Localized bridging centrality from ego betweenness

Nanda and Kotz's localized bridging centrality is the product of a node's unnormalized betweenness in its induced one-hop ego network and its bridging coefficient. The coefficient is reciprocal focal degree divided by the sum of reciprocal neighbor degrees, all measured in the original graph. It is not computed from degrees truncated to the ego network.

Usage

centrality_localized_bridging(x, ...)

Arguments

x

Network input accepted by centrality.

...

Additional arguments to centrality. normalized = TRUE divides final scores by their maximum; all-zero scores remain zero. Ego betweenness is never scaled by ego size.

Value

Named numeric vector in input node order.

Details

Each unordered pair of other ego-network vertices contributes the fraction of its shortest paths that pass through the focal vertex. Endpoints are excluded. Uses a simple unweighted undirected skeleton: either arc direction creates an edge, loops are removed and parallel edges count once. Weights, mode, inversion and cutoff are ignored. This projection is an explicit cograph convention, not a directed or weighted generalization of LBC.

Isolates and leaves score zero; the isolate value extends the undefined bridging coefficient by zero. Complete graphs score zero. Disconnected components are evaluated independently before optional maximum scaling. Empty graphs return no scores. The one-hop calculation uses the Everett-Borgatti common-neighbor shortcut in each ego network, with worst-case O(n to the fourth power) time and O(n squared) memory for dense matrix multiplication across all nodes.

References

Nanda, S. and Kotz, D. (2012). Localized Bridging Centrality. Handbook of Optimization in Complex Networks, pp. 197-224, equations 7.7-7.8. doi:10.1007/978-1-4614-0857-4_7 . This author chapter restates their 2008 LBC definition.

See also

centrality_extended_local_bridging for two-hop ego networks. centrality_local_bridging retains the distinct legacy score, inverse degree times bridging coefficient.

Examples

centrality_localized_bridging(igraph::make_graph("Zachary"))
#>         1         2         3         4         5         6         7         8 
#> 1.0638369 0.7407407 1.4256278 0.3064699 0.2580645 0.4363636 0.4363636 0.0000000 
#>         9        10        11        12        13        14        15        16 
#> 1.2620415 3.1481481 0.2580645 0.0000000 0.0000000 1.6028810 0.0000000 0.0000000 
#>        17        18        19        20        21        22        23        24 
#> 0.0000000 0.0000000 0.0000000 2.8681898 0.0000000 0.0000000 0.0000000 1.0251256 
#>        25        26        27        28        29        30        31        32 
#> 0.8888889 0.9523810 0.0000000 1.8059490 2.0481928 0.2968568 1.6546503 1.5910478 
#>        33        34 
#> 0.6822368 0.9894594