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

weighted_kshell (Garas, Schweitzer & Havlin 2012)

k-shell decomposition on the generalized degree \(k' = (k^\alpha s^\beta)^{1 / (\alpha + \beta)}\) (wks_alpha, wks_beta, both 1), after the paper's weight normalization (divide by the mean, then by the minimum, round to the nearest integer). Integer thresholds label the shells, so unit weights give the k-core number and isolates score 0. Reproduces the paper's Figure 1 example and its Table 2 core size on the netscience network. Uses edge weights.

renewed_coreness (Liu, Tang, Zhou & Do 2015)

Each link gets the diffusion importance \(D_{ij} = (|N(j) \setminus N[i]| + |N(i) \setminus N[j]|) / 2\); links below renewed_threshold (paper: 2) are removed and the k-core number of the residual graph is the renewed coreness. A clique with no outside links collapses to 0. Reproduces the paper's Figure 1 and all twelve percentages of its supplementary Table S1; the Zoo's transcription with open neighborhoods is off by one.

geodesic_kpath (Borgatti & Everett 2006)

The number of shortest paths of length at most kpath_k (default 3) that start at the node, counted with multiplicity. Note that centiserve::geokpath counts nodes within \(k\) instead, which is the paper's vertex-disjoint variant and equals m-reach.

geodesic_kpath follows mode; the other two ignore direction.

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 (matrix, igraph, network, cograph_network, tna object).

wks_alpha, wks_beta

Exponents of degree and strength in the weighted k-shell. Default 1 and 1.

...

Additional arguments passed to centrality.

renewed_threshold

Diffusion-importance threshold. Default 2.

mode

For directed networks: "all" (default), "out" (distances along out-edges), or "in".

kpath_k

Maximum path length. Default 3.

Value

Named numeric vector, one value per node.

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. Scientific Reports, 5, 13172.

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

See also

centrality_coreness, centrality_s_shell, centrality_kreach.

Examples

adj <- matrix(0, 6, 6)
adj[cbind(c(1, 1, 2, 4, 4, 5, 3), c(2, 3, 3, 5, 6, 6, 4))] <- 1
adj <- adj + t(adj)
rownames(adj) <- colnames(adj) <- LETTERS[1:6]
centrality_weighted_kshell(adj)
#> A B C D E F 
#> 2 2 2 2 2 2 
centrality_renewed_coreness(adj)
#> A B C D E F 
#> 0 0 1 1 0 0 
centrality_geodesic_kpath(adj, kpath_k = 2)
#> A B C D E F 
#> 3 3 5 5 3 3