Weighted k-shell, Renewed Coreness and Geodesic k-path

Three shell and path counts. The weighted k-shell (Garas, Schweitzer and Havlin 2012) runs the k-shell decomposition on the generalized degree $$k'_i = \left(k_i^{\alpha} s_i^{\beta}\right)^{1 / (\alpha + \beta)},$$ with strength $s_i$ after the weight normalization of the source. Renewed coreness (Liu, Tang, Zhou and Do 2015) gives each link the diffusion importance $D_{ij} = (|N(j) \setminus N[i]| + |N(i) \setminus N[j]|) / 2$, removes the links below renewed_threshold, and returns the k-core number of the residual graph. Geodesic k-path centrality (Borgatti and Everett 2006) counts the shortest paths of length at most kpath_k that start at the node, with multiplicity.

Usage

centrality_weighted_kshell(x, wks_alpha = 1, wks_beta = 1, ...)

centrality_renewed_coreness(x, renewed_threshold = 2, ...)

centrality_geodesic_kpath(x, mode = "all", kpath_k = 3, ...)

Arguments

x: Network input accepted by centrality.
wks_alpha, wks_beta: Exponents of degree and strength in the weighted k-shell (default 1 and 1).
...: Further arguments to centrality. The weighted k-shell uses weighted (use edge weights, default TRUE).
renewed_threshold: Diffusion-importance threshold for renewed coreness (default 2).
mode: For directed networks: "all" (default), "out" or "in".
kpath_k: Maximum path length for geodesic k-path (default 3).

Value

A named numeric vector with one score per node, in input node order. The weighted k-shell and renewed coreness are integer vectors.

Details

The weighted k-shell uses edge weights, and with unit weights it equals the k-core number. Isolated nodes score 0. Renewed coreness ignores weights, and a clique with no outside links scores 0. Both ignore direction. Geodesic k-path ignores weights and follows mode. The Centrality Zoo transcribes the diffusion importance with open neighborhoods, which is off by one, and the closed neighborhoods of the source are used here. centiserve::geokpath counts the nodes within $k$ hops, which is the vertex-disjoint variant of Borgatti and Everett.

References

Garas, A., Schweitzer, F., & Havlin, S. (2012). A k-shell decomposition method for weighted networks. New Journal of Physics, 14, 083030.

Liu, Y., Tang, M., Zhou, T., & Do, Y. (2015). Improving the accuracy of the k-shell method by removing redundant links: From a perspective of spreading dynamics. Scientific Reports, 5, 13172. doi:10.1038/srep13172 .

Borgatti, S. P., & Everett, M. G. (2006). A graph-theoretic perspective on centrality. Social Networks, 28(4), 466-484.

Examples

centrality_weighted_kshell(regulation_net)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#>          8          8          8          8          8          8          7 
#>   Evaluate     Create      Share 
#>          8          8          8 
centrality_renewed_coreness(regulation_net)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#>          3          3          3          3          3          3          3 
#>   Evaluate     Create      Share 
#>          3          3          3 
centrality_geodesic_kpath(regulation_net)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#>         17         18         15         21         20         20         18 
#>   Evaluate     Create      Share 
#>         20         17         18