Group Centrality (Everett-Borgatti 1999)

Group centrality measures the importance of a set of nodes \(C \subseteq V\) rather than a single node. Three variants are supported:

Usage

group_centrality(
  x,
  nodes,
  measure = c("betweenness", "closeness", "degree"),
  mode = c("all", "out", "in"),
  normalized = TRUE
)

Arguments

x: Network input (matrix, igraph, network, cograph_network, tna object).
nodes: Integer vector of node indices (1-based) or character vector of node names identifying the group \(C\).
measure: One of "betweenness", "closeness", "degree".
mode: For directed graphs with measure = "degree": "all" (both directions), "out" (outgoing), or "in" (incoming). Ignored for undirected graphs and other measures.
normalized: Logical, for "betweenness" only. If TRUE (default), divide by \((|V| - |C|)(|V| - |C| - 1)\).

Value

A single numeric scalar — the group centrality of the set nodes.

Details

betweenness: \(GBC(C) = \sum_{s,t \in V \setminus C, s \ne t} \sigma(s, t \mid C) / \sigma(s, t)\), where \(\sigma(s, t)\) is the number of shortest \(s\)-\(t\) paths and \(\sigma(s, t \mid C)\) is the number of those paths passing through at least one node in \(C\). Normalized by \(1 / ((|V| - |C|)(|V| - |C| - 1))\).
closeness: \(GCC(C) = (|V| - |C|) / \sum_{v \in V \setminus C} d(v, C)\), where \(d(v, C) = \min_{c \in C} d(v, c)\) is the shortest distance from \(v\) to any group member. Unreachable nodes contribute 0 to the denominator sum (matching NetworkX convention). For directed graphs, cograph uses \(d(v, c)\) in the original direction, equivalent to NetworkX's "reverse then multi-source".
degree: \(GDC(C) = |N(C) \setminus C| / (|V| - |C|)\), the fraction of non-group nodes adjacent to at least one group member. mode = "in" / "out" pick the corresponding directed neighborhood.

Divergence from NetworkX on betweenness

networkx.group_betweenness_centrality uses the Puzis-Yahalom-Elovici iterative algorithm, which produces results that diverge from the textbook Everett-Borgatti / Puzis 2008 "at least one node in C" definition on some graph topologies (verified via an independent Python brute-force). cograph implements the textbook formula directly; group_closeness and group_degree match NetworkX exactly.

References

Everett, M. G., & Borgatti, S. P. (1999). The centrality of groups and classes. Journal of Mathematical Sociology, 23(3), 181-201.

Puzis, R., Yahalom, R., & Elovici, Y. (2008). Augmentative data collection for betweenness centrality. In Advances in Social Networks Analysis and Mining (pp. 196-200). IEEE.

Examples

g <- igraph::make_graph("Zachary")
group_centrality(g, nodes = c(1, 2, 3), measure = "betweenness")
#> [1] 0.5754019
group_centrality(g, nodes = c(1, 2, 3), measure = "closeness")
#> [1] 0.7045455
group_centrality(g, nodes = c(1, 2, 3), measure = "degree")
#> [1] 0.6129032