <- matrix(c(0, 1, 0, 1, 0, 1, 0, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality_closeness_vitality(adj)

ClusterRank Centrality

Description

Product of clustering coefficient and sum of (neighbor degree + 1).

Usage

centrality_clusterrank(x, mode = "all", ...)

Arguments

Value

Named numeric vector of ClusterRank values.

See Also

centrality for computing multiple measures at once.

Examples

adj <- matrix(c(0, 1, 1, 1, 0, 1, 1, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality_clusterrank(adj)

Coleman-Theil hierarchy index

Description

Measures concentration of Burt's dyadic constraints over a node's contacts. Let mutual tie strength be z_ij+z_ji, and p_ij its proportion of all mutual strength incident to i. With organizational weights fixed at one, define

c_{ij}=(p_{ij}+\sum_q p_{iq}p_{qj})^2,\quad r_{ij}=c_{ij}/\operatorname{mean}_{k\in N(i)}c_{ik}.

The index is \sum_{j\in N(i)}r_{ij}\log(r_{ij})/(d_i\log(d_i)). Contacts are distinct nodes with positive mutual strength. Investment proportions use the full supplied graph, including alters' outside ties.

Usage

centrality_coleman_theil(x, ...)

Arguments

Details

Follows Burt's STRUCTURE 4.2 manual (pages 181-183): isolates score zero and nodes with one contact score one. The general formula is undefined in these two cases; these are the author's explicit conventions. JUNG's documented implementation instead returns NaN for isolates. Values range from zero for equal constraints to one for complete concentration. Input organizational/oligopoly multipliers from STRUCTURE are not implemented; they are fixed at one, as in the Zoo's formula.

Finite nonnegative weights are supported. Zero-weight ties are absent, loops are removed, and remaining parallel edges sum after generic simplification. Directed ties are combined by summing both directions; weighted=FALSE assigns unit weight to each retained edge before combining them, so reciprocity can affect mutual investment. Generic mode, shortest-path inversion and cutoff do not affect the result. Empty input returns no scores. Components are independent before global normalization.

The default output is already the unit-interval hierarchy index. normalized=TRUE additionally divides by the largest node score; an all-zero vector remains zero. Dense native arithmetic costs O(n cubed) time and O(n squared) memory. Global weight scaling precedes mutual sums. Unrepresentable positive weight or investment ranges raise an error; tiny squared constraints may underflow and use the zero-log-zero limit. Relative deviations of local constraints within 16 machine epsilons are treated as uniform; a series stabilizes the entropy near uniformity.

Value

Named numeric vector in input node order.

References

Burt, R. S. (1992). Structural Holes: The Social Structure of Competition. Harvard University Press. doi:10.4159/9780674029095.

Examples

centrality_coleman_theil(igraph::make_star(5, mode = "undirected"))

Communicability Centrality

Description

Total communicability: row sums of the matrix exponential of the adjacency matrix. Measures a node's ability to broadcast information through all paths.

Usage

centrality_communicability(x, ...)

Arguments

Value

Named numeric vector of communicability values.

See Also

centrality for computing multiple measures at once, centrality_subgraph for the diagonal-only variant.

Examples

adj <- matrix(c(0, 1, 1, 1, 0, 1, 1, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality_communicability(adj)

Communicability Betweenness Centrality

Description

Fraction of total communicability that passes through each node.

Usage

centrality_communicability_betweenness(x, ...)

Arguments

Value

Named numeric vector of communicability betweenness values.

See Also

centrality for computing multiple measures at once.

Examples

adj <- matrix(c(0, 1, 1, 1, 0, 1, 1, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality_communicability_betweenness(adj)

Community-Based Centrality, Comm Centrality and Community-Based Mediator

Description

Three community-aware measures that need a partition (membership).

Usage

centrality_community_based(x, membership = NULL, mode = "all", ...)

centrality_comm_centrality(
  x,
  membership = NULL,
  mode = "all",
  comm_r = "max_intra",
  ...
)

centrality_community_mediator(x, membership = NULL, mode = "all", ...)

Arguments

Details

community_based (Zhao, Wang, Zhang & Zhu 2015)

CbC(i) = \sum_w d_{iw} S_w / N: every link of i counts the size S_w of the community it lands in. No parameters. Reproduces Table 1 of the paper and Table 1 of Tulu et al. (2018).

comm_centrality (Gupta, Singh & Cherifi 2016)

CC(i) = (1 + \mu_C)\, \frac{k^{in}_i}{\max_{j \in C} k^{in}_j} R + (1 - \mu_C) \left(\frac{k^{out}_i}{\max_{j \in C} k^{out}_j} R\right)^2,

where k^{in}, k^{out} are the intra- and inter-community degrees, \mu_C the mean inter-link fraction in i's community, and R a scale. The default comm_r = "max_intra" is the paper's recommended R = \max_{j \in C} k^{in}_j per community; a number applies one global R. The equation uses 1 + \mu_C although the paper's prose says \mu_C; the equation is implemented. A community without intra (inter) links contributes 0 through that term.

community_mediator (Tulu, Hou & Younas 2018)

CbM(i) = H_i \, d_i / \sum_j d_j, with H_i the base-2 Shannon entropy of i's link distribution over the communities. Nodes linked to one community only score 0. Base 2 is what reproduces the paper's Table 1.

Higher = more central in all three. Under mode = "out" or "in" only out- or in-links count; edge weights are ignored.

Value

Named numeric vector, one value per node.

Conditions

Raises an error of class cograph_bad_membership when membership is not one non-missing label per node.

References

Zhao, Z., Wang, X., Zhang, W., & Zhu, Z. (2015). A community-based approach to identifying influential spreaders. Entropy, 17(4), 2228-2252.

Gupta, N., Singh, A., & Cherifi, H. (2016). Centrality measures for networks with community structure. Physica A, 452, 46-59.

Tulu, M. M., Hou, R., & Younas, T. (2018). Identifying influential nodes based on community structure to speed up the dissemination of information in complex network. IEEE Access, 6, 7390-7401.

See Also

centrality_community_hub_bridge, centrality_participation.

Examples

adj <- matrix(0, 6, 6)
adj[cbind(c(1, 1, 2, 4, 4, 5, 3), c(2, 3, 3, 5, 6, 6, 4))] <- 1
adj <- adj + t(adj)
rownames(adj) <- colnames(adj) <- LETTERS[1:6]
centrality_community_based(adj, membership = c(1, 1, 1, 2, 2, 2))
centrality_comm_centrality(adj, membership = c(1, 1, 1, 2, 2, 2))
centrality_community_mediator(adj, membership = c(1, 1, 1, 2, 2, 2))

Community Hub-Bridge Centrality

Description

Ghalmane, El Hassouni and Cherifi's (2019) score for nodes that are both hubs inside their community and bridges between communities:

CHB(i) = |C_i| \, k^{intra}_i + NNC_i \, k^{inter}_i,

where |C_i| is the number of nodes in i's own community, k^{intra}_i and k^{inter}_i its numbers of links inside and outside that community, and NNC_i the number of other communities it is linked to (eqs. 2 to 4 of the paper). Higher values mark nodes whose removal both fragments their community and cuts links between communities. A normalized variant with the same name exists in later work by the same group; this is the original raw form.

Usage

centrality_community_hub_bridge(x, membership = NULL, mode = "all", ...)

Arguments

Details

Under mode = "out" or "in" only out- or in-links count; the default ignores direction. Edge weights are ignored.

Value

Named numeric vector, one value per node.

Conditions

Raises an error of class cograph_bad_membership when membership is not one non-missing label per node.

References

Ghalmane, Z., El Hassouni, M., & Cherifi, H. (2019). Immunization of networks with non-overlapping community structure. Social Network Analysis and Mining, 9, 45.

See Also

centrality_modularity_vitality, centrality_participation.

Examples

adj <- matrix(0, 6, 6)
adj[cbind(c(1, 1, 2, 4, 4, 5, 3), c(2, 3, 3, 5, 6, 6, 4))] <- 1
adj <- adj + t(adj)
rownames(adj) <- colnames(adj) <- LETTERS[1:6]
centrality_community_hub_bridge(adj, membership = c(1, 1, 1, 2, 2, 2))

Burt's Constraint

Description

Network constraint measuring the extent to which a node's connections are redundant. Low constraint indicates access to structural holes (brokerage opportunities).

Usage

centrality_constraint(x, ...)

Arguments

Value

Named numeric vector of constraint values.

See Also

centrality for computing multiple measures at once.

Examples

adj <- matrix(c(0, 1, 1, 1, 0, 1, 1, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality_constraint(adj)

ControlRank centrality

Description

Zhou, Yu and Lu's ControlRank is the smallest eigenvalue after deleting a node's row and column from the symmetric part of the graph Laplacian. With L = D-A, this is CR_i = \lambda_{\min}(((L+L^T)/2)_{-i,-i}). D retains the original graph's degrees: the Laplacian is not recomputed on the vertex-deleted graph. Larger values receive higher rank.

Usage

centrality_controlrank(x, ...)

Arguments

Details

Uses finite nonnegative interaction weights. For directed input, A_{ij} denotes an arc from i to j and D contains outgoing strengths. This fixes the row-Laplacian orientation explicitly; transpose the input to use incoming strengths. Symmetrizing L preserves its diagonal, so this is different from constructing a Laplacian of the undirected projection. Directed scores can be negative and are not clipped. For matrix inputs with very small weights, supply directed = TRUE explicitly (or use a directed igraph object): the shared input parser's approximate symmetry detection can otherwise infer an undirected graph.

Loops are removed and zero weights are absent connections. Parallel weights follow the generic simplify rule; remaining parallel edges sum. With weighted = FALSE, each remaining edge contributes one. Mode, weight inversion for shortest paths and cutoff are ignored.

Connected undirected graphs with at least two nodes have positive scores. Disconnected undirected graphs score zero for every node because at least one component remains ungrounded. Empty graphs return no scores; singletons return zero as an explicit extension of the undefined empty minor. The source excludes isolates; the matrix formula here also applies to disconnected directed graphs, whose scores may remain negative.

This implements the spectral index, not a controller simulation, a finite-feedback convergence rate, or an optimization over controller sets. In particular, no general directed stability guarantee is inferred from these scores. The paper's multi-node selection problem is separate.

Dense eigensolves take O(n to the fourth) time and O(n squared) memory; this measure is marked costly and excluded from the default all tier. Disconnected blocks are solved separately, preserving isolated zeros before normalization. Global scaling avoids intermediate overflow. Unrepresentable weight ranges and unresolved positive spectra raise errors. Signed directed scores near zero can retain floating-point roundoff; very small raw scores can underflow. Uniform weight scaling multiplies raw scores by the same factor.

Value

Named numeric vector in input node order.

References

Zhou, J., Yu, X. and Lu, J.-A. (2019). Node Importance in Controlled Complex Networks. IEEE Transactions on Circuits and Systems II: Express Briefs, 66(3), 437-441. doi:10.1109/TCSII.2018.2845940.

Examples

centrality_controlrank(igraph::make_ring(5))
centrality_controlrank(igraph::make_star(6, mode = "undirected"))

K-Core Decomposition (Coreness)

Description

Assigns each node to its maximum k-core. A k-core is a maximal subgraph where every node has at least k connections within the subgraph.

Usage

centrality_coreness(x, mode = "all", ...)

Arguments

Value

Named numeric vector of coreness values.

See Also

centrality for computing multiple measures at once.

Examples

adj <- matrix(c(0, 1, 1, 1, 0, 1, 1, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality_coreness(adj)

Cross-Clique Connectivity

Description

Count of all cliques (not just maximal) containing each node. Measures embeddedness in dense substructures.

Usage

centrality_cross_clique(x, ...)

Arguments

Value

Named integer vector of cross-clique counts.

See Also

centrality for computing multiple measures at once.

Examples

adj <- matrix(c(0, 1, 1, 1, 0, 1, 1, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality_cross_clique(adj)

Current Flow Betweenness Centrality

Description

Betweenness based on electrical current flow rather than shortest paths. Uses the Laplacian pseudoinverse. Requires a connected graph.

Usage

centrality_current_flow_betweenness(x, ...)

Arguments

Value

Named numeric vector of current flow betweenness values.

