Percolation Centrality

Percolation centrality (Piraveenan et al. 2013) weights each shortest path through a node by the percolation state $x_s$ of its source: $$PC(v) = \frac{1}{n-2} \sum_{s \ne v \ne t} \frac{\sigma_{st}(v)}{\sigma_{st}} \frac{x_s}{\sum_i x_i - x_v},$$ where $\sigma_{st}$ is the number of shortest paths from $s$ to $t$ and $\sigma_{st}(v)$ the number through $v$. With equal states the score is the normalized betweenness.

Usage

centrality_percolation(x, states = NULL, ...)

Arguments

x: Network input accepted by centrality.
states: Percolation state of each node, between 0 and 1. The default NULL gives every node state 1.
...: Further arguments to centrality. The measure uses weighted (default TRUE), invert_weights (default NULL, which inverts for tna input only) and alpha (inversion exponent, default 1).

Value

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

Details

Edge weights are read as distances, weighted = FALSE uses hop counts, and invert_weights = TRUE uses the distance $1/w^\alpha$, so tna input is inverted by default. Paths follow edge direction on a directed network. states is matched to nodes by name when it has names and by position otherwise. Its values are clipped to $[0, 1]$ and missing values are set to 1, and a vector of the wrong length raises an error. A network with fewer than three nodes scores 0.

References

Piraveenan, M., Prokopenko, M., & Hossain, L. (2013). Percolation centrality: Quantifying graph-theoretic impact of nodes during percolation in networks. PLoS ONE, 8(1), e53095. doi:10.1371/journal.pone.0053095 .

Examples

centrality_percolation(regulation_net)
#>     Explore        Plan     Monitor       Adapt     Reflect     Discuss 
#> 0.069444444 0.215277778 0.250000000 0.208333333 0.138888889 0.006944444 
#>  Synthesize    Evaluate      Create       Share 
#> 0.090277778 0.041666667 0.180555556 0.125000000