Katz Centrality

Katz (1953) status sums the walks of every length that end at a node, each step attenuated by $\alpha$: $$c = (I - \alpha A^{T})^{-1}\mathbf{1},$$ where $A$ is the weighted adjacency matrix and $\alpha$ is katz_alpha.

Usage

centrality_katz(x, katz_alpha = 0.1, ...)

Arguments

x: Network input accepted by centrality.
katz_alpha: Attenuation factor $\alpha$ (default 0.1).
...: Further arguments to centrality.

Value

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

Details

The series converges for $\alpha < 1/\rho(A)$, where $\rho(A)$ is the spectral radius. A divergent series is detected from scores below one and raises a cograph_katz_diverged warning that names the bound; the returned values are then not Katz scores. weighted = FALSE gives every edge weight one. On a directed network the score counts walks that arrive at the node. A network of one node without a self-loop scores 1. The values equal igraph::alpha_centrality(exo = 1) with the same alpha.

References

Katz, L. (1953). A new status index derived from sociometric analysis. Psychometrika, 18(1), 39-43.

Examples

centrality_katz(regulation_net)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#>   1.084116   1.034124   1.145330   1.107873   1.126150   1.079033   1.018834 
#>   Evaluate     Create      Share 
#>   1.092721   1.078166   1.091297