Calculate Network Centrality Measures — centrality • cograph

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

x

Network input (matrix, edge-list data frame, 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 measure except the costly ones, which are held back (see include and list_centralities).

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 type = "all", i.e. every measure except the costly ones. Any custom vector of valid measure names is also accepted, and naming a costly measure there always computes it. 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). Zoo (batch 7, lowest rank-redundancy with the rest of the package per the Centrality Zoo comparison): "distance_entropy" (Stella & De Domenico 2018), "local_dimension" (Pu et al. 2014), "local_information_dimension" (Wen & Deng 2020), "neighborhood_connectivity" (Maslov & Sneppen 2002), and "modularity_vitality" (Magelinski et al. 2021; requires membership). The first three are hop-count measures and ignore edge weights. See centrality_distance_entropy, centrality_local_dimension, centrality_local_information_dimension, centrality_neighborhood_connectivity, centrality_modularity_vitality. Zoo (batch 8, the measures the Zoo comparison left "on the way"): "shapley_game1", "shapley_game2", "shapley_game3" (Michalak et al. 2013), "access_information", "hide_information" (Rosvall et al. 2005), "rumor" (Shah & Zaman 2011), "community_hub_bridge" (Ghalmane et al. 2019; requires membership), "entropy_variation_degree", "entropy_variation_betweenness" (Ai 2017), "s_shell" (Liu et al. 2017), "degree_discount", "single_discount" (Chen, Wang & Yang 2009), "ncvoterank" (Kumar & Panda 2020). All are hop-count or topology-only measures; edge weights are ignored. See the per-measure pages, e.g. centrality_shapley_game1, centrality_access_information, centrality_rumor, centrality_community_hub_bridge, centrality_entropy_variation, centrality_s_shell, centrality_degree_discount, centrality_ncvoterank. Zoo (batch 9, the remaining measures with a pinned definition): community-aware "community_based" (Zhao et al. 2015), "comm_centrality" (Gupta et al. 2016), "community_mediator" (Tulu et al. 2018), all requiring membership; dimension family "local_dimension_fixed" (Silva & Costa 2013), "fuzzy_local_dimension" (Wen & Jiang 2019), "local_volume_dimension" (Li & Deng 2021); VoteRank family "wvoterank" (Sun et al. 2019), "enrenew" (Guo et al. 2020), "voterank_plus" (Liu et al. 2021); "node_contraction", "node_contraction_improved" (Tan et al. 2006; Wang et al. 2011); "two_way_rw" (Curado et al. 2022); local measures "heatmap" (Duron 2020), "flow_coefficient" (Honey et al. 2007), "local_entropy" (Nie et al. 2016), "weighted_h_index" (Gao et al. 2019), "redundancy" (Burt 1992); "weighted_kshell" (Garas et al. 2012), "renewed_coreness" (Liu et al. 2015), "geodesic_kpath" (Borgatti & Everett 2006). Only "wvoterank", "two_way_rw" and "weighted_kshell" use edge weights. See centrality_community_based, centrality_local_dimension_fixed, centrality_wvoterank, centrality_node_contraction, centrality_two_way_rw, centrality_heatmap, centrality_weighted_kshell.

Batch 10 closes the gaps other centrality packages had and cograph did not: "local_efficiency" (Latora & Marchiori 2001), "s_core" (Eidsaa & Almaas 2013), "fragmentation" (Borgatti 2006), "kpath" (Sade 1989) and "epc" (Lin et al. 2008). "fragmentation" and "epc" are costly, so type = "all" holds them back. See centrality_local_efficiency.

Batch 11 tunes families cograph already had: "length_scaled_betweenness" (Brandes 2008), "delta_betweenness" and "delta_closeness" (Agneessens et al. 2017), "ego_betweenness" (Everett & Borgatti 2005). "gravity" gained gravity_mass and gravity_radius, and its formula was corrected – see centrality_gravity. Bounded-distance ("k-") betweenness needs no measure of its own: it is cutoff = k. See centrality_length_scaled_betweenness.

include

Character vector of costly measures to add back to a tier, or "costly" for all of them. type = "all" holds back the measures whose cost grows steeply with network size (see list_centralities), so that one call cannot take minutes by accident. Naming a measure in measures always computes it, whatever its cost. Default NULL.

