cograph (version 2.7.2)

centrality_distance_entropy: Distance Entropy

Description

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", ...)

Value

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

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.

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.

See Also

centrality for computing multiple measures at once, centrality_local_dimension for the growth-rate view of the same distance profile.

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)