Community-Based Centrality, Comm Centrality and Community-Based Mediator

Three community-aware measures that need a partition (membership).

Usage

centrality_community_based(x, membership = NULL, mode = "all", ...)

centrality_comm_centrality(
  x,
  membership = NULL,
  mode = "all",
  comm_r = "max_intra",
  ...
)

centrality_community_mediator(x, membership = NULL, mode = "all", ...)

Arguments

x: Network input (matrix, igraph, network, cograph_network, tna object).
membership: Community labels, one per node. Required; without it the function warns and returns NA.
mode: For directed networks: "all" (default), "out", or "in".
...: Additional arguments passed to centrality.
comm_r: Scale $R$ of Comm centrality: "max_intra" (default) or a single positive number. Anything else raises a cograph_bad_parameter error.

Value

Named numeric vector, one value per node.

Details

community_based (Zhao, Wang, Zhang & Zhu 2015): $CbC(i) = \sum_w d_{iw} S_w / N$: every link of $i$ counts the size $S_w$ of the community it lands in. No parameters. Reproduces Table 1 of the paper and Table 1 of Tulu et al. (2018).
comm_centrality (Gupta, Singh & Cherifi 2016): $$CC(i) = (1 + \mu_C)\, \frac{k^{in}_i}{\max_{j \in C} k^{in}_j} R + (1 - \mu_C) \left(\frac{k^{out}_i}{\max_{j \in C} k^{out}_j} R\right)^2,$$ where $k^{in}, k^{out}$ are the intra- and inter-community degrees, $\mu_C$ the mean inter-link fraction in $i$'s community, and $R$ a scale. The default comm_r = "max_intra" is the paper's recommended $R = \max_{j \in C} k^{in}_j$ per community; a number applies one global $R$. The equation uses $1 + \mu_C$ although the paper's prose says $\mu_C$; the equation is implemented. A community without intra (inter) links contributes 0 through that term.
community_mediator (Tulu, Hou & Younas 2018): $CbM(i) = H_i \, d_i / \sum_j d_j$, with $H_i$ the base-2 Shannon entropy of $i$'s link distribution over the communities. Nodes linked to one community only score 0. Base 2 is what reproduces the paper's Table 1.

Higher = more central in all three. Under mode = "out" or "in" only out- or in-links count; edge weights are ignored.

Conditions

Raises an error of class cograph_bad_membership when membership is not one non-missing label per node.

References

Zhao, Z., Wang, X., Zhang, W., & Zhu, Z. (2015). A community-based approach to identifying influential spreaders. Entropy, 17(4), 2228-2252.

Gupta, N., Singh, A., & Cherifi, H. (2016). Centrality measures for networks with community structure. Physica A, 452, 46-59.

Tulu, M. M., Hou, R., & Younas, T. (2018). Identifying influential nodes based on community structure to speed up the dissemination of information in complex network. IEEE Access, 6, 7390-7401.

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_community_based(adj, membership = c(1, 1, 1, 2, 2, 2))
#>   A   B   C   D   E   F 
#> 1.0 1.0 1.5 1.5 1.0 1.0 
centrality_comm_centrality(adj, membership = c(1, 1, 1, 2, 2, 2))
#>        A        B        C        D        E        F 
#> 2.222222 2.222222 5.777778 5.777778 2.222222 2.222222 
centrality_community_mediator(adj, membership = c(1, 1, 1, 2, 2, 2))
#>         A         B         C         D         E         F 
#> 0.0000000 0.0000000 0.1967777 0.1967777 0.0000000 0.0000000