See Also

centrality for computing multiple measures at once, centrality_betweenness for the shortest-path variant.

Examples

adj <- matrix(c(0, 1, 1, 1, 0, 1, 1, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality_current_flow_betweenness(adj)

Current Flow Closeness Centrality

Description

Information centrality based on electrical current flow through the network. Uses the pseudoinverse of the Laplacian matrix. Requires a connected graph.

Usage

centrality_current_flow_closeness(x, ...)

Arguments

Value

Named numeric vector of current flow closeness values.

See Also

centrality for computing multiple measures at once, centrality_closeness for the shortest-path variant.

Examples

adj <- matrix(c(0, 1, 1, 1, 0, 1, 1, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality_current_flow_closeness(adj)

Dangalchev Closeness Centrality

Description

Alias for residual closeness centrality: sum of 1/2^d.

Usage

centrality_dangalchev(x, mode = "all", ...)

Arguments

Value

Named numeric vector of Dangalchev closeness values.

See Also

centrality for computing multiple measures at once, centrality_residual_closeness (equivalent).

Examples

adj <- matrix(c(0, 1, 0, 1, 0, 1, 0, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality_dangalchev(adj)

Decay Centrality

Description

Sum of delta^d over all nodes, where d is the shortest path distance. Nodes near many others get higher scores. The decay_parameter controls the distance penalty.

Usage

centrality_decay(x, mode = "all", decay_parameter = 0.5, ...)

Arguments

Value

Named numeric vector of decay centrality values.

See Also

centrality for computing multiple measures at once.

Examples

adj <- matrix(c(0, 1, 0, 1, 0, 1, 0, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality_decay(adj, decay_parameter = 0.5)

Degree Centrality

Description

Number of edges connected to each node. For directed networks, centrality_indegree counts incoming edges and centrality_outdegree counts outgoing edges.

Usage

centrality_degree(x, mode = "all", ...)

centrality_indegree(x, ...)

centrality_outdegree(x, ...)

Arguments

Value

Named numeric vector of degree values.

See Also

centrality for computing multiple measures at once, centrality_strength for the weighted version.

Examples

adj <- matrix(c(0, 1, 1, 1, 0, 1, 1, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality_degree(adj)

DegreeDiscountIC and SingleDiscount Rankings

Description

Chen, Wang and Yang's (2009) degree-discount heuristics for choosing spreaders under the independent-cascade model. Nodes are selected one at a time by the largest discounted degree; after each selection every unselected neighbor v of the new seed counts one more selected neighbor, t_v, and its discounted degree becomes

dd_v = d_v - 2 t_v - (d_v - t_v)\, t_v\, p

for DegreeDiscountIC (Algorithm 4 of the paper, with propagation probability p, default 0.01), or simply d_v - t_v for SingleDiscount, where each neighbor of a new seed discounts its degree by one. Every node is placed, so the result is a full ranking, returned as a score: the first node selected scores 1, the last 1 / n.

Usage

centrality_degree_discount(x, discount_p = 0.01, ...)

centrality_single_discount(x, ...)

Arguments

Details

Ties are broken by node order, which the paper does not specify. Direction, edge weights and self-loops are ignored, as in the paper's setting. Validated against an independent implementation of the algorithm and against the reference code of the influence-maximization literature on the karate club graph.

Value

Named numeric vector in (0, 1], one score per node.

References

Chen, W., Wang, Y., & Yang, S. (2009). Efficient influence maximization in social networks. Proceedings of the 15th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 199-208.

See Also

centrality_voterank for the voting-based alternative.

Examples

adj <- matrix(0, 6, 6)
adj[cbind(c(1, 1, 2, 4, 4, 5, 3), c(2, 3, 3, 5, 6, 6, 4))] <- 1
adj <- adj + t(adj)
rownames(adj) <- colnames(adj) <- LETTERS[1:6]
centrality_degree_discount(adj)
centrality_single_discount(adj)

Diffusion Centrality

Description

Sum of scaled degrees of a node and its neighbors, measuring the node's potential for spreading information through the network.

Usage

centrality_diffusion(x, mode = "all", lambda = 1, ...)

Arguments

Details

Two methods are supported. "kandhway_kuri" (Kandhway & Kuri, 2014) computes the 1-hop binary-degree neighborhood sum and is the default for raw matrices, igraph objects, and other non-tna inputs. "power_series" computes \mathrm{rowSums}(P + P^2 + \ldots + P^n) on the weighted matrix (with diag(P) := 0 when loops = FALSE) and matches tna::centralities(., measures = "Diffusion") byte-for-byte. For tna inputs, the default switches to "power_series" to match user expectation; pass diffusion_method = "kandhway_kuri" to force the binary-degree formula.

Value

Named numeric vector of diffusion centrality values.

See Also

centrality for computing multiple measures at once.

Examples

adj <- matrix(c(0, 1, 1, 1, 0, 1, 1, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality_diffusion(adj)

Finite-horizon diffusion centrality

Description

Banerjee et al.'s diffusion centrality is DC(A;q,T) = \sum_{t=1}^{T}(qA)^t\mathbf{1}. It sums weighted walks starting at each node, allowing revisits and returns to the source. Directed edges carry information from their source to their target: the result uses row sums, regardless of mode. Transpose the input adjacency matrix to measure incoming walks.

Usage

centrality_diffusion_centrality(x, diffusion_q = 1, diffusion_steps = 3, ...)

Arguments

Details

A is the adjacency matrix with the original edge weights when weighted = TRUE, or unit edge weights otherwise. Self-loops follow loops; an undirected self-loop contributes its weight once on the diagonal. The simplify argument combines parallel edges first; any remaining parallel edges contribute additively to A. Weight inversion for shortest paths does not affect this measure.

When every entry of qA is between zero and one, scores have the paper's interpretation as expected total hearings of information. Larger weights are accepted as a mathematical weighted-walk extension of that polynomial, without a probability interpretation. Scores count repeated hearings, not distinct recipients. They need not be bounded by the number of nodes.

Default q = 1 and T = 3 are explicit cograph choices, not estimates of a diffusion process or the parameters used by the Zoo. T = 0 returns zero; T = 1 gives q times outgoing strength (degree for a binary graph). A finite horizon requires no spectral convergence condition. Numerical overflow raises an error, including when normalization is requested.

This is distinct from centrality_diffusion: its default is diffusion degree, and its TNA variant fixes q = 1 and T = n. The existing lambda and diffusion_method arguments do not affect this measure. Computation uses T matrix-vector products.

Value

Named numeric vector in input node order.

References

Banerjee, A., Chandrasekhar, A. G., Duflo, E., & Jackson, M. O. (2013). The Diffusion of Microfinance. Science, 341, 1236498. doi:10.1126/science.1236498.

Banerjee, A., Chandrasekhar, A. G., Duflo, E., & Jackson, M. O. (2019). Using Gossips to Spread Information: Theory and Evidence from Two Randomized Controlled Trials. Review of Economic Studies, 86, 2453-2490. doi:10.1093/restud/rdz008.

Examples

g <- igraph::make_graph(c(1, 2, 2, 3), directed = TRUE)
centrality_diffusion_centrality(g, diffusion_q = 0.5, diffusion_steps = 2)

Degree and Importance of Lines

Description

Liu, Xiong, Shi, Shi and Wang rank a node by its degree plus the share it can claim of the importance of the lines that touch it. A line matters when its two endpoints reach far beyond it and when no triangle offers a way round it, so the importance of the line e_{mn} is I_{e_{mn}}=U/\lambda with U=(k_m-p-1)(k_n-p-1) and \lambda=p/2+1, where p is the number of triangles one of whose edges is e_{mn}. That importance is then split between the endpoints in proportion to their own degrees, W_{v_iv_j}=I_{e_{ij}}(k_i-1)/(k_i+k_j-2), and the score is L_{v_i}=k_i+\sum_{v_j\in\Gamma_i}W_{v_iv_j} over the open neighborhood \Gamma_i. The measure is strictly two-hop local: only the degrees of a node, of its neighbors and the triangles on its incident lines enter, so it costs O(n\langle k\rangle^2) and its raw scores are component-local.

Usage

centrality_dil(x, ...)

Arguments

Details

\lambda is p/2+1, and reading it from a text layer gets it wrong. The stacked fraction extracts from the published PDF as \lambda=2p+1, in Liu et al.'s original as much as in the Almasi and Hu (2019) reproduction of it. The page image shows p over 2; so does the paper's own worked example, in printed prose, on page 210: for the seven-line network of its Fig. 1(b) it writes p=1, U=4, "\lambda=1/2+1=1.5" and I_{e_{45}}=4/1.5\approx 2.6667. The wrong reading returns 4/3 there. cograph reproduces 8/3.

U is never negative, so a score never falls below the node's degree. For a line (i,j), j belongs to N(i) but to neither N(j) nor the intersection, so p=|N(i)\cap N(j)|\le k_i-1 and both factors of U are at least zero. Since \lambda\ge 1, every I and every W is at least zero and L_{v_i}\ge k_i. Equality is common rather than exceptional: every line of a complete graph, of a star, or of any network whose lines all touch a degree-one node has U=0, so K_n scores n-1 at every node and a star scores its degree at every node.

The importance of a line is conserved when it is split. The two shares (k_i-1)/(k_i+k_j-2) and (k_j-1)/(k_i+k_j-2) sum to one, so \sum_i (L_{v_i}-k_i)=\sum_{e}I_e: the network's total excess over degree is exactly the total importance of its lines. That identity is asserted over the package's whole verification collection.

An isolated K_2 is the one undefined split, and it is resolved rather than refused. The denominator k_i+k_j-2 vanishes only when k_i=k_j=1, since both endpoints of a line have degree at least one – that is a two-node component – and there p=0 and U=(1-0-1)(1-0-1)=0, so the importance being divided is exactly zero while the split of it is 0/0. Because W is a share of I, and the two shares sum to one wherever they are defined, every admissible split of an exactly zero importance gives an exactly zero contribution: the answer does not depend on resolving the indeterminacy. cograph therefore writes the share as zero, taking the test before the division so that no 0/0 is ever evaluated, and both nodes of a K_2 score 1. The source says nothing about this case; the choice is cograph's, and it follows the precedent of centrality_lhc, whose 0/0 on a triangle-free graph is likewise written as zero because the denominator vanishes exactly where every numerator does. It deliberately does not follow centrality_iec, which returns NA on reducible input: there the closed form returns a finite number in place of an infinite one, so a value would be wrong, where here every candidate value is the same value.

Direction and weights are dropped, because the authors exclude them. Page 210 opens the derivation with "we assume that a network G=(V,E) is an undirected and unweighted network", and every quantity in the three equations is a count: a degree, a triangle census, a difference of integers. A directed, weighted or multigraph input is therefore projected onto its simple undirected skeleton – arcs symmetrized, weights and parallel edges collapsed to a single line, loops dropped – rather than refused, which is the convention every other undirected-domain measure in centrality already follows, and the projection is silent rather than warned for the same reason. There is no in/out/all reading to choose between, so the measure sits in the no-mode family and cutoff and invert_weights are ignored as well. The source states no normalization, so normalized = TRUE max-scales the finished vector as elsewhere in centrality.

Isolates, singletons and disconnected input need no special rule. An isolate has degree zero and an empty sum, so it scores zero; the single node of a one-node graph and every node of an edgeless graph score zero for the same reason, and an empty graph returns no scores. Because nothing in equations (1)-(3) reaches past a node's second neighbors, the raw scores are component-local: attaching a disjoint component leaves every existing score unchanged.

The source prints three numerical fixtures and all three are reproduced. Fig. 1 on page 210 prints I_{e_{45}}=9 at p=0 and 8/3 at p=1; Fig. 2 on page 211 prints L_{v_2}=26/9 and L_{v_5}=52/15 on a 27-node tree; and Table 3 on page 217 prints a DIL value for every one of the 21 nodes of the ARPA network, whose topology is Fig. 6 on the same page. All 21 printed values are reproduced, and the edge list read off the figure is corroborated independently by the paper's own degree column. See the batch 50 published audit in the package's verification directory.

Value

Named numeric vector in input node order, one score per node, each at least the node's degree in the simple undirected skeleton.

