Group Centrality (Everett-Borgatti 1999)

Computes the centrality of a set of nodes \(C \subseteq V\). Distances are unweighted hop counts in the direction of the edges. The measures are defined as follows.

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" (default), "closeness", or "degree".
mode: For directed graphs with measure = "degree": "all" (both directions, default), "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

Numeric scalar: the group centrality of the set nodes. Unknown node names and out-of-range indices raise an error.

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. For directed graphs, \(d(v, c)\) follows the edges from \(v\) to \(c\).
degree: \(GDC(C) = |N(C) \setminus C| / (|V| - |C|)\), the fraction of non-group nodes adjacent to at least one group member. For directed graphs, mode selects the neighborhood.

Group betweenness

Group betweenness is computed directly from the Everett and Borgatti definition, counting the shortest paths that pass through at least one node in \(C\). On some graphs the result differs from networkx.group_betweenness_centrality, which uses the iterative algorithm of Puzis, Elovici and Dolev.

References

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

Puzis, R., Elovici, Y., & Dolev, S. (2007). Fast algorithm for successive computation of group betweenness centrality. Physical Review E, 76, 056709. doi:10.1103/PhysRevE.76.056709 .

Examples

group_centrality(regulation_net, nodes = c("Plan", "Monitor"), measure = "betweenness")
#> [1] 0.3428571