Katz Centrality

Katz (1953) status index: \(C = (I - \alpha A^T)^{-1} \mathbf{1}\). Each node's score sums attenuated walks of every length back to it, with attenuation \(\alpha\) applied per step. Rankings are identical to Bonacich's alpha centrality with a uniform exogenous vector.

Usage

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

Arguments

x: Network input (matrix, igraph, network, cograph_network, tna object).
katz_alpha: Attenuation factor. Must satisfy \(\alpha < 1 / \rho(A)\) where \(\rho(A)\) is the spectral radius. Default 0.1 matches centiserve and NetworkX conventions.
...: Additional arguments passed to centrality.

Value

Named numeric vector of Katz centrality values.

Details

Equivalence is verified bit-exact against centiserve::katzcent (cograph mirrors centiserve's exact LAPACK call sequence) and at machine epsilon against igraph::alpha_centrality(exo = 1) and networkx.katz_centrality_numpy.

References

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

Examples

adj <- matrix(c(0, 1, 1, 1, 0, 1, 1, 1, 0), 3, 3)
rownames(adj) <- colnames(adj) <- c("A", "B", "C")
centrality_katz(adj)
#>    A    B    C 
#> 1.25 1.25 1.25