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": The default. Six measures: degree, strength, closeness, betweenness, eigenvector, pagerank.
"extended": The basic tier plus 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 and current_flow_closeness.
"all": Every measure except the costly ones (see include and list_centralities).

Passing measures explicitly overrides type.

measures

Character vector of measure names to compute. When NULL (default) the tier selected by type is used. "all" is a shortcut for type = "all". Unknown names raise an error. list_centralities returns every valid name together with its orientation, mode support, partition requirement, weight use and cost. The measures fall into these groups:

Core: "degree", "strength", "betweenness", "closeness", "eigenvector", "pagerank", "authority", "hub", "eccentricity", "coreness", "constraint", "transitivity", "harmonic", "alpha", "power", "subgraph".
Flow and spreading: "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 and walk-based: "communicability", "communicability_betweenness", "random_walk".
Path-based: "stress", "flow_betweenness".
Local and 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", "modularity_vitality" (Magelinski et al. 2021), "community_hub_bridge" (Ghalmane et al. 2019), "community_based" (Zhao et al. 2015), "comm_centrality" (Gupta et al. 2016), "community_mediator" (Tulu et al. 2018), and the Gould-Fernandez roles "brokerage_coordinator", "brokerage_itinerant", "brokerage_representative", "brokerage_gatekeeper", "brokerage_liaison", which also require a directed graph. See centrality_modularity_vitality and centrality_brokerage_coordinator.
Spreader identification: "gravity", "collective_influence", "local_hindex", "hindex_strength", "onion", "second_order", "infection", "nonbacktracking", "spanning_tree". See centrality_gravity.
Classical: "katz" (Katz 1953), "hubbell" (Hubbell 1965), "information" (Stephenson and Zelen 1989), "reaching_local" (Mones et al. 2012). See centrality_katz, centrality_hubbell, centrality_information, centrality_pairwisedis and centrality_reaching_local.
Psychometric: "expected_influence_1", "expected_influence_2" (Robinaugh, Millner and McNally 2016). Expected influence keeps the sign of each edge, which matters in networks with negative edges such as partial-correlation and glasso networks.
Scaling and dimension: "distance_entropy" (Stella and De Domenico 2018), "local_dimension" (Pu et al. 2014), "local_information_dimension" (Wen and Deng 2020), "neighborhood_connectivity" (Maslov and Sneppen 2002), "local_dimension_fixed" (Silva and Costa 2013), "fuzzy_local_dimension" (Wen and Jiang 2019), "local_volume_dimension" (Li and Deng 2021). See centrality_distance_entropy, centrality_local_dimension, centrality_local_information_dimension, centrality_neighborhood_connectivity and centrality_local_dimension_fixed.
Games, information and seed selection: "shapley_game1", "shapley_game2", "shapley_game3" (Michalak et al. 2013), "access_information", "hide_information" (Rosvall et al. 2005), "rumor" (Shah and Zaman 2011), "entropy_variation_degree", "entropy_variation_betweenness" (Ai 2017), "s_shell" (Liu et al. 2017), "degree_discount", "single_discount" (Chen, Wang and Yang 2009). See centrality_shapley_game1, centrality_access_information, centrality_rumor, centrality_community_hub_bridge, centrality_entropy_variation, centrality_s_shell, centrality_degree_discount and centrality_community_based.
VoteRank family: "ncvoterank" (Kumar and Panda 2020), "wvoterank" (Sun et al. 2019), "enrenew" (Guo et al. 2020), "voterank_plus" (Liu et al. 2021). See centrality_ncvoterank and centrality_wvoterank.
Contraction, walks and local structure: "node_contraction", "node_contraction_improved" (Tan et al. 2006; Wang et al. 2011), "two_way_rw" (Curado et al. 2022), "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 and Everett 2006). See centrality_node_contraction, centrality_two_way_rw, centrality_heatmap and centrality_weighted_kshell.
Efficiency, cores and percolation: "local_efficiency" (Latora and Marchiori 2001), "s_core" (Eidsaa and Almaas 2013), "fragmentation" (Borgatti 2006), "kpath" (Sade 1989), "epc" (Lin et al. 2008). See centrality_local_efficiency.
Betweenness and closeness variants: "length_scaled_betweenness" (Brandes 2008), "delta_betweenness" and "delta_closeness" (Agneessens et al. 2017), "ego_betweenness" (Everett and Borgatti 2005). Bounded-distance (k-) betweenness is betweenness with cutoff = k. See centrality_length_scaled_betweenness.

