Entropy Variation

Entropy variation (Ai 2017) is the change in the Shannon entropy of a node-level distribution \(f\) when a node and its links are removed: $$EnV_f(i) = I_f(G) - I_f(G - i), \qquad I_f(G) = -\sum_j p_j \ln p_j, \quad p_j = \frac{f(j)}{\sum_l f(l)}.$$ The distribution \(f\) is the degree or the betweenness. Higher values mark more important nodes.

Usage

centrality_entropy_variation(
  x,
  of = c("degree", "betweenness"),
  mode = "all",
  ...
)

Arguments

x

Network input accepted by centrality.

of

Distribution: "degree" (default) or "betweenness".

mode

For the degree variant on directed networks: "all" (default, in plus out), "out" or "in".

...

Further arguments to centrality. The degree variant uses loops (keep self-loops, default TRUE).

Value

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

Details

The entropy uses the natural logarithm, as in the code of the author, so scores are in nats. The difference is signed, and a negative value means that removing the node evens out the distribution. Edge weights are ignored in both variants. For the degree variant mode selects the in-, out- or total degree on a directed network, and self-loops count toward the degree, and loops = FALSE drops them. The betweenness variant ignores mode and self-loops. A deletion that leaves every value of \(f\) at zero, such as betweenness on a clique, has entropy 0.

References

Ai, X. (2017). Node importance ranking of complex networks with entropy variation. Entropy, 19(7), 303.

See also

centrality_betweenness, centrality_distance_entropy, centrality.

Examples

centrality_entropy_variation(regulation_net)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#> 0.11832094 0.12127395 0.10553447 0.12479343 0.12898833 0.10607669 0.09757111 
#>   Evaluate     Create      Share 
#> 0.09986081 0.10338637 0.09996779