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 (possible only from edge-list, cograph_network or igraph input). Options: "sum" (default), "mean", "max", "min". FALSE and "none" also sum them: the network is held as a dense weight matrix, which cannot carry parallel 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, harmonic centrality and the distance-based closeness variants (radiality, lin, decay, residual_closeness, dangalchev, generalized_closeness, harary, average_distance, barycenter, wiener, centroid, closeness_vitality, delta_closeness). 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, modularity_vitality, 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 be positive and satisfy \(w \cdot \rho(W) < 1\) for solvability; otherwise the measure warns and returns NA. Default 0.5. Only used when "hubbell" is in measures.

shapley_k

Neighbor threshold \(k\) for "shapley_game2". Default 2. See centrality_shapley_game2.

shapley_cutoff

Hop cutoff for "shapley_game3". Default 2. See centrality_shapley_game3.

s_shell_a

Exponent of the asymmetric link weights for "s_shell". A single non-negative number; default 0.5. See centrality_s_shell.

discount_p

Propagation probability for "degree_discount". Default 0.01. See centrality_degree_discount.

ncvote_theta

Weight of the plain vote in "ncvoterank". Default 0.5. See centrality_ncvoterank.

comm_r

Scale \(R\) of "comm_centrality": "max_intra" (default) or a single positive number.

ld_radius

Radius for "local_dimension_fixed", in hops. A single number of at least 1; default 2.

enrenew_depth

Renewal radius for "enrenew". Default 2.

voterank_lambda

Suppression factor for "voterank_plus". Default 0.1.

contraction_rho

\(\alpha / \beta\) for "node_contraction_improved". Default 5.

wks_alpha, wks_beta

Degree and strength exponents for "weighted_kshell". Default 1 and 1.

renewed_threshold

Diffusion-importance threshold for "renewed_coreness". Default 2.

kpath_k

Maximum path length for "geodesic_kpath". Default 3.

kpath_len

Maximum path length for "kpath". Default 3; the enumeration is exhaustive, so cost grows with the branching factor to this power.

epc_threshold

Edge removal probability for "epc". Default 0.5.

epc_runs

Number of percolation realizations for "epc". Default 1000.

epc_seed

Random seed for "epc". Default NULL, which leaves the caller's stream alone and lets the estimate vary between calls.

betweenness_delta

Decay exponent for "delta_betweenness". Default 1; 0 gives ordinary betweenness.

closeness_delta

Distance exponent for "delta_closeness". Default 1, which is harmonic over \(n - 1\).

gravity_mass

Mass in "gravity": "kshell" (default, Ma et al. 2016), "degree" (Li et al. 2019) or "legacy" for cograph's pre-2.4.8 form.

gravity_radius

Largest distance each gravity source reaches in "gravity", "extended_gravity", "mixed_gravity" or "extended_mixed_gravity": a number (default 3), "auto" for half the mean distance, or NULL for the whole graph. The auto radius uses finite positive distances, rounds to the nearest integer (ties to even), and has minimum 1; these are cograph conventions.

mdd_lambda

Exhausted-degree weight for "mdd", between 0 and 1. Default 0.7. See centrality_truss.

volume_radius

Closed neighborhood radius for "volume": a nonnegative integer or Inf, default 2. Degrees are measured in the full simple undirected graph. See centrality_volume.

diffusion_q

Multiplier between 0 and 1 for "diffusion_centrality", default 1. Independent of the existing lambda argument.

diffusion_steps

Nonnegative integer horizon for "diffusion_centrality", default 3. See centrality_diffusion_centrality for its weighted-walk definition, direction, probability interpretation and precision limits.

ds_beta

Spreading rate for "dynamics_sensitive", between zero and one, default 0.1.

ds_mu

Recovery rate for "dynamics_sensitive", between zero and one, default 1. Zero selects the SI case.

