forked from gonum/gonum
/
louvain_undirected.go
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/
louvain_undirected.go
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// Copyright ©2015 The Gonum Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package community
import (
"math"
"sort"
"golang.org/x/exp/rand"
"github.com/savalin/gonum/graph"
"github.com/savalin/gonum/graph/internal/ordered"
"github.com/savalin/gonum/graph/internal/set"
"github.com/savalin/gonum/graph/iterator"
)
// qUndirected returns the modularity Q score of the graph g subdivided into the
// given communities at the given resolution. If communities is nil, the
// unclustered modularity score is returned. The resolution parameter
// is γ as defined in Reichardt and Bornholdt doi:10.1103/PhysRevE.74.016110.
// qUndirected will panic if g has any edge with negative edge weight.
//
// Q = 1/2m \sum_{ij} [ A_{ij} - (\gamma k_i k_j)/2m ] \delta(c_i,c_j)
//
// graph.Undirect may be used as a shim to allow calculation of Q for
// directed graphs.
func qUndirected(g graph.Undirected, communities [][]graph.Node, resolution float64) float64 {
nodes := graph.NodesOf(g.Nodes())
weight := positiveWeightFuncFor(g)
// Calculate the total edge weight of the graph
// and the table of penetrating edge weight sums.
var m2 float64
k := make(map[int64]float64, len(nodes))
for _, u := range nodes {
uid := u.ID()
w := weight(uid, uid)
to := g.From(uid)
for to.Next() {
w += weight(uid, to.Node().ID())
}
m2 += w
k[uid] = w
}
if communities == nil {
var q float64
for _, u := range nodes {
uid := u.ID()
kU := k[uid]
q += weight(uid, uid) - resolution*kU*kU/m2
}
return q / m2
}
// Iterate over the communities, calculating
// the non-self edge weights for the upper
// triangle and adjust the diagonal.
var q float64
for _, c := range communities {
for i, u := range c {
uid := u.ID()
kU := k[uid]
q += weight(uid, uid) - resolution*kU*kU/m2
for _, v := range c[i+1:] {
vid := v.ID()
q += 2 * (weight(uid, vid) - resolution*kU*k[vid]/m2)
}
}
}
return q / m2
}
// louvainUndirected returns the hierarchical modularization of g at the given
// resolution using the Louvain algorithm. If src is nil, rand.Intn is used as
// the random generator. louvainUndirected will panic if g has any edge with negative edge
// weight.
//
// graph.Undirect may be used as a shim to allow modularization of directed graphs.
func louvainUndirected(g graph.Undirected, resolution float64, src rand.Source) *ReducedUndirected {
// See louvain.tex for a detailed description
// of the algorithm used here.
c := reduceUndirected(g, nil)
rnd := rand.Intn
if src != nil {
rnd = rand.New(src).Intn
}
for {
l := newUndirectedLocalMover(c, c.communities, resolution)
if l == nil {
return c
}
if done := l.localMovingHeuristic(rnd); done {
return c
}
c = reduceUndirected(c, l.communities)
}
}
// ReducedUndirected is an undirected graph of communities derived from a
// parent graph by reduction.
type ReducedUndirected struct {
// nodes is the set of nodes held
// by the graph. In a ReducedUndirected
// the node ID is the index into
// nodes.
nodes []community
undirectedEdges
// communities is the community
// structure of the graph.
communities [][]graph.Node
parent *ReducedUndirected
}
var (
reducedUndirected = (*ReducedUndirected)(nil)
_ graph.WeightedUndirected = reducedUndirected
_ ReducedGraph = reducedUndirected
)
// Communities returns the community memberships of the nodes in the
// graph used to generate the reduced graph.
func (g *ReducedUndirected) Communities() [][]graph.Node {
communities := make([][]graph.Node, len(g.communities))
if g.parent == nil {
for i, members := range g.communities {
comm := make([]graph.Node, len(members))
for j, n := range members {
nodes := g.nodes[n.ID()].nodes
if len(nodes) != 1 {
panic("community: unexpected number of nodes in base graph community")
}
comm[j] = nodes[0]
}
communities[i] = comm
}
return communities
}
sub := g.parent.Communities()
for i, members := range g.communities {
var comm []graph.Node
for _, n := range members {
comm = append(comm, sub[n.ID()]...)
}
communities[i] = comm
}
return communities
}
// Structure returns the community structure of the current level of
// the module clustering. The first index of the returned value
// corresponds to the index of the nodes in the next higher level if
// it exists. The returned value should not be mutated.
func (g *ReducedUndirected) Structure() [][]graph.Node {
return g.communities
}
// Expanded returns the next lower level of the module clustering or nil
// if at the lowest level.
func (g *ReducedUndirected) Expanded() ReducedGraph {
return g.parent
}
// reduceUndirected returns a reduced graph constructed from g divided
// into the given communities. The communities value is mutated
// by the call to reduceUndirected. If communities is nil and g is a
// ReducedUndirected, it is returned unaltered.
func reduceUndirected(g graph.Undirected, communities [][]graph.Node) *ReducedUndirected {
if communities == nil {
if r, ok := g.(*ReducedUndirected); ok {
return r
}
nodes := graph.NodesOf(g.Nodes())
// TODO(kortschak) This sort is necessary really only
// for testing. In practice we would not be using the
// community provided by the user for a Q calculation.