References

Liu, J., Xiong, Q., Shi, W., Shi, X. and Wang, K. (2016). Evaluating the importance of nodes in complex networks. Physica A: Statistical Mechanics and its Applications, 452, 209-219. doi:10.1016/j.physa.2016.02.049.

See Also

centrality_lhc and centrality_hcc for other degree-and-triangle hybrids, centrality_bridging for another measure that scores a node by the lines it carries, and list_centralities for the catalogue.

Examples

# Every line of a complete graph is shortcut by n - 2 triangles, so U is
# zero throughout and the score is the degree.
centrality_dil(igraph::make_full_graph(5))

# A triangle-free k-regular graph scores k + k(k-1)^2/2 at every node:
# 3 for a ring and 9 for the Petersen graph.
centrality_dil(igraph::make_ring(6))

# The path 1-2-3-4-5 scores 1, 2.5, 3, 2.5, 1: a line to a leaf carries
# no importance, and the two interior lines carry one each, split evenly.
centrality_dil(igraph::make_graph(c(1, 2, 2, 3, 3, 4, 4, 5),
                                  directed = FALSE))

# A triangle on two degree-three nodes is the case that needs
# lambda = p/2 + 1: I = 1 / 1.5 = 2/3, split evenly, so the two hubs
# score 3 + 1/3. Reading lambda as 2p + 1 would give 3 + 1/6.
centrality_dil(igraph::make_graph(c(1, 2, 1, 3, 2, 3, 1, 4, 2, 5),
                                  directed = FALSE))

Distance Entropy

Description

Shannon entropy of the distribution of hop distances from a node to every node it can reach (Stella & De Domenico 2018), normalized so that a uniform spread over the node's distance range scores 1:

h(i) = -\frac{1}{\log(M_i - m_i + 1)} \sum_{k = m_i}^{M_i} p_k^{(i)} \log p_k^{(i)}, \qquad p_k^{(i)} = n_k^{(i)} / R_i,

where n_k^{(i)} is the number of nodes at distance k from i, R_i the number of reachable nodes, and m_i, M_i the minimum and maximum distance. High values mark nodes whose reach is spread evenly across many network layers; a node whose reachable nodes all sit at one distance scores 0. Closeness summarizes the mean of the same distribution; distance entropy summarizes its spread.

Usage

centrality_distance_entropy(x, mode = "all", ...)

Arguments

Details

Distances are hop counts (edge weights are ignored). The original paper normalizes by \log(M_i - m_i), which is undefined when only two distinct distances occur; \log(M_i - m_i + 1) is used here so the index is bounded by 1 for a uniform distribution.

Value

Named numeric vector, one value per node, in [0, 1]. NaN for a node that reaches no other node.

References

Stella, M., & De Domenico, M. (2018). Distance entropy cartography characterises centrality in complex networks. Entropy, 20(4), 268.

See Also

centrality for computing multiple measures at once, centrality_local_dimension for the growth-rate view of the same distance profile.

Examples

path4 <- matrix(c(0,1,0,0, 1,0,1,0, 0,1,0,1, 0,0,1,0), 4, 4)
rownames(path4) <- colnames(path4) <- c("A", "B", "C", "D")
centrality_distance_entropy(path4)

Diversity Centrality

Description

Shannon entropy of the edge weight distribution per node. Measures how evenly a node distributes its connections.

Usage

centrality_diversity(x, ...)

Arguments

Value

Named numeric vector of diversity centrality values.

See Also

centrality for computing multiple measures at once.

Examples

mat <- matrix(c(0, .5, .3, .5, 0, .8, .3, .8, 0), 3, 3)
rownames(mat) <- colnames(mat) <- c("A", "B", "C")
centrality_diversity(mat)

DK-based gravity model

Description

The DK-based gravity model (DKGM) puts the degree k-shell index of both ends into a truncated squared-distance gravity sum: DKGM_i=\sum_{j\ne i,\,d(i,j)\le R}DK(i)DK(j)/d(i,j)^2. The mass DK(i)=k(i)+k_s^*(i) adds the degree to an improved shell index k_s^*(i)=k_s(i)+p(i)/(\max_k q(k)+1), where p(i) is the removal stage inside the node's shell and q(k) the number of stages the k-level needed. The stage breaks the ties that degree and k-shell leave behind: two nodes of the same shell are separated by how late the peeling reached them.

Usage

centrality_dkgm(x, dkgm_radius = 2, ...)

Arguments

Details

Stages restart at one inside every shell, but the denominator \max_k q(k)+1 is a single global maximum. A node's mass therefore depends on the whole graph: adding a disconnected component that peels in more stages lengthens that denominator and changes every raw score. This is a property of the published definition, not a cograph choice, and it distinguishes DKGM from centrality_mixed_gravity.

The paper's Algorithm 1 says "Find all nodes in G with degree k" while its stage loop ends "until All remaining nodes in G have degree > k" and its Methods define k-shell by removing "nodes whose degree k <= 1 ... Until there are no nodes in the network with degree k <= 1". Strict equality cannot terminate on a three-node path, so cograph follows the at-most reading, which is the only one consistent with the printed termination condition and which reproduces the paper's Tables 2 to 5. Removal inside a stage is simultaneous, matching the printed two-stage two-shell. Because the level starts at one, an isolate falls in the one-shell rather than the zero-shell centrality(measures = "coreness") reports; isolates carry no edges, so no other node's shell, stage or score is affected.

Uses the simple undirected unweighted skeleton, which is the source domain: either arc creates one edge, parallel edges count once and loops are removed. This projection is a cograph convention outside that domain. Edge weights, mode, cutoff, gravity_mass and path-weight inversion are ignored. Unreachable partners contribute nothing. Isolates and singleton graphs score zero; empty graphs return no scores. Optional maximum normalization applies to the complete result over all nodes. Dense all-pairs distances cost O(n cubed) time and O(n squared) memory.

Numerical verification establishes agreement with the published equations and the printed nine-node example, not parity with author software, which was not located, nor any claim about spreading performance.

Value

Named numeric vector in input node order.

References

Li, Z. and Huang, X. (2021). Identifying influential spreaders in complex networks by an improved gravity model. Scientific Reports, 11, 22194. doi:10.1038/s41598-021-01218-1.

See Also

centrality_mcgm and centrality_mixed_gravity for the other gravity masses.

Examples

centrality_dkgm(igraph::make_ring(6))
centrality_dkgm(igraph::make_star(6), dkgm_radius = 1)

Density of Maximum Neighborhood Component (DMNC)

Description

Edges divided by nodes raised to dmnc_epsilon, both taken from the largest connected component of the subgraph induced on a node's neighbors (the focal node excluded).

Usage

centrality_dmnc(x, mode = "all", dmnc_epsilon = 1.7, ...)

Arguments

Value

Named numeric vector of DMNC values.

Divergence from centiserve

centiserve::dmnc() returns different values, and not only because of its different epsilon default. Its edge count is taken with induced.subgraph(graph, which(c$membership %in% ...)), where the membership vector indexes the neighborhood subgraph but is used to subset the original graph. The two index spaces are not the same, so the edges counted are those of an unrelated vertex set. On the Zachary karate club the two disagree on 14 of 34 nodes at a matched epsilon, and reproducing that indexing exactly reproduces centiserve's output. cograph counts the edges of the component it actually found.

See Also

centrality for computing multiple measures at once, centrality_mnc for the size-only variant.

Examples

adj <- matrix(c(0, 1, 1, 1, 0, 1, 1, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality_dmnc(adj)

Dynamical importance by exact vertex deletion

Description

Restrepo, Ott & Hunt's node dynamical importance is the relative drop in adjacency spectral radius on removing that node: I_i = (\rho(A)-\rho(A_{-i}))/\rho(A) (equation 2). This function recomputes the spectral radius after every deletion. The paper's left/right eigenvector product (equation 5) is an approximation and can differ substantially on small networks; it is not used here.

Usage

centrality_dynamical_importance(x, ...)

Arguments

Details

Supports directed or undirected nonnegative weighted networks. Self-loops are always removed, as in the paper's zero-diagonal definition. Edge weights, weighted and simplify follow the same adjacency conventions as centrality_diffusion_centrality. The measure is invariant to reversing all arcs and ignores mode and path-weight inversion. Disconnected graphs use the spectral radius of the whole graph.

When the original spectral radius is zero (including any directed acyclic graph), the ratio is undefined and all vertices receive NaN. Isolates in a graph with positive spectral radius receive zero. The empty graph returns an empty vector. Strong components are evaluated separately so acyclic parts contribute exactly zero, avoiding numerical eigenvalues of nilpotent blocks. Roundoff in the final ratio is clipped to zero or one.

Repeated eigendecomposition is costly. Select this measure explicitly or use include = "dynamical_importance"; it is held back from the default type = "all" tier.

Value

Named numeric vector in input node order.

References

Restrepo, J. G., Ott, E., & Hunt, B. R. (2006). Characterizing the Dynamical Importance of Network Nodes and Links. Physical Review Letters, 97, 094102. doi:10.1103/PhysRevLett.97.094102.

Examples

centrality_dynamical_importance(igraph::make_full_graph(4))

Dynamics-sensitive centrality

Description

Liu et al.'s finite-time dynamics-sensitive (DS) centrality is S(T)=\sum_{r=0}^{T-1}\beta A[\beta A+(1-\mu)I]^r\mathbf{1}, where beta is the spreading rate and mu the recovery rate (equation 5 in the preprint). This is the full recovery-parameter family. For mu=1, it reduces to \sum_{t=1}^{T}(\beta A)^t\mathbf{1} (equation 7), also the form listed in the Centrality Zoo. For mu=0 it gives the paper's susceptible-infected case.

Usage

centrality_dynamics_sensitive(x, ds_beta = 0.1, ds_mu = 1, ds_steps = 5, ...)

Arguments

Details

Uses the simple undirected unweighted skeleton, as in the source: either direction creates an edge, parallel edges count once and loops are removed. The projection of other inputs is an explicit cograph convention. mode, edge weights and shortest-path weight inversion do not affect this measure. Isolates score zero. T=0 or beta=0 returns zero; T=1 gives beta times degree. The initial seed itself is not added to the score.

This linearized cumulative spreading score allows repeated walks and can exceed the number of nodes. It is not a bounded infection probability or an exact simulation of the nonlinear SIR/SI process. Defaults beta=0.1, mu=1 and T=5 select a parameter setting studied in the paper; they are not fitted to the input network. Any finite horizon is supported without a spectral convergence condition, subject to numerical precision. Overflow raises an error, even if normalization is requested.

Value

Named numeric vector in input node order.

References

Liu, J. G., Lin, J. H., Guo, Q., & Zhou, T. (2016). Locating influential nodes via dynamics-sensitive centrality. Scientific Reports, 6, 21380. doi:10.1038/srep21380.

See Also

centrality_diffusion_centrality.

Examples

g <- igraph::make_ring(5)
centrality_dynamics_sensitive(g, ds_beta = 0.1, ds_mu = 1, ds_steps = 5)
centrality_dynamics_sensitive(g, ds_mu = 0)

Eccentricity

Description

Maximum shortest path distance from a node to any other node. For directed networks, centrality_ineccentricity and centrality_outeccentricity use incoming and outgoing paths.

Usage

centrality_eccentricity(x, mode = "all", ...)

centrality_ineccentricity(x, ...)

centrality_outeccentricity(x, ...)

Arguments

Value

Named numeric vector of eccentricity values.

See Also

centrality for computing multiple measures at once.

Examples

adj <- matrix(c(0, 1, 0, 1, 0, 1, 0, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality_eccentricity(adj)

Effective Size (Burt's)

Description

Network effective size: degree minus redundancy. Measures non-redundant contacts in ego network.

Usage

centrality_effective_size(x, ...)

Arguments

Value

Named numeric vector of effective size values.

See Also

centrality for computing multiple measures at once, centrality_constraint for a related structural holes measure.

Examples

adj <- matrix(c(0, 1, 1, 1, 0, 1, 1, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality_effective_size(adj)

Extended hybrid characteristic centrality

Description

The extended hybrid characteristic centrality of Liu and Zheng is the closed-neighborhood sum of centrality_hcc: EHCC(u)=HCC(u)+\sum_{v\in\phi(u)}HCC(v), the focal node counted once and each neighbor of the open 1-order neighborhood once. It rewards a node whose neighbors are themselves high in both the extended degree and the E-shell hierarchy, which a node can be without being high itself.

