Suggestion: closeness vitality

[briatte]

Closeness vitality is a measure proposed in Brandes and Erlebach (2005), Section 3.6.2 (see screenshot below). It has been implemented in Python, in networkx.

The same function requires only a few lines of code with igraph:

closeness_vitality <- function(g) {
  a <- sum(igraph::distances(g))
  v <- sapply(1:igraph::vcount(g), function(v) {
    d <- igraph::distances(igraph::delete_vertices(g, v))
    a - sum(d[ !is.infinite(d) ])
  })
  names(v) <- V(g)$name
  v
}

(Ignoring infinite values in the distance matrix is only one among several options. It is the only one available through networkx, but the igraph implementation could be made to support additional options.)

As with my disparity filter submission, just let me know if you are interested, and I'll submit a proper PR.

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Related: the closeness vitality of vertex x in graph g compares the Wiener index of g to the Wiener index of that same graph after vertex x has been removed. The Wiener index (which networkx seems to have misspelt as 'Weiner') would be trivial to implement in igraph:

wiener_index <- function(g) { sum(distances(g)) }

Another strategy is to compare average path lengths instead of Wiener indexes, which is sometimes called a 'vulnerability' measure:

vulnerability <- function(g) {
  d <- mean_distance(g)
  v <- sapply(1:vcount(g), function(v) {
    g2 <- delete_vertices(g, v)
    if (all(degree(g2) > 0)) { mean_distance(g2) - d } else { NA }
  })
  names(v) <- V(g)$name
  v
}

(The function returns NA if the vertex for which vulnerability is being computed is a cut-vertex or a bridge.)

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[szhorvat]

@briatte Can you help answer the two questions I wrote in igraph/python-igraph#144 ?