Truss, mixed-degree decomposition and local social-capital measures

Five measures with explicit definitions and numerical reference checks. All use the simple, unweighted, undirected skeleton: either direction creates an edge, parallel edges count once and self-loops are removed. This projection is a cograph input convention; no directed or weighted generalization of the published measures is claimed. All isolates score 0.

Usage

centrality_truss(x, ...)

centrality_mdd(x, mdd_lambda = 0.7, ...)

centrality_bridging_coefficient(x, ...)

centrality_godfather(x, ...)

centrality_support(x, ...)

Arguments

x: Network input accepted by centrality.
...: Additional arguments to centrality. With normalized = TRUE, positive scores are divided by their maximum.
mdd_lambda: Exhausted-degree weight between 0 and 1, default 0.7.

Value

Named numeric vector in input node order.

Details

truss: Maximum truss number of an incident edge (Malliaros et al. 2016). A k-truss requires at least k-2 triangles per edge within the surviving subgraph, matching NetworkX. An edge outside any triangle has truss number 2; a complete graph on k vertices has node truss number k. Some sources instead label by the triangle threshold, producing values two smaller.
mdd: Mixed-degree decomposition (Zeng & Zhang 2013): repeatedly peel by residual degree plus mdd_lambda times exhausted degree. Nodes falling below the current shell threshold join that shell before the threshold advances. Zero recovers the k-core number; one recovers degree. Intermediate thresholds are real-valued. Default 0.7, as in the paper's worked example.
bridging_coefficient: Hwang et al.'s reciprocal-degree ratio: \((1/d_i) / \sum_{j \in N(i)} 1/d_j\). This is the coefficient itself, before multiplication by betweenness.
godfather: Jackson's Godfather index: the number of unordered pairs of neighbors with no edge between them. Equals \(d_i(d_i-1)/2\) minus the number of triangles containing i.
support: Jackson's supported relationships: the number of neighbors sharing at least one common neighbor with i. An edge is counted once even if it belongs to multiple triangles.

LocalRank (Chen et al. 2012), also listed in the Centrality Zoo, is already available as centrality_semilocal on an undirected, unweighted graph; it needs no additional numerical function.

References

Malliaros, F. D., Rossi, M. E. G., & Vazirgiannis, M. (2016). Locating influential nodes in complex networks. Scientific Reports, 6, 19307. doi:10.1038/srep19307 .

Zeng, A., & Zhang, C. J. (2013). Ranking spreaders by decomposing complex networks. Physics Letters A, 377, 1031-1035. doi:10.1016/j.physleta.2013.02.039 .

Hwang, W., Kim, T., Ramanathan, M., & Zhang, A. (2008). Bridging centrality: graph mining from element level to group level. KDD '08, 336-344. doi:10.1145/1401890.1401934 .

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 .

Chen, D., Lu, L., Shang, M. S., Zhang, Y. C., & Zhou, T. (2012). Identifying influential nodes in complex networks. Physica A, 391, 1777-1787. doi:10.1016/j.physa.2011.09.017 .

Examples

adj <- matrix(1, 4, 4)
diag(adj) <- 0
centrality_truss(adj)
#> 1 2 3 4 
#> 4 4 4 4 
centrality_mdd(adj, mdd_lambda = 0.7)
#> 1 2 3 4 
#> 3 3 3 3 
centrality_support(adj)
#> 1 2 3 4 
#> 3 3 3 3