// TarjanSCC returns the strongly connected components of the graph g using Tarjan's algorithm.
//
// A strongly connected component of a graph is a set of vertices where it's possible to reach any
// vertex in the set from any other (meaning there's a cycle between them.)
//
// Generally speaking, a directed graph where the number of strongly connected components is equal
// to the number of nodes is acyclic, unless you count reflexive edges as a cycle (which requires
// only a little extra testing.)
//
func TarjanSCC(g graph.Directed) [][]graph.Node {
nodes := g.Nodes()
t := tarjan{
succ: g.From,
indexTable: make(map[int]int, len(nodes)),
lowLink: make(map[int]int, len(nodes)),
onStack: make(internal.IntSet, len(nodes)),
}
for _, v := range nodes {
if t.indexTable[v.ID()] == 0 {
t.strongconnect(v)
}
}
return t.sccs
}
// PageRankSparse returns the PageRank weights for nodes of the sparse directed
// graph g using the given damping factor and terminating when the 2-norm of the
// vector difference between iterations is below tol. The returned map is
// keyed on the graph node IDs.
func PageRankSparse(g graph.Directed, damp, tol float64) map[int]float64 {
// PageRankSparse is implemented according to "How Google Finds Your Needle
// in the Web's Haystack".
//
// G.I^k = alpha.H.I^k + alpha.A.I^k + (1-alpha).1/n.1.I^k
//
// http://www.ams.org/samplings/feature-column/fcarc-pagerank
nodes := g.Nodes()
indexOf := make(map[int]int, len(nodes))
for i, n := range nodes {
indexOf[n.ID()] = i
}
m := make(rowCompressedMatrix, len(nodes))
var dangling compressedRow
df := damp / float64(len(nodes))
for j, u := range nodes {
to := g.From(u)
f := damp / float64(len(to))
for _, v := range to {
m.addTo(indexOf[v.ID()], j, f)
}
if len(to) == 0 {
dangling.addTo(j, df)
}
}
last := make([]float64, len(nodes))
for i := range last {
last[i] = 1
}
lastV := mat64.NewVector(len(nodes), last)
vec := make([]float64, len(nodes))
var sum float64
for i := range vec {
r := rand.NormFloat64()
sum += r
vec[i] = r
}
f := 1 / sum
for i := range vec {
vec[i] *= f
}
v := mat64.NewVector(len(nodes), vec)
dt := (1 - damp) / float64(len(nodes))
for {
lastV, v = v, lastV
m.mulVecUnitary(v, lastV) // First term of the G matrix equation;
with := dangling.dotUnitary(lastV) // Second term;
away := onesDotUnitary(dt, lastV) // Last term.
floats.AddConst(with+away, v.RawVector().Data)
if normDiff(vec, last) < tol {
break
}
}
ranks := make(map[int]float64, len(nodes))
for i, r := range v.RawVector().Data {
ranks[nodes[i].ID()] = r
}
return ranks
}
// PageRank returns the PageRank weights for nodes of the directed graph g
// using the given damping factor and terminating when the 2-norm of the
// vector difference between iterations is below tol. The returned map is
// keyed on the graph node IDs.
func PageRank(g graph.Directed, damp, tol float64) map[int]float64 {
// PageRank is implemented according to "How Google Finds Your Needle
// in the Web's Haystack".
//
// G.I^k = alpha.S.I^k + (1-alpha).1/n.1.I^k
//
// http://www.ams.org/samplings/feature-column/fcarc-pagerank
nodes := g.Nodes()
indexOf := make(map[int]int, len(nodes))
for i, n := range nodes {
indexOf[n.ID()] = i
}
m := mat64.NewDense(len(nodes), len(nodes), nil)
dangling := damp / float64(len(nodes))
for j, u := range nodes {
to := g.From(u)
f := damp / float64(len(to))
for _, v := range to {
m.Set(indexOf[v.ID()], j, f)
}
if len(to) == 0 {
for i := range nodes {
m.Set(i, j, dangling)
}
}
}
mat := m.RawMatrix().Data
dt := (1 - damp) / float64(len(nodes))
for i := range mat {
mat[i] += dt
}
last := make([]float64, len(nodes))
for i := range last {
last[i] = 1
}
lastV := mat64.NewVector(len(nodes), last)
vec := make([]float64, len(nodes))
var sum float64
for i := range vec {
r := rand.NormFloat64()
sum += r
vec[i] = r
}
f := 1 / sum
for i := range vec {
vec[i] *= f
}
v := mat64.NewVector(len(nodes), vec)
for {
lastV, v = v, lastV
v.MulVec(m, lastV)
if normDiff(vec, last) < tol {
break
}
}
ranks := make(map[int]float64, len(nodes))
for i, r := range v.RawVector().Data {
ranks[nodes[i].ID()] = r
}
return ranks
}
// HITS returns the Hyperlink-Induced Topic Search hub-authority scores for
// nodes of the directed graph g. HITS terminates when the 2-norm of the
// vector difference between iterations is below tol. The returned map is
// keyed on the graph node IDs.
func HITS(g graph.Directed, tol float64) map[int]HubAuthority {
nodes := g.Nodes()
// Make a topological copy of g with dense node IDs.
indexOf := make(map[int]int, len(nodes))
for i, n := range nodes {
indexOf[n.ID()] = i
}
nodesLinkingTo := make([][]int, len(nodes))
nodesLinkedFrom := make([][]int, len(nodes))
for i, n := range nodes {
for _, u := range g.To(n) {
nodesLinkingTo[i] = append(nodesLinkingTo[i], indexOf[u.ID()])
}
for _, v := range g.From(n) {
nodesLinkedFrom[i] = append(nodesLinkedFrom[i], indexOf[v.ID()])
}
}
indexOf = nil
w := make([]float64, 4*len(nodes))
auth := w[:len(nodes)]
hub := w[len(nodes) : 2*len(nodes)]
for i := range nodes {
auth[i] = 1
hub[i] = 1
}
deltaAuth := w[2*len(nodes) : 3*len(nodes)]
deltaHub := w[3*len(nodes):]
var norm float64
for {
norm = 0
for v := range nodes {
var a float64
for _, u := range nodesLinkingTo[v] {
a += hub[u]
}
deltaAuth[v] = auth[v]
auth[v] = a
norm += a * a
}
norm = math.Sqrt(norm)
for i := range auth {
auth[i] /= norm
deltaAuth[i] -= auth[i]
}
norm = 0
for u := range nodes {
var h float64
for _, v := range nodesLinkedFrom[u] {
h += auth[v]
}
deltaHub[u] = hub[u]
hub[u] = h
norm += h * h
}
norm = math.Sqrt(norm)
for i := range hub {
hub[i] /= norm
deltaHub[i] -= hub[i]
}
if floats.Norm(deltaAuth, 2) < tol && floats.Norm(deltaHub, 2) < tol {
break
}
}
hubAuth := make(map[int]HubAuthority, len(nodes))
for i, n := range nodes {
hubAuth[n.ID()] = HubAuthority{Hub: hub[i], Authority: auth[i]}
}
return hubAuth
}