Local efficiency, s-core, fragmentation, k-path census and EPC

Five node measures that other centrality packages expose and centrality() did not. Each is a thin wrapper on centrality.

Usage

centrality_local_efficiency(x, mode = "all", ...)

centrality_s_core(x, ...)

centrality_fragmentation(x, mode = "all", ...)

centrality_kpath(x, mode = "all", kpath_len = 3, ...)

centrality_epc(x, epc_threshold = 0.5, epc_runs = 1000, epc_seed = NULL, ...)

Arguments

x

Network input: matrix, igraph, network, cograph_network, or tna object.

mode

Direction: "all", "out" or "in".

...

Additional arguments passed to centrality.

kpath_len

Maximum path length for centrality_kpath. Default 3.

epc_threshold

Edge removal probability. Default 0.5.

epc_runs

Number of percolation realizations. Default 1000.

epc_seed

Random seed. Default NULL, which leaves the caller's stream alone and makes the estimate vary between calls.

Value

Named numeric vector, one value per node.

Details

local_efficiency (Latora & Marchiori 2001)

The global efficiency of the subgraph induced on the node's neighbors, the node itself removed: the mean of \(1 / d_{jl}\) over ordered pairs of neighbors, with distances measured inside that subgraph. Nodes with fewer than two neighbors score 0. High values mark a node whose neighborhood survives its loss. Matches igraph::local_efficiency() and brainGraph::efficiency(type = "local").

s_core (Eidsaa & Almaas 2013)

The weighted k-core: the largest strength threshold \(s\) whose maximal subgraph of nodes with strength at least \(s\) still contains the node. Unit weights give the k-core number exactly. Uses edge weights.

fragmentation (Borgatti 2006)

Distance-weighted fragmentation of the network after deleting the node: \(1 - \sum 1/d_{ij} / ((n-1)(n-2))\) over the ordered pairs that remain. Higher means a more disruptive removal. Matches keyplayer::fragment() on unweighted input.

kpath (Sade 1989)

The number of simple paths of length at most kpath_len (default 3) that the node lies on, endpoints included; length 1 alone reproduces degree. Matches the per-vertex column sums of sna::kpath.census(). Enumeration is exhaustive, so cost grows with branching factor to the power kpath_len.

epc (Lin et al. 2008)

Edge percolated component: each edge survives with probability 1 - epc_threshold, and the score is the mean size of the node's component over epc_runs realizations, as a share of the network. cytoHubba and centiserve::epc() divide by the node count alone, so their number is epc_runs times this one; the ranking is the same. A Monte Carlo estimate – pass epc_seed for a reproducible value.

local_efficiency, fragmentation and kpath follow mode; s_core and epc read the undirected skeleton.

References

Latora, V., & Marchiori, M. (2001). Efficient behavior of small-world networks. Physical Review Letters, 87(19), 198701.

Eidsaa, M., & Almaas, E. (2013). s-core network decomposition: A generalization of k-core analysis to weighted networks. Physical Review E, 88(6), 062819.

Borgatti, S. P. (2006). Identifying sets of key players in a social network. Computational and Mathematical Organization Theory, 12(1), 21-34.

Sade, D. S. (1989). Sociometrics of Macaca mulatta III: n-path centrality in grooming networks. Social Networks, 11(3), 273-292.

Lin, C.-Y., Chin, C.-H., Wu, H.-H., Chen, S.-H., Ho, C.-W., & Ko, M.-T. (2008). Hubba: hub objects analyzer. Nucleic Acids Research, 36, W438-W443.

See also

centrality_coreness, centrality_weighted_kshell, centrality_geodesic_kpath, network_local_efficiency.

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_local_efficiency(adj)
#>         A         B         C         D         E         F 
#> 1.0000000 1.0000000 0.3333333 0.3333333 1.0000000 1.0000000 
centrality_s_core(adj)
#> A B C D E F 
#> 2 2 2 2 2 2 
centrality_fragmentation(adj)
#>         A         B         C         D         E         F 
#> 0.2833333 0.2833333 0.6000000 0.6000000 0.2833333 0.2833333 
centrality_kpath(adj, kpath_len = 2)
#>  A  B  C  D  E  F 
#>  6  6 10 10  6  6 
centrality_epc(adj, epc_runs = 50, epc_seed = 1)
#>         A         B         C         D         E         F 
#> 0.4733333 0.4866667 0.5400000 0.5033333 0.4766667 0.4066667