Distance Entropy

Shannon entropy of the distribution of hop distances from a node to every node it can reach (Stella & De Domenico 2018), normalized so that a uniform spread over the node's distance range scores 1: $$h(i) = -\frac{1}{\log(M_i - m_i + 1)} \sum_{k = m_i}^{M_i} p_k^{(i)} \log p_k^{(i)}, \qquad p_k^{(i)} = n_k^{(i)} / R_i,$$ where $n_k^{(i)}$ is the number of nodes at distance $k$ from $i$, $R_i$ the number of reachable nodes, and $m_i, M_i$ the minimum and maximum distance. High values mark nodes whose reach is spread evenly across many network layers; a node whose reachable nodes all sit at one distance scores 0. Closeness summarizes the mean of the same distribution; distance entropy summarizes its spread.

Usage

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

Arguments

x: Network input (matrix, igraph, network, cograph_network, tna object).
mode: For directed networks: "all" (default), "out" (distances along out-edges), or "in".
...: Additional arguments passed to centrality.

Value

Named numeric vector, one value per node, in [0, 1]. NaN for a node that reaches no other node.

Details

Distances are hop counts (edge weights are ignored). The original paper normalizes by $\log(M_i - m_i)$, which is undefined when only two distinct distances occur; $\log(M_i - m_i + 1)$ is used here so the index is bounded by 1 for a uniform distribution.

References

Stella, M., & De Domenico, M. (2018). Distance entropy cartography characterises centrality in complex networks. Entropy, 20(4), 268.

Examples

path4 <- matrix(c(0,1,0,0, 1,0,1,0, 0,1,0,1, 0,0,1,0), 4, 4)
rownames(path4) <- colnames(path4) <- c("A", "B", "C", "D")
centrality_distance_entropy(path4)
#>         A         B         C         D 
#> 1.0000000 0.9182958 0.9182958 1.0000000