Distance Entropy

Distance entropy (Stella and De Domenico 2018) is the Shannon entropy of the distribution of hop distances from a node to the nodes it reaches, scaled by the logarithm of the number of distance values in its range. With $p_k$ the share of reachable nodes at distance $k$, and $m_i$ and $M_i$ the smallest and largest distance, $$h_i = -\frac{1}{\log(M_i - m_i + 1)} \sum_{k = m_i}^{M_i} p_k \log p_k.$$

Usage

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

Arguments

x: Network input accepted by centrality.
mode: For directed networks: "all" (default), "out" or "in".
...: Further arguments to centrality, such as normalized.

Value

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

Details

Distances are hop counts, so edge weights are ignored. On a directed network mode sets the direction of the paths. Scores lie between 0 and 1. A node whose reachable nodes all lie at one distance scores 0, and a node that reaches no other node returns NaN. The source divides by $\log(M_i - m_i)$, which is zero when the range holds two distances. The implementation divides by $\log(M_i - m_i + 1)$, so a uniform distribution scores 1.

References

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

Examples

centrality_distance_entropy(regulation_net)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#>  0.9910761  0.9182958  0.7642045  0.9182958  0.9910761  0.9910761  0.9910761 
#>   Evaluate     Create      Share 
#>  0.9910761  0.9182958  0.9910761