Usage

centrality_ehcc(x, ...)

Arguments

Details

Everything recorded on centrality_hcc carries over unchanged: the source's \arg\max/\arg\min typo in step 3 of the E-shell procedure, the original-graph reading of k^{ex} and k^{ex}_{max} against the residual-graph peel, the global and therefore not component-local normalizers, the hcc_delta domain [0,1], the 0/0 of an edgeless graph written as zero, the simple undirected unweighted skeleton, and the ignored weights, mode, cutoff and inversion. Because HCC lies in [0,2], EHCC lies in [0,2(1+k_{max})], and an isolate scores exactly its own HCC.

Value

Named numeric vector in input node order.

References

Liu, J. and Zheng, J. (2023). Identifying important nodes in complex networks based on extended degree and E-shell hierarchy decomposition. Scientific Reports, 13, 3197. doi:10.1038/s41598-023-30308-5.

See Also

centrality_hcc for the summand and list_centralities for the catalogue.

Examples

# On a regular graph every node scores 2, so EHCC is 2 (1 + k).
centrality_ehcc(igraph::make_ring(6))

# The star's center collects every leaf's score as well as its own.
centrality_ehcc(igraph::make_star(6, mode = "undirected"))

Eigenvector Centrality

Description

Influence-based centrality where a node's score depends on the scores of its neighbors. Nodes connected to other high-scoring nodes get higher scores.

Usage

centrality_eigenvector(x, ...)

Arguments

Value

Named numeric vector of eigenvector centrality values.

See Also

centrality for computing multiple measures at once, centrality_pagerank for a random walk variant.

Examples

adj <- matrix(c(0, 1, 1, 1, 0, 1, 1, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality_eigenvector(adj)

Entropy Centrality

Description

Graph-theoretic entropy based on shortest path distribution in the residual graph after removing the node.

Usage

centrality_entropy(x, mode = "all", ...)

Arguments

Value

Named numeric vector of entropy centrality values.

See Also

centrality for computing multiple measures at once.

Examples

adj <- matrix(c(0, 1, 1, 1, 0, 1, 1, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality_entropy(adj)

Entropy Variation

Description

Ai's (2017) vitality measure: the change in the Shannon entropy of a node-level distribution when a node and its links are removed,

EnV_f(i) = I_f(G) - I_f(G - i), \qquad I_f(G) = -\sum_j p_j \log p_j, \quad p_j = \frac{f(j)}{\sum_l f(l)},

with f the degree ("entropy_variation_degree", in-, out- or total degree by mode) or the betweenness ("entropy_variation_betweenness"). Natural logarithm, as in the author's code. The difference is signed: a positive value means the remaining network is less even without the node, a negative value that removing it evens the distribution out. Higher = more important.

Usage

centrality_entropy_variation(
  x,
  of = c("degree", "betweenness"),
  mode = "all",
  ...
)

Arguments

Details

The degree variant is computed in closed form. The betweenness variant recomputes betweenness once per node and costs O(n \cdot nm); it ignores edge weights. Self-loops are counted as igraph counts them. When a deletion leaves every f at zero (for instance betweenness on a clique) that entropy is taken as 0.

Validated against the author's own R code path (iCalEnV() from the paper's repository) to 10^{-15} and against the quantiles of Table 2 of the paper on its 4234-node Snake Idioms network.

Value

Named numeric vector, one value per node, in nats.

References

Ai, X. (2017). Node importance ranking of complex networks with entropy variation. Entropy, 19(7), 303.

See Also

centrality for computing multiple measures at once.

Examples

star5 <- matrix(0, 5, 5)
star5[1, 2:5] <- 1; star5[2:5, 1] <- 1
rownames(star5) <- colnames(star5) <- LETTERS[1:5]
centrality_entropy_variation(star5)
centrality_entropy_variation(star5, of = "betweenness")

Exogenous centrality

Description

Measures a node's contribution to the base centrality of all other nodes, following Everett and Borgatti (2010): E(i)=\sum_{j\ne i}[C_G(j)-C_{G-i}(j)]. The focal node's own base score is excluded. Three base measures are supported, each calculated without normalization before deletion:

reverse_closeness

Default. For a graph H with m remaining nodes, C_H(j)=\sum_{k\ne j}\max(N-d_H(j,k),0), where N is the ORIGINAL input size, including isolates. Unreachable pairs contribute zero. This implements the fixed-size adjustment in section 3.3; it is distinct from ordinary reciprocal-farness closeness.

betweenness

Raw shortest-path betweenness with endpoints excluded. Each unordered pair counts once on undirected graphs; directed pairs count separately. Exogenous contributions can be negative when removing a node increases the remaining nodes' betweenness.

degree

Simple degree in the chosen base direction. On an undirected graph the exogenous result equals degree. On a directed graph, outgoing base degree produces incoming exogenous degree, and incoming base degree produces outgoing exogenous degree.

Usage

centrality_exogenous(
  x,
  mode = "all",
  exogenous_base = "reverse_closeness",
  ...
)

Arguments

Details

Uses the simple binary topology: parallel connections count once, self-loops are removed and weights/path inversion are ignored. Mode "all" projects onto the undirected skeleton; "out" and "in" use directed paths when the input is directed. For undirected input all three modes agree. Empty input returns an empty vector; isolates and singletons score zero. Original size includes other components, so adding an isolate can change reverse-closeness scores of connected nodes even though the isolate's own contribution is zero.

normalized = TRUE applies cograph's final division by a positive maximum, retaining negative values; it does not normalize the base measure, nor apply the paper's theoretical normalization. If the maximum is nonpositive, values remain unchanged. Arbitrary normalized base scores, such as unit-length eigenvectors, are not supported.

Numerical verification uses independent NetworkX base scores, explicit path enumeration and analytical graphs. Some numerical entries in the paper's Florentine tables could not be reproduced from NetworkX's graph plus the Pucci isolate; this implementation follows the stated definition and does not claim complete published-table or UCINET parity.

Betweenness and reverse-closeness require repeated all-pairs distances, with worst-case O(N^4) time using the current dense kernels. The measure is therefore in the costly tier even when the degree base is selected.

Value

Named numeric vector in input node order.

References

Everett, M. G., & Borgatti, S. P. (2010). Induced, endogenous and exogenous centrality. Social Networks, 32(4), 339-344. doi:10.1016/j.socnet.2010.06.004.

Examples

centrality_exogenous(igraph::make_ring(4), exogenous_base = "betweenness")

Expected Centrality

Description

Sum of neighbor degrees. Simple but effective influence proxy.

Usage

centrality_expected(x, mode = "all", ...)

Arguments

Value

Named numeric vector of expected centrality values.

See Also

centrality for computing multiple measures at once.

Examples

adj <- matrix(c(0, 1, 1, 1, 0, 1, 1, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality_expected(adj)

Expected Force centrality

Description

Computes Lawyer's Expected Force after exactly two transmission events without recovery. For each seed, enumerate ordered sequences of two infected-to-susceptible edge transmissions. Each sequence produces a three-node infected cluster with D outgoing edges to susceptible nodes. Normalize these D values across all sequences and take their Shannon entropy using natural logarithms (Lawyer 2015, equation 1).

Usage

centrality_expected_force(x, ...)

Arguments

Details

Different event orders or transmitting parents remain distinct even when they infect the same three nodes. A seed and two adjacent neighbors of an undirected triangle form four sequences, not one. Boundary edges are counted individually even when they reach the same susceptible node. This is not entropy over distinct infected sets or over boundary-degree categories, and is not a probability-weighted epidemic simulation.

Uses the simple unweighted graph, retaining direction. In directed graphs, only outgoing infected-to-susceptible arcs transmit or contribute boundary degree, following the paper's directed extension. Loops and duplicate arcs are removed after generic processing. Weights, mode, inversion and cutoff do not affect the result. The weighted extension and horizons other than two events are outside this implementation.

Zero-degree outcomes use the zero-log-zero entropy limit. If no sequence can perform two transmissions, or every resulting cluster has zero onward force, cograph returns zero. The latter is an explicit extension of the paper's undefined all-zero normalization, not author-code parity. Isolates and components of at most three nodes therefore score zero. A single positive-force outcome also has entropy zero. Empty input returns no scores. The measure is local and does not establish epidemic probability, outbreak size or predictive accuracy on the supplied graph.

Native computation groups three-node clusters by boundary degree while preserving their event multiplicities. Worst-case time is O(n cubed), memory O(n squared), including dense graph preparation. Scores remain independent between components before maximum normalization.

Value

Named numeric vector in input node order.

References

Lawyer, G. (2015). Understanding the influence of all nodes in a network. Scientific Reports, 5, 8665. doi:10.1038/srep08665.

See Also

centrality_modified_expected_force for degree adjustment. centrality_expected computes a different quantity, the sum of neighbor degrees.

Examples

centrality_expected_force(igraph::make_graph("Zachary"))

Expected Influence (one-step)

Description

Signed-weight sum of a node's edges (Robinaugh, Millner & McNally 2016). The appropriate centrality for networks with positive and negative edges (partial-correlation, glasso, signed correlation networks) where treating negative edges as positive magnitudes can be misleading.

Usage

centrality_expected_influence_1(x, mode = "out", ...)

Arguments

Value

Named numeric vector of expected-influence values (signed).

References

Robinaugh DJ, Millner AJ, McNally RJ (2016). Identifying highly influential nodes in the complicated grief network. Journal of Abnormal Psychology, 125(6), 747-757.

See Also

centrality_expected_influence_2 for the two-step variant, centrality_strength for the weighted-degree analogue.

Examples

# Signed weight matrix (partial correlations, for example)
W <- matrix(c( 0.0,  0.5, -0.3,  0.2,
               0.5,  0.0,  0.4, -0.1,
              -0.3,  0.4,  0.0,  0.6,
               0.2, -0.1,  0.6,  0.0), 4, 4, byrow = TRUE)
rownames(W) <- colnames(W) <- c("A", "B", "C", "D")
centrality_expected_influence_1(W)

Expected Influence (two-step)

Description

Two-step signed-weight sum: a node's own expected influence (EI1) plus the weighted sum of its neighbors' EI1 (Robinaugh, Millner & McNally 2016). Captures both the node's direct influence and the influence it exerts indirectly via highly-connected neighbors.

Usage

centrality_expected_influence_2(x, mode = "out", ...)

Arguments

Value

Named numeric vector of two-step expected-influence values.

References

Robinaugh DJ, Millner AJ, McNally RJ (2016). Identifying highly influential nodes in the complicated grief network. Journal of Abnormal Psychology, 125(6), 747-757.

See Also

centrality_expected_influence_1 for the one-step variant.

Examples

W <- matrix(c( 0.0,  0.5, -0.3,  0.2,
               0.5,  0.0,  0.4, -0.1,
              -0.3,  0.4,  0.0,  0.6,
               0.2, -0.1,  0.6,  0.0), 4, 4, byrow = TRUE)
rownames(W) <- colnames(W) <- c("A", "B", "C", "D")
centrality_expected_influence_2(W)

Extended neighborhood coreness

Description

Bae and Kim's extended neighborhood coreness sums the neighborhood coreness of every immediate neighbor: C_{nc+}(i)=\sum_{j\in N(i)}\sum_{l\in N(j)}k_s(l). Equivalently, the score is A^2 k_s. Here k_s is the core-number vector of the original simple undirected graph. Core numbers are not recomputed inside each neighborhood.

Usage

centrality_extended_coreness(x, ...)

Arguments

Details

Every length-two walk contributes its endpoint's core number, including returns to the focal node and repeated endpoints reached via different neighbors. This is not a sum over distinct nodes at distance two. Isolates score zero. On a tree, it equals the sum of neighboring degrees; on a d-regular graph it equals d cubed. A larger score means more access to core-rich neighborhoods; numerical equivalence does not imply superior spreading prediction for every network.

Uses the simple undirected unweighted skeleton: either direction creates an edge, parallel edges count once and loops are removed. This projection is a cograph convention for inputs outside the published domain. Weights, mode and shortest-path weight inversion do not affect the score.

