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,
mode = "all",
normalized = FALSE,
weighted = TRUE,
directed = NULL,
loops = TRUE,
simplify = "sum",
digits = NULL,
sort_by = NULL,
cutoff = -1,
invert_weights = NULL,
alpha = 1,
damping = 0.85,
personalized = NULL,
transitivity_type = "local",
isolates = "nan",
lambda = 1,
diffusion_method = NULL,
k = 3,
states = NULL,
decay_parameter = 0.5,
dmnc_epsilon = 1.7,
membership = NULL,
katz_alpha = 0.1,
hubbell_weight = 0.5,
tna_network = NULL,
psych_network = NULL,
...
)Arguments
- x
Network input (matrix, edge-list data frame, igraph, network, cograph_network, tna object)
- type
Character scalar selecting a curated tier of measures when
measuresis not supplied. One of:"basic"(default) 6 canonical measures:
degree,strength,closeness,betweenness,eigenvector,pagerank."extended"Basic plus commonly-reported second-tier measures: harmonic, coreness, eccentricity, radiality, lin, decay, load, stress, katz, alpha, power, authority, leverage, constraint, effective_size, bridging, transitivity, subgraph, diffusion, laplacian, kreach, current_flow_betweenness, current_flow_closeness.
"all"Every available measure.
Passing
measuresexplicitly overridestype.- measures
Character vector of specific measure names to compute. When
NULL(default) the tier selected bytypeis used. Accepts"all"as a shortcut for every measure. Any custom vector of valid measure names is also accepted. Core (igraph-backed): "degree", "strength", "betweenness", "closeness", "eigenvector", "pagerank", "authority", "hub", "eccentricity", "coreness", "constraint", "transitivity", "harmonic", "alpha", "power", "subgraph". Native: "diffusion", "leverage", "kreach", "laplacian", "load", "current_flow_closeness", "current_flow_betweenness", "voterank", "percolation". Distance-based: "radiality", "lin", "decay", "residual_closeness", "dangalchev", "generalized_closeness", "harary", "average_distance", "barycenter", "wiener", "closeness_vitality". Spectral/walk: "communicability", "communicability_betweenness", "random_walk". Path-based: "stress", "flow_betweenness". Local/neighborhood: "lobby", "entropy", "semilocal", "clusterrank", "bottleneck", "centroid", "mnc", "dmnc", "lac", "topological_coefficient", "bridging", "local_bridging", "effective_size", "diversity", "cross_clique", "markov". Influence: "integration", "expected", "gilschmidt". Directed-only: "salsa", "leaderrank", "trophic_level", "pairwisedis", "prestige_domain", "prestige_domain_proximity". Community-aware (requiremembership): "participation", "within_module_z", "gateway", "brokerage_coordinator", "brokerage_itinerant", "brokerage_representative", "brokerage_gatekeeper", "brokerage_liaison" (the last 5 also require a directed graph; seecentrality_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). Seecentrality_katz,centrality_hubbell,centrality_information,centrality_pairwisedis,centrality_reaching_local. Psychometric (signed-weight): "expected_influence_1", "expected_influence_2" (Robinaugh, Millner & McNally 2016). Expected influence keeps signed edge contributions, which is important when edges can be negative (partial-correlation, glasso, signed correlation networks).- mode
For directed networks: "all", "in", or "out". Affects measures whose output columns carry a mode suffix, including degree, strength, closeness, eccentricity, coreness, harmonic, diffusion, leverage, k-reach, distance-based measures, community-aware measures, and expected influence.
- normalized
Logical. Normalize values by dividing by max. Most measures are scaled to 0-1; signed expected-influence measures can retain negative values under psychometric normalization. For closeness, this is passed directly to igraph.
- weighted
Logical. Use edge weights if available. Default TRUE.
- directed
Logical or NULL. If NULL (default), auto-detect from matrix symmetry. Set TRUE to force directed, FALSE to force undirected.
- loops
Logical. If TRUE (default), keep self-loops. Set to FALSE to remove them before calculation.
- simplify
How to combine multiple edges between the same node pair. Options: "sum" (default), "mean", "max", "min", or FALSE/"none" to keep multiple edges.
- digits
Integer or NULL. Round all numeric columns to this many decimal places. Default NULL (no rounding).
- sort_by
Character or NULL. Column name to sort results by (descending order). Default NULL (original node order).
- cutoff
Maximum path length to consider for betweenness, closeness, and harmonic centrality. Default -1 (no limit). Set to a positive value for faster computation on large networks at the cost of accuracy.
- invert_weights
Logical or NULL. For path- and distance-based measures (for example betweenness, closeness, harmonic, eccentricity, k-reach, radiality, decay, stress, flow betweenness, and related variants), should weights be inverted so that higher weights mean shorter paths? Default NULL auto-detects: TRUE for tna objects (transition probabilities), FALSE otherwise (matching igraph/sna). Set explicitly to TRUE for strength/frequency weights (qgraph style) or FALSE for distance/cost weights.
- alpha
Numeric. Exponent for weight transformation when
invert_weights = TRUE. Distance is computed as1 / 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 matchestna::centralities(., "Clustering")byte-for-byte. Auto-set to"onnela"whentna_network = TRUEand 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 matchestna::centralities(., measures = "Diffusion")whenloops = FALSE. Default NULL auto-detects:"power_series"for tna objects (transition probabilities),"kandhway_kuri"otherwise.- k
Path length parameter for geodesic k-path centrality. Default 3.
- states
Named numeric vector of percolation states (0-1) for percolation centrality. Each value represents how "activated" or "infected" a node is. Default NULL (all nodes get state 1, equivalent to betweenness).
- decay_parameter
Numeric. Decay parameter for decay and generalized closeness centrality. Default 0.5. Must be between 0 and 1.
- dmnc_epsilon
Numeric. Epsilon exponent for DMNC (Density of Maximum Neighborhood Component). Default 1.7 as recommended by Lin et al. (2008). centiserve uses 1.67 (four-community assumption). Must be between 1 and 2.
- membership
Integer vector of community assignments (one per node) for community-aware measures: participation, within_module_z, gateway, and the Gould-Fernandez brokerage roles. Default NULL. Required when requesting these measures.
- katz_alpha
Attenuation factor for Katz centrality. Must satisfy \(\alpha < 1 / \rho(A)\). Default 0.1 (matches centiserve and NetworkX conventions). Only used when
"katz"is inmeasures.- hubbell_weight
Weight factor \(w\) for Hubbell centrality. Must satisfy \(w \cdot \rho(W) \le 1\) for solvability. Default 0.5. Only used when
"hubbell"is inmeasures.- tna_network
Logical or NULL. Umbrella switch that forces tna-style conventions across all measures.
NULL(default) auto-detects from the input class — TRUE iffxis atnaor related sequence-network object.TRUEforces tna conventions even on raw matrices:invert_weights = TRUE,loops = FALSE,diffusion_method = "power_series",transitivity_type = "onnela".FALSEsuppresses all tna defaults even for tna inputs, giving the cograph defaults verbatim. Precedence: any arg the user passes explicitly always wins overtna_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. WhenTRUE, normalized expected influence is divided by the maximum absolute expected-influence value, preserving sign and bounding the result from -1 to 1.FALSEkeeps the generic cograph normalization convention.- ...
Additional arguments (currently unused)
Value
A data frame with columns:
node: Node labels/namesOne column per measure, with mode suffix for directional measures (e.g.,
degree_in,closeness_all)
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).
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 2.943751e-20
#> 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