The remaining measures are described under Details.

include

Character vector of costly measures to add to a tier, or "costly" for all of them. type = "all" holds back the measures whose cost grows steeply with network size (the costly column of list_centralities). A measure named in measures is always computed, whatever its cost. Unknown names raise a cograph_unknown_measure error. Default NULL.

mode

For directed networks: "all" (default), "in", or "out". Affects the mode-aware measures, whose output columns carry a mode suffix. These include degree, strength, closeness, eccentricity, coreness, harmonic, diffusion, leverage, kreach, the distance-based measures, most community-aware measures, and expected influence (the mode_aware column of list_centralities).

normalized

Logical. If TRUE, each measure is divided by its maximum, which scales non-negative measures to 0-1. A measure whose maximum is not positive is left unchanged. Closeness follows igraph's normalization instead, multiplying each value by the number of nodes the node reaches, and harmonic centrality is divided by \(n - 1\) only. Under psych_network = TRUE expected influence is divided by its maximum absolute value and keeps its sign. Default FALSE.

weighted

Logical. Use edge weights if available. Default TRUE. With FALSE every measure works on the binary adjacency, every edge with weight one.

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. tna_network = TRUE changes the default to FALSE.

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, because 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). A name that is not a column of the result raises an error.

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). A positive value ignores longer paths, which shortens computation on large networks and changes the values.

invert_weights

Logical or NULL. Whether path- and distance-based measures (for example betweenness, closeness, harmonic, k-reach, radiality, decay, stress, flow betweenness and related variants) invert the weights, so that higher weights mean shorter paths. A message reports the inversion. Eccentricity always reads the raw edge weights as distances. The default NULL is TRUE for tna objects (transition probabilities) and FALSE otherwise, as in igraph and sna. TRUE suits strength or frequency weights (the qgraph convention) and FALSE suits distance or 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

Non-negative numeric vector of reset probabilities for personalized PageRank, one value per node. A named vector is matched to the node names, an unnamed one is taken in node order. The vector is rescaled to sum to 1. Default NULL (standard PageRank).

transitivity_type

Type of transitivity to calculate: "local" (default), "global", "undirected", "localundirected", "barrat" (weighted), "weighted", or "onnela". The first six follow the conventions of igraph::transitivity(). "global" and "undirected" give one graph-level value, repeated on every row. "onnela" computes the weighted clustering coefficient of Zhang and Horvath (2005) on the symmetrized matrix x + t(x), the form tna uses under that name, and matches tna::centralities(x, "Clustering"). tna_network = TRUE changes the default to "onnela".

isolates

Value of local transitivity at nodes where it is undefined: "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

Distance bound for "kreach", the number of nodes reachable within k steps. Default 3.

states

Numeric vector of percolation states for percolation centrality, one per node. Each value represents how "activated" or "infected" a node is. A named vector is matched to the node labels, missing nodes get state 1, and values are clipped to 0-1. Default NULL gives every node state 1, which makes percolation equal to betweenness divided by \((n-1)(n-2)\).

decay_parameter

Numeric. Decay parameter for decay and generalized closeness centrality, strictly between 0 and 1; smaller values discount distant nodes more. Other values raise a cograph_bad_parameter error when one of these measures is requested. Default 0.5.

dmnc_epsilon

Numeric. Epsilon exponent for DMNC (Density of Maximum Neighborhood Component), a single positive number. Default 1.7, as recommended by Lin et al. (2008). centiserve uses 1.67. Other values raise a cograph_bad_parameter error when "dmnc" is requested.

membership

Integer vector of community assignments (one per node) for the community-aware measures listed under measures. Default NULL. Without it those measures return NA with a warning of classes cograph_bad_membership and cograph_undefined_measure. "map_equation" uses it when supplied.

katz_alpha

Attenuation factor for Katz centrality. The Katz series converges only for \(\alpha < 1 / \rho(A)\). Otherwise the measure raises a cograph_katz_diverged warning. Default 0.1, the centiserve and NetworkX convention. 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 draws from the session's random number stream, so the estimate varies between calls. A seed makes the estimate reproducible, and the caller's random number state is restored afterwards.

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", which uses a unit focal mass and a partner mass equal to degree times k-shell index.

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 automatic radius averages the finite positive distances, rounds half to even, and is at least 1.

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. It is separate from lambda, which scales "diffusion".

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 a cograph_no_converge warning, which always happens on a bipartite component with unequal vertex classes. 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. A value of one gives the classical degree and zero drops the node's own degree. Values outside \([0,1]\) raise an error. 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 in the source, which reports 2-3 as optimal. At one the ball contains only the neighbors. At or above the diameter the score no longer changes. Values below one and non-integers raise an error. 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 of the degree ratio in the trust value of "trust_pagerank", the \(k\) of the source's equation (6). The similarity ratio receives weight \(1-k\). A single number in \([0,1]\), default 0.85, the value selected in section 3.3 of the source. One drops the similarity 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 in the source. \(C\) changes the result. 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 tolerance is relative 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 a cograph_no_converge warning.

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 value NetworkToolbox::rspbc() recommends. Small values approach the random-walk limit. Values of 1 and above move the measure 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. 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) is TRUE when x is a tna, group_tna, ctna, ftna or atna object (or a group of these) and FALSE otherwise. TRUE applies the tna conventions to any input, setting invert_weights = TRUE, loops = FALSE, diffusion_method = "power_series" and transitivity_type = "onnela". FALSE keeps the cograph defaults for tna inputs as well. An argument passed explicitly always takes precedence 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, which keeps its sign and bounds it between -1 and 1. FALSE keeps the generic cograph normalization.