Value

Named numeric vector in input node order.

References

Bae, J., & Kim, S. (2014). Identifying and ranking influential spreaders in complex networks by neighborhood coreness. Physica A, 395, 549-559. doi:10.1016/j.physa.2013.10.047.

Examples

centrality_extended_coreness(igraph::make_ring(6))

Extended gravity centrality

Description

Ma et al.'s extended gravity score is the sum of the immediate neighbors' raw gravity scores: G^+(i)=\sum_{j\in N(i)}G(j), where G(j)=\sum_{l:0<d(j,l)\le r}k_s(j)k_s(l)/d(j,l)^2. Core numbers and hop distances are calculated on the original simple undirected graph. The radius applies around each neighbor j; it is not a radius around the focal node i. A contribution can therefore reach r+1 hops from i, and paths from a neighbor back to i also contribute.

Usage

centrality_extended_gravity(x, gravity_radius = 3, ...)

Arguments

Details

Default radius three is the setting used in the original paper. NULL or infinity includes every reachable partner, excluding the gravity source itself. Radius zero and isolates score zero. The outer neighbor sum has no distance penalty. All inner scores remain raw until the final optional max normalization.

Uses the simple undirected unweighted skeleton, with either direction creating an edge, parallel edges counted once and loops removed. This projection is a cograph convention for other inputs. Edge weights, mode, gravity_mass and path-weight inversion do not affect this measure: its masses are always k-shell indices. Computation includes all-pairs hop distances, so it can be expensive for large graphs.

Value

Named numeric vector in input node order.

References

Ma, L. L., Ma, C., Zhang, H. F., & Wang, B. H. (2016). Identifying influential spreaders in complex networks based on gravity formula. Physica A, 451, 205-212. doi:10.1016/j.physa.2015.12.162.

See Also

centrality_gravity.

Examples

centrality_extended_gravity(igraph::make_ring(6), gravity_radius = 3)

Extended local bridging centrality

Description

Macker's two-hop localized bridging centrality multiplies betweenness of the focal node in its induced closed two-hop neighborhood by its bridging coefficient. Degrees for that coefficient come from the original graph. The ego network includes every edge between the selected vertices. Its shortest paths can be up to four edges long; this is not global betweenness with a path-length cutoff of two. Betweenness uses unordered pairs, excludes endpoints, and is not normalized by ego-network size.

Usage

centrality_extended_local_bridging(x, ...)

Arguments

Details

Uses the same simple undirected unweighted projection and zero conventions as centrality_localized_bridging. Macker's separate weighted model uses link quality for degree and costs for paths; that model is outside this implementation. Native breadth-first path counts cost O(sum over ego networks of n_ego times (n_ego + m_ego)), at worst O(n to the fourth power), with O(n squared) memory. This measure is marked costly and must be selected explicitly or through include.

Value

Named numeric vector in input node order.

References

Macker, J. P. (2016). An improved local bridging centrality model for distributed network analytics. MILCOM, pp. 600-605. doi:10.1109/MILCOM.2016.7795393.

Examples

centrality_extended_local_bridging(igraph::make_graph("Zachary"))

Extended mixed gravitational centrality

Description

Extended mixed gravitational centrality (EMGC), also called IGC+, sums the raw MGC scores of immediate neighbors: EMGC_i=\sum_{j\in N(i)}MGC_j. Each inner MGC score uses its own source node j's core number, partner degrees, and original-graph hop distances. The inner radius is centered on j, so a contribution can reach r+1 hops from i. Paths from j back to i are included. The outer neighbor sum has no distance or mass factor.

Usage

centrality_extended_mixed_gravity(x, gravity_radius = 3, ...)

Arguments

Details

Follows the reproduction in Li and Huang (2022), equation 8, attributed to Wang et al. (2018); the original full equations and software have not been inspected. Uses the same skeleton and radius conventions as centrality_mixed_gravity. Default inner radius three follows the reproduced definition; radius one matches the Zoo's literal inner neighbor sum. Optional maximum normalization occurs only after summing raw neighbor scores. Isolates and radii below one score zero. Empty and singleton graphs give no scores and zero, respectively. Dense O(n cubed) time and O(n squared) memory. Verification of these numerical equations does not establish author-software parity or predictive superiority.

Value

Named numeric vector in input node order.

References

Wang, J., Li, C. and Xia, C. (2018). Improved centrality indicators to characterize the nodal spreading capability in complex networks. Applied Mathematics and Computation, 334, 388-400. doi:10.1016/j.amc.2018.04.028.

Li, Z. and Huang, X. (2022). Identifying influential spreaders by gravity model considering multi-characteristics of nodes. Scientific Reports, 12, 9879. doi:10.1038/s41598-022-14005-3.

Examples

centrality_extended_mixed_gravity(igraph::make_ring(6))

Flow Betweenness Centrality

Description

Max-flow based betweenness centrality.

Usage

centrality_flow_betweenness(x, ...)

Arguments

Value

Named numeric vector of flow betweenness values.

See Also

centrality for computing multiple measures at once, centrality_betweenness for shortest-path variant.

Examples

adj <- matrix(c(0, 1, 1, 1, 0, 1, 1, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality_flow_betweenness(adj)

Gateway Coefficient

Description

Inter-community brokerage weighted by centrality. Combines participation with degree information. Requires community membership.

Usage

centrality_gateway(x, membership = NULL, mode = "all", ...)

Arguments

Value

Named numeric vector of gateway coefficient values (0-1).

See Also

centrality for computing multiple measures at once, centrality_participation for the simpler participation coefficient.

Examples

adj <- matrix(c(0,1,1,0,0, 1,0,1,0,0, 1,1,0,1,0, 0,0,1,0,1, 0,0,0,1,0), 5, 5)
rownames(adj) <- colnames(adj) <- LETTERS[1:5]
centrality_gateway(adj, membership = c(1, 1, 1, 2, 2))

Generalized Closeness Centrality

Description

Sum of alpha^d over all nodes. Generalization of decay centrality matching tidygraph's implementation.

Usage

centrality_generalized_closeness(x, mode = "all", decay_parameter = 0.5, ...)

Arguments

Value

Named numeric vector of generalized closeness values.

See Also

centrality for computing multiple measures at once, centrality_decay (equivalent formulation).

Examples

adj <- matrix(c(0, 1, 0, 1, 0, 1, 0, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality_generalized_closeness(adj)

Gil-Schmidt Power Index

Description

Sum of 1/d(v,w) normalized by (n-1). Variant of closeness using harmonic mean of distances.

Usage

centrality_gilschmidt(x, mode = "all", ...)

Arguments

Value

Named numeric vector of Gil-Schmidt power index values.

See Also

centrality for computing multiple measures at once, centrality_harmonic for a related measure.

Examples

adj <- matrix(c(0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0), 4, 4)
rownames(adj) <- colnames(adj) <- c("A", "B", "C", "D")
centrality_gilschmidt(adj)

Global structure model centrality

Description

The Global Structure Model (GSM) of Ullah et al. (2021) is GSM(i)=\exp(k_s(i)/N)\sum_{j\ne i}k_s(j)/d_{ij}, where k_s denotes original graph core numbers and d denotes hop distances. It combines a focal coreness factor with distance-discounted coreness of other nodes. N is the total original node count, including isolates.

Usage

centrality_global_structure(x, ...)

Arguments

Details

Both GSM and centrality_hybrid_global_structure use the simple undirected skeleton, ignoring weights, mode, path inversion and distance cutoffs. Loops are removed and parallel connections count once. Only reachable partners contribute; this is an explicit disconnected-graph extension. Isolates and singletons score zero, empty input returns an empty vector. Other components can affect results through the global node count and, for H-GSM, its global mean. These are not independent per-component calculations.

Production uses native coreness and all-pairs distance kernels, with worst-case O(N^3) time and O(N^2) memory. Numerical verification uses independent NetworkX cores/distances and exhaustive small-graph oracles. Agreement with a numerical definition does not establish author-software parity or superior epidemic-spreading predictions.

Value

Named numeric vector in input node order.

References

Ullah, A., Wang, B., Sheng, J., Long, J., Khan, N., & Sun, Z. (2021). Identification of nodes influence based on global structure model in complex networks. Scientific Reports, 11, 6173. doi:10.1038/s41598-021-84684-x.

Examples

centrality_global_structure(igraph::make_ring(4))

Graph regularization centrality

Description

Dal Col and Petronetto's graph regularization centrality is GRC_i = 1/[(I+\gamma L)^{-1}]_{ii}, where L is the unnormalized weighted graph Laplacian. The ith column of this inverse minimizes \|s-e_i\|^2+\gamma s^T Ls. A larger score indicates that smoothing retains less of a unit impulse at its source vertex. This implements the centrality with unit impulses; the author's separate signal option returns smoothed signal values and is not this centrality.

Usage

centrality_graph_regularization(x, grc_gamma = 1, ...)

Arguments

Details

grc_gamma accepts any finite nonnegative number, default one. At zero every score is one. Isolates also score one. Within a component of n vertices scores lie between one and n, approaching n as gamma grows without bound. Adding disconnected components does not change existing raw scores. Edge weights and gamma act multiplicatively; uniform weight scaling changes scores unless gamma is adjusted inversely.

Uses finite nonnegative edge weights when weighted = TRUE. Zero weights are absent connections. Unweighted inputs use the simple undirected skeleton. Loops are removed. For weighted directed inputs, opposite arcs are added. The generic simplify argument combines parallel edges first; remaining weighted parallel edges are added. These projections are explicit cograph conventions for the published undirected domain. Generic mode, shortest-path weight inversion and cutoff do not affect the result.

The native dense spectral calculation separates each component's constant eigenvector and evaluates the remaining filter in log space. This supports extreme finite gamma and uniform weight scales without forming their product. Unresolvable weight ranges or positive spectral condition numbers above 1/(64 times machine epsilon) raise an error. Runtime is O(n cubed) and memory O(n squared) per component. Empty graphs return an empty vector.

The author software approximates the same filter with ten Chebyshev terms. This function evaluates the defining inverse to numerical precision; default author-software values need not coincide. Optional normalized = TRUE divides scores by their global maximum.

Value

Named numeric vector in input node order.

References

Dal Col, A., & Petronetto, F. (2023). Graph regularization centrality. Physica A, 628, 129188. doi:10.1016/j.physa.2023.129188.

Dal Col, A. (2023). GRC. Mendeley Data, version 1. doi:10.17632/ns63f5dj86.1.

Examples

centrality_graph_regularization(igraph::make_ring(4), grc_gamma = 0.5)

Gravity centrality

Description

G(i) = \sum_j m_i m_j / d_{ij}^{2}, optionally truncated at gravity_radius. The published members of the family differ only in the mass and the reach:

Usage

centrality_gravity(
  x,
  mode = "all",
  gravity_mass = "kshell",
  gravity_radius = 3,
  ...
)

Arguments

Details

Gravity centrality (Ma, Ma, Zhang & Wang 2016)

k-shell mass, radius 3 – the default.

Gravity model (Li, Ren, Ma, Liu, Zhang & Zhou 2019, eq. 1)

gravity_mass = "degree", gravity_radius = NULL.

Local gravity model (same paper, eq. 2)

gravity_mass = "degree", gravity_radius = "auto", which uses their empirical half-mean-distance heuristic (eq. 5). cograph rounds to the nearest integer (ties to even), with minimum 1, using finite positive distances on disconnected graphs. These rounding and disconnected-graph rules are cograph conventions.

Value

Named numeric vector, one value per node.

Change in 2.4.8

Before 2.4.8 this measure computed \sum_j k_j s_j / d_{ij}^2: the product of degree and k-shell on the partner, no mass at all on the focal node, and no truncation. That is not the formula of Li et al. (2019) that its help page cited, and dropping the focal mass changes the ranking rather than the scale. The default is now Ma et al. (2016). gravity_mass = "legacy" with gravity_radius = NULL reproduces the earlier values exactly.

References

Ma, L.-L., Ma, C., Zhang, H.-F., & Wang, B.-H. (2016). Identifying influential spreaders in complex networks based on gravity formula. Physica A, 451, 205-212.

