Katz centrality — katz_centrality • netmem

Katz centrality (1953), which counts the paths that arrive to a node, discounting the longer ones.

Usage

katz_centrality(
  A,
  alpha = 0.1,
  beta = 1,
  type = c("in", "out"),
  digraph = TRUE,
  weighted = FALSE
)

Arguments

A: A square matrix
alpha: The attenuation of the longer paths
beta: The status that every node has independently of the network
type: Whether the paths considered are the ones that arrive (in, default) or leave (out) the node
digraph: Whether the matrix is directed or undirected
weighted: Whether the matrix is weighted

Value

This function returns the Katz centrality of the nodes.

Details

The centrality is \(x = \beta (I - \alpha A^T)^{-1} 1\), so a path of length \(k\) contributes \(\alpha^k\). The attenuation alpha should be smaller than the inverse of the leading eigenvalue of the matrix, otherwise the sum does not converge.

References

Katz, L. (1953). A new status index derived from sociometric analysis. Psychometrika, 18(1), 39–43. doi:10.1007/BF02289026

Author

Alejandro Espinosa-Rada

Examples

A <- matrix(c(
  0, 1, 1, 1, 0,
  1, 0, 0, 0, 0,
  1, 0, 0, 0, 1,
  1, 0, 0, 0, 0,
  0, 0, 1, 0, 0
), byrow = TRUE, ncol = 5)
rownames(A) <- letters[1:nrow(A)]
colnames(A) <- rownames(A)

katz_centrality(A, alpha = 0.1, digraph = FALSE)
#>        a        b        c        d        e 
#> 1.351802 1.135180 1.247657 1.135180 1.124766