Local Efficiency, s-Core, Fragmentation, k-Path and EPC

Local efficiency (Latora and Marchiori 2001) is the mean of $1/d_{jl}$ over ordered pairs of a node's neighbors, with distances measured inside the subgraph induced on those neighbors. The s-core index (Eidsaa and Almaas 2013) is the largest strength threshold $s$ whose s-core contains the node. Fragmentation (Borgatti 2006) is the distance-weighted fragmentation of the network after the node is deleted, $$F_{-v} = 1 - \frac{\sum_{i \ne j} 1/d_{ij}}{(n-1)(n-2)}.$$ The k-path count (Sade 1989) is the number of simple paths of length at most kpath_len that pass through or end at the node. The edge percolated component (EPC; Lin et al. 2008) is the mean size of the node's component, as a share of all nodes, when each edge is kept with probability 1 - epc_threshold.

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 accepted by centrality.
mode: Direction for directed networks: "all" (default), "out" or "in".
...: Further arguments to centrality. Local efficiency and fragmentation use weighted (default TRUE), invert_weights (default NULL, which inverts for tna input only) and alpha (inversion exponent, default 1). The s-core index uses weighted.
kpath_len: Maximum path length for the k-path count. Default 3. Length 1 gives the degree in the undirected skeleton.
epc_threshold: Probability that an edge is removed in one realization. Default 0.5.
epc_runs: Number of percolation realizations. Default 1000.
epc_seed: Random seed. The default NULL uses the caller's random stream, so the estimate varies between calls. A seed gives a reproducible value and leaves the caller's stream unchanged.

Value

A named numeric vector with one score per node, in input node order.

Details

Local efficiency, fragmentation and the k-path count follow mode, and mode = "all" treats edges as undirected. Local efficiency and fragmentation read edge weights as distances, so with weights below one local efficiency exceeds one and fragmentation is negative; invert_weights = TRUE converts weights to distances $1/w^\alpha$. On unweighted input both lie between 0 and 1, and a node with fewer than two neighbors has local efficiency 0. Fragmentation is NaN on a network with fewer than three nodes. The s-core index symmetrizes the network by the stronger of the two directions and equals the k-core number when all weights are one. The k-path count and EPC ignore weights, and EPC uses the undirected skeleton. EPC is a Monte Carlo estimate, and cytoHubba and centiserve::epc() report the same quantity multiplied by the number of runs.

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. doi:10.1103/PhysRevE.88.062819 .

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, a framework of interactome hubs identification for network biology. Nucleic Acids Research, 36, W438-W443. doi:10.1093/nar/gkn257 .

Examples

centrality_local_efficiency(regulation_net)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#>   2.951812   3.504006   3.592003   2.642620   2.314935   3.058385   5.142414 
#>   Evaluate     Create      Share 
#>   4.300894   2.473077   4.373003 
centrality_s_core(regulation_net)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#>       0.99       1.08       1.08       0.99       0.92       0.99       0.77 
#>   Evaluate     Create      Share 
#>       1.08       1.08       1.08 
centrality_fragmentation(regulation_net, weighted = FALSE)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#>  0.1944444  0.2083333  0.2222222  0.2083333  0.1944444  0.1944444  0.1805556 
#>   Evaluate     Create      Share 
#>  0.1944444  0.2083333  0.1944444 
centrality_kpath(regulation_net, kpath_len = 2)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#>         37         47         58         46         36         38         30 
#>   Evaluate     Create      Share 
#>         40         48         40 
centrality_epc(regulation_net, epc_runs = 100, epc_seed = 1)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#>      0.931      0.956      0.953      0.938      0.920      0.942      0.922 
#>   Evaluate     Create      Share 
#>      0.946      0.952      0.956