Alpha Centrality

Alpha centrality (Bonacich and Lloyd 2001) gives every node an exogenous score of one plus the scores of the nodes that point to it: $$x = \alpha A^{T} x + e, \qquad x = (I - \alpha A^{T})^{-1} e.$$ The attenuation is fixed at $\alpha = 1$ and $e$ is a vector of ones, the defaults of igraph::alpha_centrality().

Usage

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

Arguments

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

Value

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

Details

$A$ holds the edge weights, or ones with weighted = FALSE, with the diagonal set to zero. The alpha argument of centrality is the weight-inversion exponent, which this measure does not read. The scores are positive when the spectral radius of $A$ is below one and can be negative otherwise. A singular system or a negative edge weight raises an error of class cograph_singular_system.

On a directed network mode = "in" sums over incoming ties as above and equals igraph::alpha_centrality(), mode = "out" uses $A$ in place of $A^{T}$ and so sums over outgoing ties, and mode = "all" (default) uses the symmetrized weights $A + A^{T}$. On an undirected network the three modes agree.

References

Bonacich, P., & Lloyd, P. (2001). Eigenvector-like measures of centrality for asymmetric relations. Social Networks, 23(3), 191-201. doi:10.1016/S0378-8733(01)00038-7 .

Examples

centrality_alpha(regulation_net)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#> -0.1294030 -2.1294525 -2.2733148 -1.2166591  0.6012820 -0.3834963  0.6523346 
#>   Evaluate     Create      Share 
#> -2.0816623 -2.0691360 -2.3130493