Bridging Capital

Bridging capital (Jackson 2020, section 3.3) credits node $i$ with the expected number of walks of length at most $T$ that are lost when one entry $P_{ij}$ of the transmission matrix is deleted, summed over $j$ and weighted by source-destination values $v_{st}$: $$Brid_i = \sum_j \sum_{s,t} v_{st} \sum_{h=1}^{T} \left[P^h - (P - P_{ij} E_{ij})^h\right]_{st}.$$

Usage

centrality_bridging_capital(x, bridging_steps = 2, bridging_values = NULL, ...)

Arguments

x: Network input accepted by centrality.
bridging_steps: Horizon $T$, a nonnegative integer. Default 2.
bridging_values: Nonnegative n by n matrix of source-destination values $v$. NULL (default) sets every value to one. Row and column names, when present, are matched to the node names.
...: Further arguments to centrality. The measure uses weighted (use edge weights, default TRUE) and normalized (divide by the maximum, default FALSE).

Value

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

Details

Edge weights are the transmission probabilities in $P$ and must lie between zero and one, and an unweighted edge has probability one. A weight above one raises an error. Direction and loops are kept, and the two entries of an undirected edge are deleted separately. Walks may repeat nodes and edges, and a walk that uses the deleted entry several times is counted once. Isolated nodes score zero, and bridging_steps = 0 gives zero scores. The function computes the expected walk count EInf of the source paper.

References

Jackson, M. O. (2020). A typology of social capital and associated network measures. Social Choice and Welfare, 54, 311-336. doi:10.1007/s00355-019-01189-3 .

Examples

centrality_bridging_capital(regulation_net)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#>     1.4617     3.4556     1.7107     2.1084     0.5485     1.7608     0.9969 
#>   Evaluate     Create      Share 
#>     2.0890     2.3416     2.9270