Entropy Variation

Ai's (2017) vitality measure: the change in the Shannon entropy of a node-level distribution when a node and its links are removed, $$EnV_f(i) = I_f(G) - I_f(G - i), \qquad I_f(G) = -\sum_j p_j \log p_j, \quad p_j = \frac{f(j)}{\sum_l f(l)},$$ with \(f\) the degree ("entropy_variation_degree", in-, out- or total degree by mode) or the betweenness ("entropy_variation_betweenness"). Natural logarithm, as in the author's code. The difference is signed: a positive value means the remaining network is less even without the node, a negative value that removing it evens the distribution out. Higher = more important.

Usage

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

Arguments

x

Network input (matrix, igraph, network, cograph_network, tna object).

of

Which distribution: "degree" (default) or "betweenness".

mode

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

...

Additional arguments passed to centrality.

Value

Named numeric vector, one value per node, in nats.

Details

The degree variant is computed in closed form. The betweenness variant recomputes betweenness once per node and costs \(O(n \cdot nm)\); it ignores edge weights. Self-loops are counted as igraph counts them. When a deletion leaves every \(f\) at zero (for instance betweenness on a clique) that entropy is taken as 0.

Validated against the author's own R code path (iCalEnV() from the paper's repository) to \(10^{-15}\) and against the quantiles of Table 2 of the paper on its 4234-node Snake Idioms network.

References

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

See also

centrality for computing multiple measures at once.

Examples

star5 <- matrix(0, 5, 5)
star5[1, 2:5] <- 1; star5[2:5, 1] <- 1
rownames(star5) <- colnames(star5) <- LETTERS[1:5]
centrality_entropy_variation(star5)
#>        A        B        C        D        E 
#> 1.386294 0.143841 0.143841 0.143841 0.143841 
centrality_entropy_variation(star5, of = "betweenness")
#> A B C D E 
#> 0 0 0 0 0