// brandes is the common code for Betweenness and EdgeBetweenness. It corresponds
// to algorithm 1 in http://algo.uni-konstanz.de/publications/b-vspbc-08.pdf with
// the accumulation loop provided by the accumulate closure.
func brandes(g graph.Graph, accumulate func(s graph.Node, stack internal.NodeStack, p map[int][]graph.Node, delta, sigma map[int]float64)) {
var (
nodes = g.Nodes()
stack internal.NodeStack
p = make(map[int][]graph.Node, len(nodes))
sigma = make(map[int]float64, len(nodes))
d = make(map[int]int, len(nodes))
delta = make(map[int]float64, len(nodes))
queue internal.NodeQueue
)
for _, s := range nodes {
stack = stack[:0]
for _, w := range nodes {
p[w.ID()] = p[w.ID()][:0]
}
for _, t := range nodes {
sigma[t.ID()] = 0
d[t.ID()] = -1
}
sigma[s.ID()] = 1
d[s.ID()] = 0
queue.Enqueue(s)
for queue.Len() != 0 {
v := queue.Dequeue()
stack.Push(v)
for _, w := range g.From(v) {
// w found for the first time?
if d[w.ID()] < 0 {
queue.Enqueue(w)
d[w.ID()] = d[v.ID()] + 1
}
// shortest path to w via v?
if d[w.ID()] == d[v.ID()]+1 {
sigma[w.ID()] += sigma[v.ID()]
p[w.ID()] = append(p[w.ID()], v)
}
}
}
for _, v := range nodes {
delta[v.ID()] = 0
}
// S returns vertices in order of non-increasing distance from s
accumulate(s, stack, p, delta, sigma)
}
}
// Betweenness returns the non-zero betweenness centrality for nodes in the unweighted graph g.
//
// C_B(v) = \sum_{s ≠ v ≠ t ∈ V} (\sigma_{st}(v) / \sigma_{st})
//
// where \sigma_{st} and \sigma_{st}(v) are the number of shortest paths from s to t,
// and the subset of those paths containing v respectively.
func Betweenness(g graph.Graph) map[int]float64 {
// Brandes' algorithm for finding betweenness centrality for nodes in
// and unweighted graph:
//
// http://www.inf.uni-konstanz.de/algo/publications/b-fabc-01.pdf
// TODO(kortschak): Consider using the parallel algorithm when
// GOMAXPROCS != 1.
//
// http://htor.inf.ethz.ch/publications/img/edmonds-hoefler-lumsdaine-bc.pdf
// Also note special case for sparse networks:
// http://wwwold.iit.cnr.it/staff/marco.pellegrini/papiri/asonam-final.pdf
var (
cb = make(map[int]float64)
nodes = g.Nodes()
stack internal.NodeStack
p = make(map[int][]graph.Node, len(nodes))
sigma = make(map[int]float64, len(nodes))
d = make(map[int]int, len(nodes))
delta = make(map[int]float64, len(nodes))
queue internal.NodeQueue
)
for _, s := range nodes {
stack = stack[:0]
for _, w := range nodes {
p[w.ID()] = p[w.ID()][:0]
}
for _, t := range nodes {
sigma[t.ID()] = 0
d[t.ID()] = -1
}
sigma[s.ID()] = 1
d[s.ID()] = 0
queue.Enqueue(s)
for queue.Len() != 0 {
v := queue.Dequeue()
stack.Push(v)
for _, w := range g.From(v) {
// w found for the first time?
if d[w.ID()] < 0 {
queue.Enqueue(w)
d[w.ID()] = d[v.ID()] + 1
}
// shortest path to w via v?
if d[w.ID()] == d[v.ID()]+1 {
sigma[w.ID()] += sigma[v.ID()]
p[w.ID()] = append(p[w.ID()], v)
}
}
}
for _, v := range nodes {
delta[v.ID()] = 0
}
// S returns vertices in order of non-increasing distance from s
for stack.Len() != 0 {
w := stack.Pop()
for _, v := range p[w.ID()] {
delta[v.ID()] += sigma[v.ID()] / sigma[w.ID()] * (1 + delta[w.ID()])
}
if w.ID() != s.ID() {
if d := delta[w.ID()]; d != 0 {
cb[w.ID()] += d
}
}
}
}
return cb
}