Exogenous Centrality

Exogenous centrality (Everett and Borgatti 2010) is the total change in a base centrality of the other nodes when the node is deleted: $$E(i) = \sum_{j \ne i} \left[C_G(j) - C_{G-i}(j)\right].$$ The default base is reverse closeness, $C_H(j) = \sum_{k \ne j} \max(N - d_H(j,k), 0)$, with $N$ the number of nodes of the original network. The other bases are raw betweenness and degree.

Usage

centrality_exogenous(
  x,
  mode = "all",
  exogenous_base = "reverse_closeness",
  ...
)

Arguments

x: Network input accepted by centrality.
mode: Direction of the base measure: "all" (default), "out" or "in".
exogenous_base: Base centrality: "reverse_closeness" (default), "betweenness" or "degree".
...: Further arguments to centrality, such as normalized.

Value

A named numeric vector with one score per node, in input node order.

Details

The measure uses the simple binary network, so weights, loops and parallel edges are ignored. mode = "all" uses the undirected skeleton, and "out" and "in" use directed paths and degrees. On an undirected network the three modes agree and the degree base returns the degree. On a directed network the out-degree base returns the in-degree and the in-degree base the out-degree. Exogenous betweenness can be negative when a deletion raises the betweenness of the remaining nodes. An isolated node scores 0. normalized = TRUE divides by the largest score.

References

Everett, M. G., & Borgatti, S. P. (2010). Induced, endogenous and exogenous centrality. Social Networks, 32(4), 339-344. doi:10.1016/j.socnet.2010.06.004 .

Examples

centrality_exogenous(regulation_net)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#>         77         78         79         78         77         77         76 
#>   Evaluate     Create      Share 
#>         77         78         77