ds_steps

Nonnegative integer horizon for "dynamics_sensitive", default 5. See centrality_dynamics_sensitive.

cda_alpha

Degree-versus-strength weight for "cda", between zero and one; default 0.5. See centrality_cda.

icc_alpha

Shortest-path multiplicity exponent for "improved_closeness", between zero and one; default 0.2.

exogenous_base

Base for "exogenous": reverse_closeness (default), betweenness or degree. See centrality_exogenous.

wlr_alpha

Finite in-degree exponent for "weighted_leaderrank", default one. See centrality_weighted_leaderrank.

alr_h_mode

H-index convention for "adaptive_leaderrank": all (default), out or in. See centrality_adaptive_leaderrank.

grc_gamma

Finite nonnegative regularization strength for "graph_regularization", default one. See centrality_graph_regularization.

rwd_decay

Finite first-arrival discount in \([0,1)\) for "random_walk_decay", default 0.5.

rwd_node_weights

Nonnegative starting weights for "random_walk_decay"; NULL means ones. See centrality_random_walk_decay.

linerank_aggregation

LineRank endpoint aggregation: probability (default) or weight. See centrality_linerank.

bridging_steps

Nonnegative bridging-capital walk horizon, default two.

bridging_values

Optional source-destination value matrix for centrality_bridging_capital; NULL uses ones.

proximal_variant

Proximal betweenness role: source (default), target, sum, or union. See centrality_proximal_betweenness.

exf_alpha

Modified Expected Force degree factor, default two, finite and greater than one.

beta_direction

BG-index orientation, positive (default) or negative. See centrality_beta_measure.

ninl_order

Nonnegative NINL iteration count, default three.

ninl_radius

NINL hop radius, NULL for ceiling of mean path length. See centrality_ninl for disconnected graphs and overrides.

map_flow

Map equation flow model, unrecorded (default) or recorded.

map_convention

Map equation coding convention, paper (default) or infomap. See centrality_map_equation.

sr_prior

SpectralRank diagonal prior, default zero; scalar or one value per node. See centrality_spectralrank.

mcgm_radius

MCGM hop cutoff, default two; NULL includes all reachable nodes.

mcgm_alpha

MCGM coefficient, NULL for the published adaptive rule. See centrality_mcgm for disconnected-graph conventions.

dkgm_radius

DKGM hop cutoff, default two as in the paper's printed example; NULL or infinity includes all reachable nodes and "auto" applies the paper's half-mean-distance rule with cograph rounding. See centrality_dkgm.

nd_order

Steps of neighbors summed by "neighbor_distance", a nonnegative whole number, default two; zero returns nd_mass. See centrality_neighbor_distance.

nd_decay

Per-step decay for "neighbor_distance", a finite number, default 0.2 as in the source.

nd_mass

Benchmark centrality summed by "neighbor_distance": degree (default) or coreness.

ira_mass

Node centrality allocated by "ira" and "iira": coreness (default, the k-shell index both sources use in their worked examples) or degree. See centrality_ira.

ira_alpha

Exponent on the "ira" mass, a finite number, default one as in the source.

ira_tol

Stopping tolerance for "ira" on the largest absolute change between iterates, a positive finite number, default 1e-6 as in the source.

ira_max_iter

Iteration bound for "ira", a whole number of at least one, default 1000. Reaching it raises cograph_no_converge, which a bipartite component with unequal vertex classes always does. See centrality_ira.

iira_beta

Spreading rate for "iira", a number in \((0,1]\), default 0.2 as in the source.

iira_steps

Iterations for "iira", a nonnegative whole number, default 50 as in the source; zero returns the initial unit resource. See centrality_iira.

hcc_delta

Weight on a node's own degree in the extended degree used by "hcc" and "ehcc", a single number in \([0,1]\), default 0.5 as in the source; one recovers the classical degree and zero drops the node's own degree entirely. Values outside \([0,1]\) are refused. See centrality_hcc.

