Katz centrality

Katz sums walks of every length with a geometric attenuation, so the status vector solves a single linear system. Convergence of the series needs alpha below the reciprocal of the spectral radius of A.

Usage

Katz(A, alpha = 0.1)

Arguments

A

Square adjacency matrix.

alpha

Positive attenuation factor.

Value

List with estimate (largest score), centrality, total, alpha, n, method.

Details

Formula: x = (I - alpha A)^{-1} 1 = sum_{k >= 0} alpha^k A^k 1.

References

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

Examples

Katz(matrix(c(0, 1, 1, 0, 0, 1, 1, 0, 0), 3, 3, byrow = TRUE))
#> $estimate
#> [1] 1.223458
#> 
#> $centrality
#> [1] 1.223458 1.112235 1.122346
#> 
#> $total
#> [1] 3.458038
#> 
#> $alpha
#> [1] 0.1
#> 
#> $n
#> [1] 3
#> 
#> $method
#> [1] "x = (I - alpha A)^{-1} 1, Katz (1953)"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"