centrality: Calculate Network Centrality Measures

In cograph: Analysis and Visualization of Complex Networks

Calculate Network Centrality Measures

Description

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

Usage

centrality(
  x,
  type = c("basic", "extended", "all"),
  measures = 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,
  tna_network = NULL,
  psych_network = NULL,
  ...
)

Arguments

x	Network input (matrix, igraph, network, cograph_network, tna object)
type	Character scalar selecting a curated tier of measures when measures is not supplied. One of: "basic" (default) 6 canonical measures: degree, strength, closeness, betweenness, eigenvector, pagerank. "extended" Basic plus commonly-reported second-tier measures: harmonic, coreness, eccentricity, radiality, lin, decay, load, stress, katz, alpha, power, authority, leverage, constraint, effective_size, bridging, transitivity, subgraph, diffusion, laplacian, kreach, current_flow_betweenness, current_flow_closeness. "all" Every available measure. Passing measures explicitly overrides type.
measures	Character vector of specific measure names to compute. When NULL (default) the tier selected by type is used. Accepts "all" as a shortcut for every measure. Any custom vector of valid measure names is also accepted. Core (igraph-backed): "degree", "strength", "betweenness", "closeness", "eigenvector", "pagerank", "authority", "hub", "eccentricity", "coreness", "constraint", "transitivity", "harmonic", "alpha", "power", "subgraph". Native: "diffusion", "leverage", "kreach", "laplacian", "load", "current_flow_closeness", "current_flow_betweenness", "voterank", "percolation". Distance-based: "radiality", "lin", "decay", "residual_closeness", "dangalchev", "generalized_closeness", "harary", "average_distance", "barycenter", "wiener", "closeness_vitality". Spectral/walk: "communicability", "communicability_betweenness", "random_walk". Path-based: "stress", "flow_betweenness". Local/neighborhood: "lobby", "entropy", "semilocal", "clusterrank", "bottleneck", "centroid", "mnc", "dmnc", "lac", "topological_coefficient", "bridging", "local_bridging", "effective_size", "diversity", "cross_clique", "markov". Influence: "integration", "expected", "gilschmidt". Directed-only: "salsa", "leaderrank", "trophic_level", "pairwisedis", "prestige_domain", "prestige_domain_proximity". Community-aware (require membership): "participation", "within_module_z", "gateway", "brokerage_coordinator", "brokerage_itinerant", "brokerage_representative", "brokerage_gatekeeper", "brokerage_liaison" (the last 5 also require a directed graph; see centrality_brokerage_coordinator). Zoo (batch 2): "gravity", "collective_influence", "local_hindex", "hindex_strength", "onion", "second_order", "infection", "nonbacktracking", "spanning_tree". Classical (batch 3, reference-validated): "katz" (Katz 1953), "hubbell" (Hubbell 1965), "information" (Stephenson-Zelen 1989), "reaching_local" (Mones et al. 2012). See centrality_katz, centrality_hubbell, centrality_information, centrality_pairwisedis, centrality_reaching_local. Psychometric (signed-weight): "expected_influence_1", "expected_influence_2" (Robinaugh, Millner & McNally 2016). Expected influence keeps signed edge contributions, which is important when edges can be negative (partial-correlation, glasso, signed correlation networks).
mode	For directed networks: "all", "in", or "out". Affects measures whose output columns carry a mode suffix, including degree, strength, closeness, eccentricity, coreness, harmonic, diffusion, leverage, k-reach, distance-based measures, community-aware measures, and expected influence.
normalized	Logical. Normalize values by dividing by max. Most measures are scaled to 0-1; signed expected-influence measures can retain negative values under psychometric normalization. For closeness, this is passed directly to igraph.
weighted	Logical. Use edge weights if available. Default TRUE.
directed	Logical or NULL. If NULL (default), auto-detect from matrix symmetry. Set TRUE to force directed, FALSE to force undirected.
loops	Logical. If TRUE (default), keep self-loops. Set to FALSE to remove them before calculation.
simplify	How to combine multiple edges between the same node pair. Options: "sum" (default), "mean", "max", "min", or FALSE/"none" to keep multiple edges.
digits	Integer or NULL. Round all numeric columns to this many decimal places. Default NULL (no rounding).
sort_by	Character or NULL. Column name to sort results by (descending order). Default NULL (original node order).
cutoff	Maximum path length to consider for betweenness, closeness, and harmonic centrality. Default -1 (no limit). Set to a positive value for faster computation on large networks at the cost of accuracy.
