Randomized Shortest Paths Betweenness Centrality

Randomized shortest paths betweenness (Kivimaki et al. 2016) scores a node by the expected number of visits it receives from absorbing walks between every ordered source-target pair, under a Boltzmann distribution that favors low-cost walks with inverse temperature $\beta$. Large $\beta$ approaches shortest-path betweenness, and $\beta \to 0$ approaches a random-walk quantity that is proportional to degree on an undirected graph. With $Z = (I - W)^{-1}$ and $W = (D^{-1}A) \circ \exp(-\beta C)$: $$bet_i = \sum_{s,t} \left(\frac{z_{si}}{z_{st}} - \frac{z_{ti}}{z_{tt}}\right) z_{it}$$

Usage

centrality_rsp_betweenness(x, ...)

Arguments

x: Network input accepted by centrality.
...: Further arguments to centrality. The measure uses rsp_beta (inverse temperature, default 0.01), rsp_cost ("inverse" (default) or "weight") and weighted (use edge weights, default TRUE).

Value

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

Details

Direction and edge weights are used and loops are dropped. mode, cutoff and invert_weights have no effect. The cost $C$ is $1/w$ with rsp_cost = "inverse" and $w$ with "weight", and with weighted = FALSE every arc costs one. Following the source, a pair with no connecting path contributes zero, so scores are component-local, and nodes with no out-edges score zero. A rsp_beta at or below zero raises a cograph_bad_parameter error, and negative or non-finite weights raise a cograph_bad_input error. The measure needs one dense matrix inverse, so it is costly and is computed only when requested by name or through include.

References

Kivimaki, I., Lebichot, B., Saramaki, J. and Saerens, M. (2016). Two betweenness centrality measures based on Randomized Shortest Paths. Scientific Reports, 6, 19668. doi:10.1038/srep19668 .

Examples

centrality_rsp_betweenness(regulation_net)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#>   79.71481   22.23475  132.02683   91.88715   79.40471   43.82322   23.30359 
#>   Evaluate     Create      Share 
#>   51.38749  102.91337   69.90697