Bonacich Power Centrality

Bonacich (1987) power centrality with exponent $\beta = 1$ sums the walks leaving a node, each further step weighted by $\beta$: $$c = \gamma\,(I - A)^{-1} A \mathbf{1},$$ where $A$ is the binary adjacency matrix and $\gamma$ scales the scores so that their squares sum to $n$.

Usage

centrality_power(x, mode = "all", ...)

Arguments

x: Network input accepted by centrality.
mode: For directed networks: "all" (default), "out" or "in".
...: Further arguments to centrality.

Value

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

Details

The exponent is fixed at 1. Edge weights and self-loops are ignored. On a directed network mode = "out" sums over out-ties and equals igraph::power_centrality(exponent = 1), mode = "in" sums over in-ties, and mode = "all" (default) uses the undirected skeleton, the binary matrix with a tie wherever either direction has one. On an undirected network the three modes agree. Where $I - A$ is singular, as on a network that contains an isolated edge, a cograph_singular_system error is raised, and a network without edges gives NaN. The scores can be negative, as on regulation_net, and normalized = TRUE leaves scores that are all negative unchanged.

References

Bonacich, P. (1987). Power and centrality: A family of measures. American Journal of Sociology, 92(5), 1170-1182. doi:10.1086/228631 .

Examples

centrality_power(regulation_net)
#>       Explore          Plan       Monitor         Adapt       Reflect 
#> -1.217161e+00 -6.756602e-16 -6.085806e-01 -1.217161e+00 -1.825742e+00 
#>       Discuss    Synthesize      Evaluate        Create         Share 
#> -1.217161e+00 -1.217161e+00 -6.085806e-01 -5.855722e-16 -3.941351e-16