// Probably we should use a function to map the
// communities in the test sets to the remapped order.
sort.Sort(ordered.ByID(nodes))
communities = make([][]graph.Node, len(nodes))
for i := range nodes {
communities[i] = []graph.Node{node(i)}
}
weight := positiveWeightFuncFor(g)
r := ReducedUndirected{
nodes: make([]community, len(nodes)),
undirectedEdges: undirectedEdges{
edges: make([][]int, len(nodes)),
weights: make(map[[2]int]float64),
},
communities: communities,
}
communityOf := make(map[int64]int, len(nodes))
for i, n := range nodes {
r.nodes[i] = community{id: i, nodes: []graph.Node{n}}
communityOf[n.ID()] = i
}
for _, u := range nodes {
uid := u.ID()
ucid := communityOf[uid]
var out []int
to := g.From(uid)
for to.Next() {
vid := to.Node().ID()
vcid := communityOf[vid]
if vcid != ucid {
out = append(out, vcid)
}
if ucid < vcid {
// Only store the weight once.
r.weights[[2]int{ucid, vcid}] = weight(uid, vid)
}
}
r.edges[ucid] = out
}
return &r
}
// Remove zero length communities destructively.
var commNodes int
for i := 0; i < len(communities); {
comm := communities[i]
if len(comm) == 0 {
communities[i] = communities[len(communities)-1]
communities[len(communities)-1] = nil
communities = communities[:len(communities)-1]
} else {
commNodes += len(comm)
i++
}
}
r := ReducedUndirected{
nodes: make([]community, len(communities)),
undirectedEdges: undirectedEdges{
edges: make([][]int, len(communities)),
weights: make(map[[2]int]float64),
},
}
r.communities = make([][]graph.Node, len(communities))
for i := range r.communities {
r.communities[i] = []graph.Node{node(i)}
}
if g, ok := g.(*ReducedUndirected); ok {
// Make sure we retain the truncated
// community structure.
g.communities = communities
r.parent = g
}
weight := positiveWeightFuncFor(g)
communityOf := make(map[int64]int, commNodes)
for i, comm := range communities {
r.nodes[i] = community{id: i, nodes: comm}
for _, n := range comm {
communityOf[n.ID()] = i
}
}
for ucid, comm := range communities {
var out []int
for i, u := range comm {
uid := u.ID()
r.nodes[ucid].weight += weight(uid, uid)
for _, v := range comm[i+1:] {
r.nodes[ucid].weight += 2 * weight(uid, v.ID())
}
to := g.From(uid)
for to.Next() {
vid := to.Node().ID()
vcid := communityOf[vid]
found := false
for _, e := range out {
if e == vcid {
found = true
break
}
}
if !found && vcid != ucid {
out = append(out, vcid)
}
if ucid < vcid {
// Only store the weight once.
r.weights[[2]int{ucid, vcid}] += weight(uid, vid)
}
}
}
r.edges[ucid] = out
}
return &r
}
// Node returns the node with the given ID if it exists in the graph,
// and nil otherwise.
func (g *ReducedUndirected) Node(id int64) graph.Node {
if g.has(id) {
return g.nodes[id]
}
return nil
}
// has returns whether the node exists within the graph.
func (g *ReducedUndirected) has(id int64) bool {
return 0 <= id && id < int64(len(g.nodes))
}
// Nodes returns all the nodes in the graph.
func (g *ReducedUndirected) Nodes() graph.Nodes {
nodes := make([]graph.Node, len(g.nodes))
for i := range g.nodes {
nodes[i] = node(i)
}
return iterator.NewOrderedNodes(nodes)
}
// From returns all nodes in g that can be reached directly from u.
func (g *ReducedUndirected) From(uid int64) graph.Nodes {
out := g.edges[uid]
nodes := make([]graph.Node, len(out))
for i, vid := range out {
nodes[i] = g.nodes[vid]
}
return iterator.NewOrderedNodes(nodes)
}
// HasEdgeBetween returns whether an edge exists between nodes x and y.
func (g *ReducedUndirected) HasEdgeBetween(xid, yid int64) bool {
if xid == yid || !isValidID(xid) || !isValidID(yid) {
return false
}
if xid > yid {
xid, yid = yid, xid
}
_, ok := g.weights[[2]int{int(xid), int(yid)}]
return ok
}
// Edge returns the edge from u to v if such an edge exists and nil otherwise.
