Relative-Entropy Integrated Evaluation

The relative-entropy integrated evaluation (Chen, Wang and Luo 2016) combines several centrality indexes into one score. Each index is turned into a distribution over the nodes, and the score is the distribution with the smallest total relative entropy to all of them, which is their normalized geometric mean: $$w_i = \frac{\prod_{j=1}^{m} u_{ji}^{1/m}} {\sum_{l} \prod_{j=1}^{m} u_{jl}^{1/m}}$$ A positive index (larger is more important) maps to $u_i = C_i / \sum_j C_j$, and a negative index (smaller is more important) maps to the normalized complement $1 - C_i / \sum_j C_j$.

Usage

centrality_relative_entropy(
  x,
  re_indexes = c("degree", "closeness", "betweenness", "constraint"),
  re_negative = NULL,
  ...
)

Arguments

x: Network input accepted by centrality.
re_indexes: Character vector of constituent indexes, without repeats. The default c("degree", "closeness", "betweenness", "constraint") is the source's four-index set.
re_negative: Character vector naming the members of re_indexes treated as negative. The default NULL uses the source's declarations, which make constraint and largest_component negative. character(0) treats every index as positive.
...: Further arguments to centrality, such as normalized (divide by the maximum, default FALSE).

Value

A named numeric vector with one score per node, in input node order, summing to one. With normalized = TRUE the scores are divided by their maximum and no longer sum to one.

Details

The measure is computed on the simple undirected skeleton of the network, so direction, weights and loops are ignored. The available indexes are "degree", "closeness" (the reciprocal of the distance sum over reachable partners), "betweenness" (summed over ordered pairs), "constraint" (the source's equation 6, whose outer sum runs over every other node), "n_components" and "largest_component" (the components left after deleting the node). The scores sum to one, and a node that scores zero on any index scores zero overall. An index that is zero at every node, such as betweenness on a complete graph, raises a cograph_undefined_index error when the measure is requested by name. Inside type = "all" the same case gives a cograph_undefined_measure warning and an NA column. The source's equation 6 differs from Burt's constraint as computed by igraph::constraint().

References

Chen, B., Wang, Z. and Luo, C. (2016). Integrated evaluation approach for node importance of complex networks based on relative entropy. Journal of Systems Engineering and Electronics, 27(6), 1219-1226. doi:10.21629/JSEE.2016.06.10 .

Examples

centrality_relative_entropy(regulation_net)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#> 0.09263748 0.11362201 0.12675208 0.12014870 0.10357082 0.09332242 0.07081870 
#>   Evaluate     Create      Share 
#> 0.08726193 0.10712914 0.08473674