lhc_radius

Radius of the ball \(\Phi(v)\) summed over by "lhc", the \(d\) of the source's equation (1); a single whole number of at least one, default 2 as the source sets it. The source sweeps it and reports 2-3 as optimal. At one the ball collapses to the neighbors; at or above the diameter the score stops moving. Values below one and non-integers are refused. See centrality_lhc.

tpr_alpha

Jump probability of the trust-PageRank iteration used by "trust_pagerank", a single number strictly between zero and one, default 0.85 as the source sets it below its equation (7). See centrality_trust_pagerank.

tpr_k

Weight the trust-value puts on the degree ratio rather than the similarity ratio in "trust_pagerank", the \(k\) of the source's equation (6); a single number in \([0,1]\), default 0.85, the value the source's section 3.3 selects from a Kendall-against-SIR sweep. One drops the similarity entirely and zero drops the degree.

tpr_decay

Attenuation factor of the similarity recursion used by "trust_pagerank", the \(C\) of the source's equation (4); a single number in \((0,1]\), default 1 as the source fixes it. The source's claim that \(C\) does not affect the result holds only for a homogeneous recursion and not for this one; see centrality_trust_pagerank.

tpr_tol

Convergence tolerance on the largest relative change of either trust-PageRank recursion, a single positive number, default 1e-14. The source fixes no iteration count because it does not need one: both recursions have unique fixed points. The test is relative rather than absolute because the similarities on one graph span many orders of magnitude; see centrality_trust_pagerank.

tpr_max_iter

Iteration bound for both trust-PageRank recursions, a whole number of at least one, default 1000. Reaching it raises cograph_no_converge.

rsp_beta

Inverse temperature of the randomized-shortest-paths model used by "rsp_betweenness", a single finite number strictly above zero, default 0.01. The source fixes no default; 0.01 is the value NetworkToolbox::rspbc() recommends, and it sits near the random-walk limit, so raise it towards 1 and beyond to move the reading towards shortest paths. See centrality_rsp_betweenness.

rsp_cost

How an edge weight becomes a traversal cost for "rsp_betweenness": "inverse" (default) for \(C=1/w\), reading a weight as an affinity, or "weight" for \(C=w\), reading it as a distance. The source leaves the cost matrix free; both settings give unit cost per arc on a binary graph. See centrality_rsp_betweenness.

re_indexes

Constituent indexes integrated by "relative_entropy", default the source's four distinctiveness indexes; the vocabulary also holds "n_components" and "largest_component". See centrality_relative_entropy.

re_negative

Which of re_indexes are negative indexes, NULL for the source's own declarations. See centrality_relative_entropy.

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)

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.

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.