Li, Z., Ren, T., Ma, X., Liu, S., Zhang, Y., & Zhou, T. (2019). Identifying influential spreaders by gravity model. Scientific Reports, 9, 8387.

See Also

centrality_coreness, centrality_kreach, centrality.

Examples

adj <- matrix(0, 6, 6)
adj[cbind(c(1, 1, 2, 4, 4, 5, 3), c(2, 3, 3, 5, 6, 6, 4))] <- 1
adj <- adj + t(adj)
rownames(adj) <- colnames(adj) <- LETTERS[1:6]
centrality_gravity(adj)
centrality_gravity(adj, gravity_mass = "degree", gravity_radius = NULL)

Harary Centrality

Description

Sum of 1/d^2 over all reachable node pairs. Robust to disconnected graphs.

Usage

centrality_harary(x, mode = "all", ...)

Arguments

Value

Named numeric vector of Harary centrality values.

See Also

centrality for computing multiple measures at once.

Examples

adj <- matrix(c(0, 1, 0, 1, 0, 1, 0, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality_harary(adj)

Harmonic Centrality

Description

Sum of inverse shortest path distances to all other nodes. Unlike closeness, harmonic centrality handles disconnected graphs naturally (unreachable nodes contribute 0 instead of making the measure undefined).

Usage

centrality_harmonic(x, mode = "all", ...)

centrality_inharmonic(x, ...)

centrality_outharmonic(x, ...)

Arguments

Value

Named numeric vector of harmonic centrality values.

See Also

centrality for computing multiple measures at once, centrality_closeness for the traditional variant.

Examples

adj <- matrix(c(0, 1, 1, 1, 0, 1, 1, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality_harmonic(adj)

Hybrid characteristic centrality

Description

Liu and Zheng's hybrid characteristic centrality adds a local and a global characteristic on the same scale: HCC(u)=k^{ex}(u)/k^{ex}_{max}+pos(u)/pos_{max}. The local half is the extended degree k^{ex}(u)=\delta k(u)+(1-\delta)\sum_{v\in\phi(u)}k(v), the node's own degree blended with its neighbors'; the global half is the E-shell position index, the round in which a repeated minimum-extended-degree peel removes the node. Both terms are divided by their largest value, so each lies in [0,1] and the raw score lies in [0,2].

Usage

centrality_hcc(x, ...)

Arguments

Details

The E-shell hierarchy decomposition is not a k-shell decomposition, although the Centrality Zoo describes it as "a variant of k-shell decomposition". There is no outer loop over a shell index and no repeat-until-stable inner loop: each round removes exactly the set of remaining nodes attaining the current minimum extended degree, recomputes the extended degrees on what is left, and tags the removed nodes with the round number. The position indexes therefore run 1,\dots,pos_{max} with every value attained, rather than being shell numbers, and the two procedures disagree on the source's own figure 1.

The source's printed algorithm contains a typo, and cograph implements the correction its own tables require. Step 3 of the E-shell procedure prints S_p=\arg\max_{u\in G_p}\{k^{ex}(u)\} while the same sentence calls S_p "the set of minimum nodes", the preceding paragraph says "the nodes with minimum extended degree are found and deleted", and the paper's table 2 heads its column "Minimum extended degree" with the increasing values 2, 2.5, 3, 4.5, 5, 6. The minimum reading reproduces every printed row; the literal maximum reading is a different measure.

The peel recomputes but equation (4) does not. Step 6 updates the extended degrees on the residual graph after every removal, which is what the printed table 2 minima require. The k^{ex}(u) of equation (4), and the k^{ex}_{max} it is divided by, are nevertheless the original-graph values: the paper's own worked line HCC(a)=4.5/11+4/6 uses the original 4.5 and the original maximum 11, and its node d settles the question, since its original 9.5 gives the printed 1.86 while its residual 6 at removal time would give 1.55.

Raw scores are not component-local. k^{ex}_{max} and pos_{max} are single global constants, so adding a disconnected component – an isolate included – can change every score, and not merely by a common factor, because the two terms rescale independently. The source does not discuss disconnected graphs.

Degenerate cases are cograph decisions, not the source's. An isolate has extended degree zero, which for \delta\in[0,1] is the global minimum, so it always leaves in the first round with pos=1. On an edgeless graph every extended degree is zero and k^{ex}_{max}=0, making the first term 0/0; it is written as zero, which leaves the E-shell term alone. One round then removes everything, so every node of an edgeless graph – a singleton included – scores exactly 1. Empty graphs return no scores.

hcc_delta defaults to the source's 0.5 and is restricted to the source's stated domain [0,1], where \delta=1 recovers the classical degree and \delta=0 drops the node's own degree entirely. Values outside that interval are refused with a cograph_bad_parameter error rather than extended: they make the extended degree negative on some graphs, and then equation (4) divides by a nonpositive maximum, which the source never contemplates.

Uses the simple undirected unweighted skeleton, the source's stated domain: either arc creates one edge, parallel edges count once and loops are removed. Edge weights, mode, cutoff and path-weight inversion are ignored, and directed input is symmetrized rather than read as a directed case, which the source does not define. The source states no further normalization; normalized = TRUE max-scales the finished vector as elsewhere in centrality, on top of the two divisions equation (4) already performs. Cost is one dense matrix-vector product per peeling round, so O(n^3) in the worst case rather than the O(n+m) a sparse min-heap would give.

Numerical verification establishes agreement with the definition and with the values the source prints for its figure 1, not parity with author software, which does not exist, and not any claim about spreading performance.

Value

Named numeric vector in input node order.

References

Liu, J. and Zheng, J. (2023). Identifying important nodes in complex networks based on extended degree and E-shell hierarchy decomposition. Scientific Reports, 13, 3197. doi:10.1038/s41598-023-30308-5.

See Also

centrality_ehcc for the neighborhood sum of this score, centrality_dkgm for another shell-and-degree hybrid, and list_centralities for the catalogue.

Examples

# Every node of a regular graph has the same extended degree, so one
# round removes the whole graph and every node scores 1 + 1 = 2.
centrality_hcc(igraph::make_ring(6))

# A star peels its leaves first and its center second.
centrality_hcc(igraph::make_star(6, mode = "undirected"))

# delta = 1 is the classical degree in the extended-degree slot.
centrality_hcc(igraph::make_star(6, mode = "undirected"), hcc_delta = 1)

Heatmap, Flow Coefficient, Local Entropy, Weighted h-index, Redundancy

Description

Five local measures.

Usage

centrality_heatmap(x, mode = "all", ...)

centrality_flow_coefficient(x, ...)

centrality_local_entropy(x, mode = "all", ...)

centrality_weighted_h_index(x, mode = "all", ...)

centrality_redundancy(x, ...)

Arguments

Details

heatmap (Duron 2020)

Farness minus the mean farness of the neighbors, C(v) = f(v) - \frac{1}{k_v} \sum_{u \in N(v)} f(u), with f the sum of hop distances to reachable nodes. Lower is more central. Isolates score NaN. Reproduces Table 1 of the paper.

flow_coefficient (Honey et al. 2007)

Among ordered pairs of distinct neighbors, the fraction joined by a two-step path through the node but not by a direct link, as implemented in the Brain Connectivity Toolbox. On an undirected graph it equals one minus the clustering coefficient; it carries new information only on directed graphs. Nodes with fewer than two neighbors score 0.

local_entropy (Nie et al. 2016)

-\sum_{j \in N(i)} k_j \ln k_j, as printed by the sources. Always non-positive and more negative for larger, denser neighborhoods, so lower is more central; isolates score 0, the maximum. The original article is closed access; the formula is that of the Zoo and of Omar and Plapper's 2021 survey, which agree.

weighted_h_index (Gao et al. 2019)

h-index of the multiset in which each neighbor j contributes the topological weight k_i k_j repeated k_j times. Edge weights on the input play no role.

redundancy (Burt 1992; Borgatti 1997)

Mean degree of the node's neighbors within its ego network, 2 t_i / k_i; equal to degree minus effective size. Higher = fewer structural holes. Reproduces Borgatti's worked example.

heatmap, local_entropy and weighted_h_index follow mode; the others ignore direction. Edge weights are ignored.

Value

Named numeric vector, one value per node.

References

Duron, C. (2020). Heatmap centrality: A new measure to identify super- spreader nodes in scale-free networks. PLOS ONE, 15(7), e0235690.

Honey, C. J., Kotter, R., Breakspear, M., & Sporns, O. (2007). Network structure of cerebral cortex shapes functional connectivity on multiple time scales. PNAS, 104(24), 10240-10245.

Nie, T., Guo, Z., Zhao, K., & Lu, Z.-M. (2016). Using mapping entropy to identify node centrality in complex networks. Physica A, 453, 290-297.

Gao, L., Yu, S., Li, M., Shen, Z., & Gao, Z. (2019). Weighted h-index for identifying influential spreaders. Symmetry, 11(10), 1263.

Borgatti, S. P. (1997). Structural holes: Unpacking Burt's redundancy measures. Connections, 20(1), 35-38.

See Also

centrality_effective_size, centrality_transitivity.

Examples

star5 <- matrix(0, 5, 5)
star5[1, 2:5] <- 1; star5[2:5, 1] <- 1
rownames(star5) <- colnames(star5) <- LETTERS[1:5]
centrality_heatmap(star5)
centrality_weighted_h_index(star5)
centrality_redundancy(star5)

Hubbell Centrality

Description

Hubbell (1965) input-output centrality: C = (I - w W)^{-1} \mathbf{1}, where W is the (weighted) adjacency matrix and w is a weight factor that must satisfy w \cdot \rho(W) < 1 for the system to be solvable.

Usage

centrality_hubbell(x, hubbell_weight = 0.5, ...)

Arguments

Details

Bit-exact match against centiserve::hubbell when edge weights are passed explicitly (cograph mirrors centiserve's full-inverse LAPACK call path).

Value

Named numeric vector of Hubbell centrality values (or NA if the system is not solvable).

Note on centiserve equivalence

centiserve::hubbell(g, weights = NULL) silently resets all edge weights to 1, ignoring the graph's weight attribute. To reproduce cograph's values with centiserve on a weighted graph, pass weights = igraph::E(g)$weight explicitly.

References

Hubbell, C. H. (1965). An input-output approach to clique identification. Sociometry, 28(4), 377-399.

See Also

centrality, centrality_katz.

Examples

# Small weighted path graph; spectral radius permits weightfactor = 0.5
adj <- matrix(0, 4, 4)
adj[1,2] <- adj[2,1] <- adj[2,3] <- adj[3,2] <- adj[3,4] <- adj[4,3] <- 0.3
rownames(adj) <- colnames(adj) <- LETTERS[1:4]
centrality_hubbell(adj, hubbell_weight = 0.5)

Hybrid global structure model centrality

Description

Mukhtar et al.'s H-GSM (2023) uses s_i=\exp(k_s(i)k_i/N), a=\lceil\log_2(N^{-1}\sum_i s_i)\rceil, and H\text{-}GSM(i)=s_i\sum_{j\ne i}s_j/d_{ij}^{a}. k_i is simple degree, k_s(i) is original coreness, and d is hop distance. The ceiling exponent is computed from the mean self-influence over ALL original nodes, including isolates whose self-influence is one. The factor s_i alone is not the final centrality score.

Usage

centrality_hybrid_global_structure(x, ...)

Arguments

Details

Topology and disconnected-graph conventions are shared with centrality_global_structure. The adaptive exponent is used exactly as specified, including its discontinuities at powers of two; it is not smoothed or replaced by a fixed exponent.

Self-influence, its mean and final sums are evaluated in logarithmic form. Raw scores exceeding double precision raise an error. With normalized = TRUE, final scores are computed directly as exponentials of log-score differences, so normalized results remain available even when raw scores overflow. Extremely small normalized ratios may underflow to zero. Normalization is applied to the complete score, not separately to self-influence or neighbor contributions.

Value

Named numeric vector in input node order.

References

Mukhtar, M. F., et al. (2023). Integrating local and global information to identify influential nodes in complex networks. Scientific Reports, 13, 11411. doi:10.1038/s41598-023-37570-7.