...

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). Distances use the raw edge weights whatever invert_weights says, and hop counts with weighted = FALSE.
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, the sum of inverse distances. It stays finite on disconnected graphs (supports mode: in/out/all)
diffusion: Diffusion degree centrality. With diffusion_method = "kandhway_kuri" it is the scaled degree of the node plus the scaled degrees of its neighbors (supports mode: in/out/all, lambda scaling)
leverage: Leverage centrality. Influence over neighbors based on relative degree differences (supports mode: in/out/all)
kreach: K-reach centrality. Number of nodes reachable within k steps (supports mode: in/out/all)
alpha: Alpha centrality. Influence through paths, attenuated by length, with a unit exogenous contribution at every node (supports mode: in/out/all)
power: Bonacich power centrality. Influence based on connections to other influential nodes (supports mode: in/out/all)
subgraph: Subgraph centrality. Participation in closed walks, with shorter walks weighted more heavily
laplacian: Laplacian centrality with the local formula of Qi et al. (2012). Matches NetworkX and centiserve::laplacian()
load: Load centrality. Share of shortest-path load passing through the node, with load split evenly at each branching point
current_flow_closeness: Information centrality. Closeness based on electrical current flow (requires a connected graph)
current_flow_betweenness: Random-walk betweenness. Betweenness based on electrical current flow (requires a connected graph)
voterank: VoteRank. Influential spreaders selected by iterative voting. The value is the election order rescaled so that the first elected node scores 1 and the last scores \(1/n\)
percolation: Percolation centrality. Shortest-path betweenness weighted by the node states in states. With equal states it is betweenness divided by \((n-1)(n-2)\)
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 \(\delta^d\) over all nodes, the node itself included, with \(\delta\) = decay_parameter.
residual_closeness: Residual closeness. Sum of \(1/2^d\) over all nodes, the node itself included.
dangalchev: Dangalchev closeness. Same values as residual_closeness.
generalized_closeness: Generalized closeness. Same formula as decay, using decay_parameter.
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. It is 1 when the distances are spread evenly and 0 when all lie at one distance.
local_dimension: Growth exponent of the ball around a node (slope of \(\ln B_i(r)\) on \(\ln r\)). Lower values mark more influential nodes.
local_information_dimension: Entropy-weighted local dimension over boxes up to half the node's eccentricity. Higher values mark more influential nodes.
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 values mark community hubs and negative values mark bridges (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 values mark well-connected nodes.
hide_information: Mean bits others need to find the node. High values mark hidden nodes.
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, where higher values mark more influential nodes; slope of the degree volume, where lower values mark more important nodes.
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 values mark more central nodes.
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 values mark more central nodes.
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. This differs from igraph::local_efficiency(), which 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; epc_seed makes it reproducible.
length_scaled_betweenness: Betweenness with each separated pair weighted by \(1 / d(s,t)\).
delta_betweenness: Betweenness with the pair weight \((h(s,t) - 1)^{-\delta}\) (betweenness_delta), where \(h(s,t)\) is the number of edges on a shortest path.
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. 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

The community-aware measures warn and return an all-NA column when membership is missing, and the brokerage roles do the same on an undirected graph. "relative_entropy" has no value when one of its constituent indexes is zero at every node. Named in measures or include, it then raises a cograph_undefined_index error. Supplied by a tier, it gives a cograph_undefined_measure warning and an all-NA column, and the other measures of the tier are still computed. "flow_betweenness" requires the igraph package. Without igraph, naming it raises a cograph_needs_igraph error, and a tier gives the same warning and all-NA column.

Examples

centrality(regulation_net)
#>          node degree_all strength_all closeness_all betweenness eigenvector
#> 1     Explore          6         1.39     0.5154639         5.0   0.7209674
#> 2        Plan          7         1.90     0.4484305        15.5   0.2010163
#> 3     Monitor          8         1.87     0.5714286        18.0   1.0000000
#> 4       Adapt          6         1.77     0.3787879        15.0   0.8380103
#> 5     Reflect          6         1.39     0.5208333        10.0   0.7619198
#> 6     Discuss          5         1.53     0.3460208         0.5   0.5077694
#> 7  Synthesize          4         0.77     0.4830918         6.5   0.1980065
#> 8    Evaluate          5         1.71     0.4032258         3.0   0.4995074
#> 9      Create          7         1.64     0.5263158        13.0   0.6689427
#> 10      Share          6         1.95     0.3521127         9.0   0.5849834
#>      pagerank
#> 1  0.11846147
#> 2  0.03640953
#> 3  0.18376724
#> 4  0.12356096
#> 5  0.12513119
#> 6  0.06803638
#> 7  0.03760071
#> 8  0.07386400
#> 9  0.13821279
#> 10 0.09495573