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)
#>    node degree_all strength_all closeness_all betweenness  eigenvector
#> 1    Ac         33          129    0.01754386   26.342857 1.000000e+00
#> 2    Ad         20           36    0.01754386   42.541520 1.096110e-01
#> 3    Fi         24           51    0.01666667   35.721634 1.789565e-01
#> 4    Ik         14           24    0.01666667   25.844874 1.551369e-02
#> 5    Vx         26           43    0.01960784   90.717124 7.238902e-02
#> 6    Rt         20           37    0.01785714   63.135739 1.159931e-01
#> 7    Km         11           16    0.01639344   18.175108 2.804725e-02
#> 8    Gj         19           31    0.01818182  114.599049 3.265786e-02
#> 9    Bd         12           18    0.01612903   21.769264 9.607736e-03
#> 10   Ce         10           13    0.01612903   16.648629 4.473504e-03
#> 11   Oq         14           20    0.01754386   34.151726 2.293068e-02
#> 12   Ya         13           19    0.01612903   18.216122 1.758656e-02
#> 13   Mo         12           17    0.01587302   38.264502 1.003629e-01
#> 14   Hj         12           19    0.01754386   85.816522 2.013125e-02
#> 15   Tv         10           13    0.01666667   25.916306 1.320877e-02
#> 16   Eg         10           12    0.01639344   22.335171 5.783916e-03
#> 17   Pr         11           18    0.01666667   23.974060 7.602231e-02
#> 18   Qs         15           19    0.01785714   76.910851 1.511764e-02
#> 19   Xz         14           18    0.01639344   22.280159 8.533484e-03
#> 20   Np          8            8    0.01666667   12.044048 1.549052e-02
#> 21   Dg         13           13    0.01886792   29.240901 6.260099e-03
#> 22   Hk         16           25    0.01818182   72.176441 1.201845e-01
#> 23   Wy         11           16    0.01639344   34.014358 1.054705e-03
#> 24   Jl         15           18    0.01818182   67.359085 5.009801e-02
#> 25   Fh         21           55    0.01818182   78.588877 2.817489e-01
#> 26   Zb          7            8    0.01538462    9.583333 5.429715e-05
#> 27   Eh          7           13    0.01428571   34.325000 1.163185e-03
#> 28   Be         14           16    0.01851852  105.250898 2.203252e-03
#> 29   Df          8           10    0.01562500   11.026190 4.800876e-06
#> 30   Cf         12           15    0.01724138  119.109163 1.243207e-02
#> 31   Su          6            9    0.01369863   33.154401 1.028472e-04
#> 32   Ln          7            8    0.01408451    5.749708 1.376854e-02
#> 33   Gi          3            4    0.01351351    0.000000 0.000000e+00
#> 34   Uw          4            7    0.01250000    0.000000 0.000000e+00
#>       pagerank
#> 1  0.285861728
#> 2  0.052985644
#> 3  0.077591140
#> 4  0.024836857
#> 5  0.057364714
#> 6  0.042552472
#> 7  0.016655998
#> 8  0.025444498
#> 9  0.014321668
#> 10 0.010679742
#> 11 0.016087378
#> 12 0.016588876
#> 13 0.031180263
#> 14 0.019051413
#> 15 0.012644206
#> 16 0.010425091
#> 17 0.022794289
#> 18 0.019784870
#> 19 0.013229134
#> 20 0.008466679
#> 21 0.010345027
#> 22 0.040383192
#> 23 0.009067435
#> 24 0.020529375
#> 25 0.070538086
#> 26 0.005635780
#> 27 0.010080122
#> 28 0.009378924
#> 29 0.004957518
#> 30 0.017877547
#> 31 0.005136500
#> 32 0.007628879
#> 33 0.005483193
#> 34 0.004411765

# 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)
#>   node degree_all strength_all closeness_all betweenness eigenvector  pagerank
#> 1    A          2            2           0.5           0           1 0.3333333
#> 2    B          2            2           0.5           0           1 0.3333333
#> 3    C          2            2           0.5           0           1 0.3333333

# Specific measures
centrality(adj, measures = c("degree", "betweenness"))
#>   node degree_all betweenness
#> 1    A          2           0
#> 2    B          2           0
#> 3    C          2           0

# Directed network with normalization
centrality(adj, mode = "in", normalized = TRUE)
#>   node degree_in strength_in closeness_in betweenness eigenvector pagerank
#> 1    A         1           1            1           0           1        1
#> 2    B         1           1            1           0           1        1
#> 3    C         1           1            1           0           1        1

# Sort by pagerank
centrality(adj, sort_by = "pagerank", digits = 3)
#>   node degree_all strength_all closeness_all betweenness eigenvector pagerank
#> 1    A          2            2           0.5           0           1    0.333
#> 2    B          2            2           0.5           0           1    0.333
#> 3    C          2            2           0.5           0           1    0.333

# PageRank with custom damping
centrality(adj, measures = "pagerank", damping = 0.9)
#>   node  pagerank
#> 1    A 0.3333333
#> 2    B 0.3333333
#> 3    C 0.3333333

# Harmonic centrality (better for disconnected graphs)
centrality(adj, measures = "harmonic")
#>   node harmonic_all
#> 1    A            2
#> 2    B            2
#> 3    C            2

# Global transitivity
centrality(adj, measures = "transitivity", transitivity_type = "global")
#>   node transitivity
#> 1    A            1
#> 2    B            1
#> 3    C            1