Examples

centrality_hybrid_global_structure(igraph::make_ring(4))

Immediate Effects Centrality

Description

Friedkin's immediate effects centrality scores a node by how quickly the rest of the network's influence reaches it. Actors whose effects travel over long sequences of interpersonal influence are more dependent on intervening actors than those whose effects travel over short ones, so the measure is the reciprocal of the mean length of the influence sequences that end at a node. Writing W for the row-stochastic influence matrix, c for its left eigenvector at eigenvalue one, Z=(I-W+\mathbf{1}c')^{-1} for the fundamental matrix, Z_{dg} for Z with its off-diagonal entries set to zero and E for the all-ones matrix, the mean lengths are M=(I-Z+EZ_{dg})\,\mathrm{diag}(1/c) and the score is c_{IEC}(j)=(n-1)/\sum_{i\neq j}m_{ij}. M is the mean first passage time matrix of the chain, so the sum runs down column j and a high score marks a node the network reaches fast.

Usage

centrality_iec(x, ...)

Arguments

Details

The influence matrix carries a unit self-loop, and the self-loop is load-bearing. The source builds W by setting the diagonal of the adjacency matrix to one and dividing each row by its sum, w_{ij}=a_{ij}/\sum_j a_{ij} with a_{ii}=1, a construction it attributes to French (1956) and states twice on page 1494, once in the body and once in the note to Table 1. Its footnote 10 says why the diagonal is there: a strong network with w_{ii}>0 must be regular, meaning aperiodic, and its footnote 9 gives the two-cycle counterexample that a zero diagonal admits. An implementation that drops the self-loop is not computing this measure on a different scale, it is computing a different measure.

This is not cograph's centrality_markov, and the difference is not a rescaling. The two are both built from mean first passage times and are easy to confuse – cograph's own candidate ledger confused them for several rounds – but they differ twice over. markov normalizes A without adding the diagonal, and it divides the column sum by n, counting the excluded diagonal entry, where equation (20) divides by n-1. The second difference is a constant factor n/(n-1) and cannot reorder anything; the first can and does. On the five-node star markov gives 1.25, 0.161, 0.161, 0.161, 0.161 where iec gives 0.5, 0.08, 0.08, 0.08, 0.08, and the two rank the nodes differently on 2 of the 21 connected five-node graphs. Both are kept: markov is the older behavior that existing results depend on, iec is Friedkin's published measure.

Reducible input is refused, not extended. Equation (11) needs an irreducible chain. Without one the eigenvector of equation (9) has a dimension per closed class, so c is not determined, and \mathrm{diag}(1/c) is undefined wherever c vanishes. The danger is that the closed form does not announce the failure: for i and j in different blocks z_{ij}=0, and equation (11) then returns the entirely finite m_{ij}=z_{jj}/c_j in place of an infinite mean first passage time. Rather than publish a finite wrong number, cograph tests the chain first and returns NA at every node with a cograph_undefined_measure warning. In practice the test is connectedness of an undirected graph and strong connectedness of a directed one, since the mandated self-loops settle aperiodicity for free. Friedkin restricts his own analysis to regular networks and never defines the measure outside them. centrality_rsp_betweenness answers on disconnected input because its source states a rule for an unreachable pair; this one states none, and a component-wise reading would additionally have to invent whether the n-1 of equation (20) counts the component or the network.

A singleton is NA and an empty graph returns no scores. Equation (20) divides by n-1, which is zero when n=1; the same NA and the same warning follow. An isolate never appears on its own, because a graph containing one is reducible and is already NA everywhere.

Direction is kept; weights, loops and parallel edges are not. W is a matrix of directed influence, row i being what actor i attends to, so a directed input is used as it stands and the measure needs a strongly connected one. There is no in/out/all variant to choose between, so mode, cutoff and invert_weights are ignored. Weights are dropped, deliberately: a_{ii}=1 is calibrated against a_{ij}=1, so multiplying every weight by a constant would silently re-weight each actor's self-reliance against the network, and the source demonstrates only the binary case. Loops in the input are absorbed by the mandated unit diagonal and parallel edges collapse, since a_{ij}=1 "wherever a line exists between two points". The source states no normalization, so normalized = TRUE max-scales the finished vector as elsewhere in centrality.

The source prints a complete numerical fixture. Table 1, pages 1492-1494, gives this measure to three decimals for every node of all 21 connected non-isomorphic five-node graphs. All 105 printed values are reproduced by this implementation; see the batch 49 published audit in the package's verification directory.

Value

Named numeric vector in input node order, NA at every node when the influence chain is reducible or the graph has one node.

References

Friedkin, N. E. (1991). Theoretical foundations for centrality measures. American Journal of Sociology, 96(6), 1478-1504. doi:10.1086/229694.

See Also

centrality_markov for the older, and different, mean-first-passage measure, centrality_random_walk for another chain-based score, and list_centralities for the catalogue.

Examples

# On a complete graph W = J/n, so Z = I, every mean first passage time is
# n, and the score is (n - 1) / (n (n - 1)) = 1/n. Friedkin's Table 1
# prints .200 for the five-node case.
centrality_iec(igraph::make_full_graph(5))

# The five-node star is row 1 of that table: .500 at the center and .080
# at each leaf.
centrality_iec(igraph::make_star(5, mode = "undirected"))

# A disconnected graph has no answer: the influence chain is reducible,
# so every node is NA and a warning says why.
two <- matrix(0, 4, 4)
two[1, 2] <- two[2, 1] <- two[3, 4] <- two[4, 3] <- 1
tryCatch(centrality_iec(two), warning = conditionMessage)

Improved iterative resource allocation (IIRA)

Description

IIRA is centrality_ira with the receiver's share scaled by how much of a spreading process that receiver could actually carry: a_{ij}=[1-(1-\beta)^{k_i}]\,\theta_i (\sum_{u\in\Gamma(j)}\theta_u)^{-1}, where k_i is the degree of i and \beta the spreading rate. The recursion and the initial condition I(0)=(1,\dots,1) are unchanged; there is no \alpha exponent, and the denominator keeps the plain masses.

Usage

centrality_iira(
  x,
  ira_mass = "coreness",
  iira_beta = 0.2,
  iira_steps = 50,
  ...
)

Arguments

Details

The scores are tiny and only their order means anything. The factor \psi_i=1-(1-\beta)^{k_i} is strictly below one, so every column of A sums to less than one, the spectral radius is below one, and I(t)\to 0 geometrically. The source runs exactly t=50 steps and prints an I(50) of order 10^{-20}; cograph returns that raw vector, so the printed example is reproducible, and normalized = TRUE max-scales it into [0,1] for reading. Never compare raw IIRA scores across connected components: each component decays at its own rate, so after iira_steps steps they sit on different exponential scales. A large iira_steps underflows to zero.

The Centrality Zoo entry is not this formula. Section 2.185 prints p_{ij}=(1-(1-\beta)^{d_i})a_{ij}c_i/\sum_k a_{ik}c_k, which pairs the numerator's index with the denominator's own neighborhood; the source pairs them with opposite sets. As printed, the Zoo's row sums are \psi_i c_i d_i/\sum_{k\in N(i)}c_k, so its matrix is stochastic in neither direction although the entry calls it stochastic, and it does not reproduce the source's printed matrix or its printed I(50). cograph implements the source.

Uses the simple undirected unweighted skeleton, which is the source domain: either arc creates one edge, parallel edges count once and loops are removed. Edge weights, mode, cutoff and path-weight inversion are ignored. An isolate has an empty neighbor sum and \psi=0, so it scores zero from the first step; that is the value of the source's empty sum, not an accidental zero. iira_steps = 0 returns the initial I(0), a vector of ones. Empty graphs return no scores. Cost is one dense n^2 matrix plus iira_steps matrix-vector products.

The version of record was not read: what was read is the author preprint arXiv:1505.03214v1, whose method section, worked example and figures carry the definition reproduced here. Numerical verification establishes agreement with those equations and with every value printed in the preprint's figure 2 example, not parity with author software, which does not exist, and not any claim about spreading performance.

Value

Named numeric vector in input node order.

References

Zhong, L.-F., Liu, J.-G. and Shang, M.-S. (2015). Iterative resource allocation based on propagation feature of node for identifying the influential nodes. Physics Letters A, 379(38), 2272-2276. doi:10.1016/j.physleta.2015.05.021.

See Also

centrality_ira for the measure this improves, and list_centralities for the catalogue.

Examples

# The source's figure 2, whose printed I(50) is
# 8.19e-20, 4.32e-20, 4.32e-20, 6.7e-21, 6.7e-21
fig2 <- igraph::make_graph(c(1, 2, 1, 3, 2, 3, 1, 4, 1, 5),
                           directed = FALSE)
centrality_iira(fig2)

# Only the order carries meaning, so max-scale for reading
centrality_iira(fig2, normalized = TRUE)

Improved closeness centrality

Description

Luan et al.'s improved closeness is ICC(i)=(n-1)/\sum_{j\ne i}d_{ij}/\sigma_{ij}^{\alpha}, where d is the hop distance and sigma counts shortest paths. Multiple shortest paths reduce the effective distance to a partner. At alpha zero this is ordinary normalized closeness on a connected graph; on a tree it is independent of alpha because each pair has one shortest path. Scores need not be bounded by one.

Usage

centrality_improved_closeness(x, icc_alpha = 0.2, ...)

Arguments

Details

Uses the simple undirected unweighted skeleton: either direction creates an edge, parallel edges count once and self-loops are removed. Weights, mode and path-weight inversion do not affect the result. These are explicit cograph projections to the published domain.

In a disconnected graph, every node has an unreachable partner and therefore scores zero under the global infinite-distance convention. Singletons score zero by an explicit cograph convention for the otherwise undefined zero-over-zero expression. For within-component scores, supply each component separately. Empty input returns an empty vector.

Breadth-first traversal counts shortest paths in logarithmic form, avoiding overflow when the number of paths exceeds double precision. Extremely small effective-distance terms can underflow to zero, but direct-neighbor terms remain one and keep the denominator positive. Computation costs O(n times (n+m)) with an additional dense adjacency representation. Default alpha 0.2 is a setting studied in the source, not an estimate or a guarantee of optimal spreading predictions.

Value

Named numeric vector in input node order.

References

Luan, Y., Bao, Z., & Zhang, H. (2021). Identifying Influential Spreaders in Complex Networks by Considering the Impact of the Number of Shortest Paths. Journal of Systems Science and Complexity, 34, 2168-2181. doi:10.1007/s11424-021-0111-7.

Examples

centrality_improved_closeness(igraph::make_ring(4), icc_alpha = 0.2)

Improved global structure model centrality

Description

The IGSM definition reproduced in Mukhtar et al. (2023), equation 5, is IGSM(i)=\exp(k_i/N)\sum_{j\ne i}k_j/d_{ij}^{a}, with a=\lceil\log_2(\overline{k})\rceil. The original method is attributed to Zhu and Wang (2022); the exact equation used here was checked in the later primary experimental paper, not its original full text. IGSM uses simple degrees rather than GSM's core numbers, and its distance exponent depends on global mean degree, including isolates.

Usage

centrality_improved_global_structure(x, ...)

Arguments

Details

Topology, normalization and disconnected-graph conventions follow centrality_global_structure. For a positive mean degree below one, the exponent may be zero or negative; it is not clamped. With a negative exponent, more distant reachable partners contribute more, an explicit consequence of extending the equation to sparse disconnected inputs. Unreachable partners still contribute zero. Edgeless graphs score zero by an explicit extension because the logarithm of zero in the exponent is otherwise undefined.

This implements IGSM itself, without an additional nearest-neighbor aggregation for the extended IGSM variant.

Value

Named numeric vector in input node order.

References

Zhu, J.-C., & Wang, L.-W. (2022). An extended improved global structure model for influential node identification in complex networks. Chinese Physics B, 31, 068904. doi:10.1088/1674-1056/ac380d.

Examples

centrality_improved_global_structure(igraph::make_ring(4))

Information Centrality (Stephenson-Zelen)

Description

Information centrality (Stephenson & Zelen 1989) measures a node's importance in terms of the "information" contained in all paths (not only shortest) passing through it. Defined via the inverse of a Laplacian-like matrix, yielding per-node IC_i = 1 / (C_{ii} + (\mathrm{tr}(C) - 2 R_i) / n) where C = A^{-1} and R_i is the row sum of C.

