Wiener Index Centrality

The Wiener centrality of a node is the sum of its shortest-path distances to the nodes it reaches: $$W(i) = \sum_{j \ne i,\, d_{ij} < \infty} d_{ij}.$$ High values mark peripheral nodes. On a connected undirected network half the sum of the scores is the Wiener index of the network (Wiener 1947).

Usage

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

Arguments

x

Network input accepted by centrality.

mode

Direction for directed networks: "all" (default), "out" or "in".

...

Further arguments to centrality. The measure uses weighted (default TRUE), invert_weights (default NULL, which inverts for tna input only), alpha (inversion exponent, default 1) and cutoff (largest distance counted, default -1 for no limit).

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 converts a weight \(w\) to the distance \(1/w^\alpha\). mode = "all" treats edges as undirected, "out" uses distances from the node and "in" distances to it. Unreachable nodes add nothing, so on a disconnected network a node in a small component also scores low.

References

Wiener, H. (1947). Structural determination of paraffin boiling points. Journal of the American Chemical Society, 69(1), 17-20. doi:10.1021/ja01193a005 .

See also

centrality_closeness, centrality_average_distance, centrality.

Examples

centrality_wiener(regulation_net)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#>       1.94       2.23       1.75       2.64       1.92       2.89       2.07 
#>   Evaluate     Create      Share 
#>       2.48       1.90       2.84