invert_weights	Logical or NULL. For path- and distance-based measures (for example betweenness, closeness, harmonic, eccentricity, k-reach, radiality, decay, stress, flow betweenness, and related variants), should weights be inverted so that higher weights mean shorter paths? Default NULL auto-detects: TRUE for tna objects (transition probabilities), FALSE otherwise (matching igraph/sna). Set explicitly to TRUE for strength/frequency weights (qgraph style) or FALSE for distance/cost weights.
alpha	Numeric. Exponent for weight transformation when invert_weights = TRUE. Distance is computed as 1 / weight^alpha. Default 1. Higher values increase the influence of weight differences on path lengths.
damping	PageRank damping factor. Default 0.85. Must be between 0 and 1.
personalized	Named numeric vector for personalized PageRank. Default NULL (standard PageRank). Values should sum to 1.
transitivity_type	Type of transitivity to calculate: "local" (default), "global", "undirected", "localundirected", "barrat" (weighted), "weighted", or "onnela". The first six dispatch to igraph::transitivity(); "onnela" computes the Onnela / Holme weighted clustering coefficient on the symmetrized matrix (wcc(x + t(x))) and matches tna::centralities(., "Clustering") byte-for-byte. Auto-set to "onnela" when tna_network = TRUE and the user did not pass an explicit value.
isolates	How to handle isolate nodes in transitivity calculation: "nan" (default) returns NaN, "zero" returns 0.
lambda	Diffusion scaling factor for diffusion centrality. Default 1. Only used when diffusion_method = "kandhway_kuri".
diffusion_method	Character or NULL. Selects the diffusion-centrality formula. "kandhway_kuri" (Kandhway & Kuri, 2014) computes the 1-hop binary-degree neighborhood sum \lambda d_v + \lambda \sum_{u \in N(v)} d_u. "power_series" computes the matrix power series \mathrm{rowSums}(P + P^2 + \ldots + P^n) on the (optionally diagonal-zeroed) weighted matrix and matches tna::centralities(., measures = "Diffusion") when loops = FALSE. Default NULL auto-detects: "power_series" for tna objects (transition probabilities), "kandhway_kuri" otherwise.
k	Path length parameter for geodesic k-path centrality. Default 3.
states	Named numeric vector of percolation states (0-1) for percolation centrality. Each value represents how "activated" or "infected" a node is. Default NULL (all nodes get state 1, equivalent to betweenness).
decay_parameter	Numeric. Decay parameter for decay and generalized closeness centrality. Default 0.5. Must be between 0 and 1.
dmnc_epsilon	Numeric. Epsilon exponent for DMNC (Density of Maximum Neighborhood Component). Default 1.7 as recommended by Lin et al. (2008). centiserve uses 1.67 (four-community assumption). Must be between 1 and 2.
membership	Integer vector of community assignments (one per node) for community-aware measures: participation, within_module_z, gateway, and the Gould-Fernandez brokerage roles. Default NULL. Required when requesting these measures.
katz_alpha	Attenuation factor for Katz centrality. Must satisfy \alpha < 1 / \rho(A). Default 0.1 (matches centiserve and NetworkX conventions). Only used when "katz" is in measures.
hubbell_weight	Weight factor w for Hubbell centrality. Must satisfy w \cdot \rho(W) \le 1 for solvability. Default 0.5. Only used when "hubbell" is in measures.
tna_network	Logical or NULL. Umbrella switch that forces tna-style conventions across all measures. NULL (default) auto-detects from the input class — TRUE iff x is a tna or related sequence-network object. TRUE forces tna conventions even on raw matrices: invert_weights = TRUE, loops = FALSE, diffusion_method = "power_series", transitivity_type = "onnela". FALSE suppresses all tna defaults even for tna inputs, giving the cograph defaults verbatim. Precedence: any arg the user passes explicitly always wins over tna_network.
psych_network	Logical or NULL. Switch for signed psychometric network conventions. NULL (default) auto-detects TRUE when a signed weighted network is evaluated with expected-influence measures. When TRUE, normalized expected influence is divided by the maximum absolute expected-influence value, preserving sign and bounding the result from -1 to 1. FALSE keeps the generic cograph normalization convention.
...	Additional arguments (currently unused)

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).

Value

A data frame with columns:

• node: Node labels/names

• One column per measure, with mode suffix for directional measures (e.g., degree_in, closeness_all)

Examples

# Basic usage with matrix
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")

cograph documentation built on May 31, 2026, 5:06 p.m.