Usage

centrality_information(x, ...)

Arguments

Details

Bit-exact match against sna::infocent on connected undirected graphs (cograph mirrors sna's exact construction and call sequence).

Value

Named numeric vector of information centrality values.

References

Stephenson, K., & Zelen, M. (1989). Rethinking centrality: Methods and examples. Social Networks, 11(1), 1-37.

See Also

centrality, centrality_current_flow_closeness.

Examples

adj <- matrix(c(0,1,1,0, 1,0,1,1, 1,1,0,1, 0,1,1,0), 4, 4)
rownames(adj) <- colnames(adj) <- LETTERS[1:4]
centrality_information(adj)

Integration Centrality

Description

Distance-based influence: sum of 1 - (d-1)/max(d) over all nodes.

Usage

centrality_integration(x, mode = "all", ...)

Arguments

Value

Named numeric vector of integration centrality values.

See Also

centrality for computing multiple measures at once.

Examples

adj <- matrix(c(0, 1, 0, 1, 0, 1, 0, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality_integration(adj)

Iterative resource allocation (IRA)

Description

Every node starts with one unit of resource and hands it to its neighbors in proportion to the receiver's centrality, repeatedly, until the amounts stop moving. The share node j sends to a neighbor i is a_{ij}=\theta_i^{\alpha}/\sum_{u\in\Gamma(j)}\theta_u^{\alpha}, the recursion is I(t+1)=AI(t) from I(0)=(1,\dots,1), and the steady state I ranks the spreaders. Because every non-isolate column of A sums to one, the total resource is conserved: \sum_i I_i(t)=n at every step on a graph with no isolates, and each connected component keeps its own vertex count.

Usage

centrality_ira(
  x,
  ira_mass = "coreness",
  ira_alpha = 1,
  ira_tol = 1e-06,
  ira_max_iter = 1000,
  ...
)

Arguments

Details

The equilibrium has a closed form. Writing s_i=\sum_{u\in\Gamma(i)} \theta_u^{\alpha}, the limit is I_i\propto\theta_i^{\alpha}s_i within each component, scaled so the component's scores sum to its size. On the source's own figure 1(a) that reproduces the printed [15/8, 5/4, 5/4, 5/16, 5/16] exactly. cograph nevertheless iterates, because the iteration is what the source defines and what its table reports, and because the closed form is a limit that need not exist; see the next paragraph.

The iteration does not always converge, and cograph says so. A is the transition matrix of a reversible walk, so on a bipartite component it has an eigenvalue of exactly -1. The coefficient of that eigenvector in I(0)=(1,\dots,1) is the difference in size between the component's two vertex classes, so the iteration settles into a period-two cycle, never meets ira_tol, and returns a value that depends on the parity of the last step. The three-star alternates for ever between (3,1/3,1/3,1/3) and (1,1,1,1), while the four-path, whose classes are equal, converges to (2/3,4/3,4/3,2/3). Neither the source nor the Centrality Zoo mentions this. cograph runs the source's own rule, stops at ira_max_iter, raises a cograph_no_converge warning naming the largest remaining change, and returns I at ira_max_iter. It does not silently report that iterate as an equilibrium, and it does not substitute the average of the two alternating iterates, which would converge but is not the source's rule. Every graph in the source's own figure 1 carries a triangle and converges.

The Centrality Zoo (section 2.204) states the transpose, p_{ij}=a_{ij}c_j^{\alpha}/\sum_k a_{ik}c_k^{\alpha}, and asks for the principal left eigenvector of P. That is the same object up to scale on a graph where the limit exists, but it is not the source's finite iteration: it sidesteps the parity problem instead of reporting it, and it carries no \sum_i I_i=n scale.

Uses the simple undirected unweighted skeleton, which is the source domain: either arc creates one edge, parallel edges count once and loops are removed. Edge weights, mode, cutoff and path-weight inversion are ignored. An isolate is in nobody's neighborhood, so it receives nothing and its own unit is not passed on: it scores zero from the first step, which is the value of the source's empty sum and not an accidental zero, and it is the reason \sum_i I_i=n is stated only for graphs with no isolates. Empty graphs return no scores. Cost is one dense n^2 matrix plus one matrix-vector product per iteration.

Numerical verification establishes agreement with the source equations and with every value printed in the source's table 1, not parity with author software, which does not exist, and not any claim about spreading performance.

Value

Named numeric vector in input node order.

References

Ren, Z.-M., Zeng, A., Chen, D.-B., Liao, H. and Liu, J.-G. (2014). Iterative resource allocation for ranking spreaders in complex networks. EPL (Europhysics Letters), 106(4), 48005. doi:10.1209/0295-5075/106/48005.

See Also

centrality_iira for the improved variant, and list_centralities for the catalogue.

Examples

# The source's figure 1(a): a triangle with two pendants on one corner.
# The printed steady state is 15/8, 5/4, 5/4, 5/16, 5/16.
fig1a <- igraph::make_graph(c(1, 2, 1, 3, 2, 3, 1, 4, 1, 5),
                            directed = FALSE)
centrality_ira(fig1a)

# The source's other mass, and a nonlinear exponent
centrality_ira(fig1a, ira_mass = "degree", ira_alpha = 2)

Katz Centrality

Description

Katz (1953) status index: C = (I - \alpha A^T)^{-1} \mathbf{1}. Each node's score sums attenuated walks of every length back to it, with attenuation \alpha applied per step. Rankings are identical to Bonacich's alpha centrality with a uniform exogenous vector.

Usage

centrality_katz(x, katz_alpha = 0.1, ...)

Arguments

Details

Equivalence is verified bit-exact against centiserve::katzcent (cograph mirrors centiserve's exact LAPACK call sequence) and at machine epsilon against igraph::alpha_centrality(exo = 1) and networkx.katz_centrality_numpy.

Value

Named numeric vector of Katz centrality values.

References

Katz, L. (1953). A new status index derived from sociometric analysis. Psychometrika, 18(1), 39-43.

See Also

centrality, centrality_eigenvector, centrality_pagerank.

Examples

adj <- matrix(c(0, 1, 1, 1, 0, 1, 1, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality_katz(adj)

KED method centrality

Description

The KED method of Chen, Xiao, Zeng and Zhang combines how many local paths leave a node with how diverse they are: KED(i)=k_i\,(1+H_i)\,\exp(K_i/N), where K_i=\sum_{j\in N(i)}k_j is the sum of the neighbors' degrees, H_i=\bigl(\sum_{j\in N(i)}-p_j\log p_j\bigr)/\log k_i with p_j=k_j/K_i is the normalized entropy of the neighbor-degree distribution, and N is the number of nodes in the whole graph. The source calls K_i the local path number and H_i the path diversity: two nodes of equal degree with equally many second neighbors are separated by how evenly their neighbors carry those paths.

Usage

centrality_ked(x, ...)

Arguments

Details

H_i is a ratio of two logarithms in the same base – equation (2) divides the entropy by the entropy of the uniform distribution on k_i outcomes – so the base cancels and no choice of base is being made. H_i lies in [0,1] and is exactly one when the neighbor degrees are all equal, which is the source's 1\le E_i\le 2.

The measure takes no parameters. Equation (6) is a bare product; the exponents \alpha and \beta the Centrality Zoo attributes to Chen et al. appear nowhere in the paper, and none is offered here.

Two degenerate cases are cograph decisions, not the source's. A node with one neighbor has p=1, so its entropy is zero, and its normalizer \log k_i is zero too: H_i is 0/0 and is written as zero, giving E_i=1. That is the value approached from k_i=2 as one neighbor's share vanishes, and the one that gives a node with a single path the least path diversity; the alternative reading of 0/0 as "the entropy equals its own maximum, so H=1" would double every leaf's score. An isolate has both sums empty; H_i is written as zero there as well, and the score is zero whatever finite E_i is chosen, because k_i multiplies the product. Empty graphs return no scores.

Raw scores are not comparable across graphs of different order. N in D_i is the vertex count of the whole network, as the source's own table 1 defines it, so adding a disconnected component – an isolate included – changes every score, and unlike a plain rescaling it can also change the ranking, because \exp(K_i/N) shrinks the large K_i more than the small.

The source's stated range 1\le D_i\le e is not general. It holds exactly when K_i\le N, which is true of the sparse toy networks of its figure 1 and false on dense graphs: every node of K_5 has K_i=16 against N=5, so D_i=e^{3.2}. cograph implements the formula, not the range claim. Scores can therefore be large; K_i/N\le (n-1)^2/n, so nothing overflows below about 710 vertices even on a complete graph, and an overflow beyond that raises an error rather than returning Inf.

This is not the Centrality Zoo's formula. Zoo section 2.215 writes c_{KED}(i)=k_i E_i^\alpha D_i^\beta with E_i=\bigl(\sum_{j}-p_j\log p_j\bigr)/\log k_i and D_i=\exp(K_i/\max_l K_l): it drops the 1+ from E_i and divides by the largest cluster degree instead of by N. On the source's own figure 1 that reading gives 13.5914 and 6.5672 where the paper prints 25.9187 and 19.2212, which cograph reproduces. The \max_l K_l denominator is a plausible misreading, since it makes the paper's stated 1\le D_i\le e hold, but it reproduces neither printed number. cograph implements the paper and offers no Zoo variant.

Uses the simple undirected unweighted skeleton, the source's undirected domain: either arc creates one edge, parallel edges count once and loops are removed. Edge weights, mode, cutoff and path-weight inversion are ignored. The source also defines a directed variant (its equation 3, replacing the neighborhood by the out-neighborhood and k_i by k_i^{out}); that variant is not implemented, so a directed input is symmetrized rather than being read as the paper's directed case. The source states no normalization; normalized = TRUE max-scales the finished vector as elsewhere in centrality. Cost is two sparse matrix-vector products, O(n + m).

Numerical verification establishes agreement with the two scores the source prints for its figure 1, not parity with author software, which does not exist, and not any claim about spreading performance.

Value

Named numeric vector in input node order.

References

Chen, D.-B., Xiao, R., Zeng, A. and Zhang, Y.-C. (2014). Path diversity improves the identification of influential spreaders. Europhysics Letters, 104(6), 68006. doi:10.1209/0295-5075/104/68006.

See Also

centrality_lnc and centrality_neighbor_distance for other neighbor-degree sums, centrality_entropy for a plain neighborhood entropy, and list_centralities for the catalogue.

Examples

# Every node of a ring has two neighbors of degree two, so the
# neighbor degrees are even, H is one and the score is 4 exp(4 / n)
centrality_ked(igraph::make_ring(8))

# A star: the center's neighbors are all leaves, so H is one again,
# and the center scores exactly 2 (n - 1) times a leaf, here 10
centrality_ked(igraph::make_star(6, mode = "undirected"))

Geodesic K-Path Centrality

Description

Count of nodes reachable within shortest path distance k. Measures how many nodes a given node can reach quickly.

Usage

centrality_kreach(x, mode = "all", k = 3, ...)

Arguments

Value

Named numeric vector of k-reach centrality values.

See Also

centrality for computing multiple measures at once.

Examples

adj <- matrix(c(0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0), 4, 4)
rownames(adj) <- colnames(adj) <- c("A", "B", "C", "D")
centrality_kreach(adj, k = 2)

Local Average Connectivity (LAC)

Description

Average degree of neighbors within the neighborhood subgraph. Measures how interconnected a node's neighbors are. Proposed by Li et al. (2011) for identifying essential proteins in PPI networks.

Usage

centrality_lac(x, mode = "all", ...)

Arguments

Value

Named numeric vector of LAC values.

References

Li, M., Wang, J., Chen, X., Wang, H., & Pan, Y. (2011). A local average connectivity-based method for identifying essential proteins from the network level. Computational Biology and Chemistry, 35(3), 143-150.

See Also

centrality for computing multiple measures at once, centrality_dmnc for another neighborhood density measure.

Examples

adj <- matrix(c(0, 1, 1, 1, 0, 1, 1, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality_lac(adj)

Laplacian Centrality

Description

Energy drop from the graph Laplacian when a node is removed (Qi et al. 2012). Measures a node's importance to the overall network energy.

Usage

centrality_laplacian(x, ...)

Arguments