// The node v must be directly reachable from u as defined by the From method.
func (g *ReducedUndirected) Edge(uid, vid int64) graph.Edge {
return g.WeightedEdgeBetween(uid, vid)
}
// WeightedEdge returns the weighted edge from u to v if such an edge exists and nil otherwise.
// The node v must be directly reachable from u as defined by the From method.
func (g *ReducedUndirected) WeightedEdge(uid, vid int64) graph.WeightedEdge {
return g.WeightedEdgeBetween(uid, vid)
}
// EdgeBetween returns the edge between nodes x and y.
func (g *ReducedUndirected) EdgeBetween(xid, yid int64) graph.Edge {
return g.WeightedEdgeBetween(xid, yid)
}
// WeightedEdgeBetween returns the weighted edge between nodes x and y.
func (g *ReducedUndirected) WeightedEdgeBetween(xid, yid int64) graph.WeightedEdge {
if xid == yid || !isValidID(xid) || !isValidID(yid) {
return nil
}
if yid < xid {
xid, yid = yid, xid
}
w, ok := g.weights[[2]int{int(xid), int(yid)}]
if !ok {
return nil
}
return edge{from: g.nodes[xid], to: g.nodes[yid], weight: w}
}
// Weight returns the weight for the edge between x and y if Edge(x, y) returns a non-nil Edge.
// If x and y are the same node the internal node weight is returned. If there is no joining
// edge between the two nodes the weight value returned is zero. Weight returns true if an edge
// exists between x and y or if x and y have the same ID, false otherwise.
func (g *ReducedUndirected) Weight(xid, yid int64) (w float64, ok bool) {
if !isValidID(xid) || !isValidID(yid) {
return 0, false
}
if xid == yid {
return g.nodes[xid].weight, true
}
if xid > yid {
xid, yid = yid, xid
}
w, ok = g.weights[[2]int{int(xid), int(yid)}]
return w, ok
}
// undirectedLocalMover is a step in graph modularity optimization.
type undirectedLocalMover struct {
g *ReducedUndirected
// nodes is the set of working nodes.
nodes []graph.Node
// edgeWeightOf is the weighted degree
// of each node indexed by ID.
edgeWeightOf []float64
// m2 is the total sum of
// edge weights in g.
m2 float64
// weight is the weight function
// provided by g or a function
// that returns the Weight value
// of the non-nil edge between x
// and y.
weight func(xid, yid int64) float64
// communities is the current
// division of g.
communities [][]graph.Node
// memberships is a mapping between
// node ID and community membership.
memberships []int
// resolution is the Reichardt and
// Bornholdt γ parameter as defined
// in doi:10.1103/PhysRevE.74.016110.
resolution float64
// moved indicates that a call to
// move has been made since the last
// call to shuffle.
moved bool
// changed indicates that a move
// has been made since the creation
// of the local mover.
changed bool
}
// newUndirectedLocalMover returns a new undirectedLocalMover initialized with
// the graph g, a set of communities and a modularity resolution parameter. The
// node IDs of g must be contiguous in [0,n) where n is the number of nodes.
// If g has a zero edge weight sum, nil is returned.
func newUndirectedLocalMover(g *ReducedUndirected, communities [][]graph.Node, resolution float64) *undirectedLocalMover {
nodes := graph.NodesOf(g.Nodes())
l := undirectedLocalMover{
g: g,
nodes: nodes,
edgeWeightOf: make([]float64, len(nodes)),
communities: communities,
memberships: make([]int, len(nodes)),
resolution: resolution,
weight: positiveWeightFuncFor(g),
}
// Calculate the total edge weight of the graph
// and degree weights for each node.
for _, u := range l.nodes {
uid := u.ID()
w := l.weight(uid, uid)
to := g.From(uid)
for to.Next() {
w += l.weight(uid, to.Node().ID())
}
l.edgeWeightOf[uid] = w
l.m2 += w
}
if l.m2 == 0 {
return nil
}
// Assign membership mappings.
for i, c := range communities {
for _, u := range c {
l.memberships[u.ID()] = i
}
}
return &l
}
// localMovingHeuristic performs the Louvain local moving heuristic until
// no further moves can be made. It returns a boolean indicating that the
// undirectedLocalMover has not made any improvement to the community
// structure and so the Louvain algorithm is done.
func (l *undirectedLocalMover) localMovingHeuristic(rnd func(int) int) (done bool) {
for {
l.shuffle(rnd)
for _, n := range l.nodes {
dQ, dst, src := l.deltaQ(n)
if dQ <= 0 {
continue
}
l.move(dst, src)
}
if !l.moved {
return !l.changed
}
}
}
// shuffle performs a Fisher-Yates shuffle on the nodes held by the
// undirectedLocalMover using the random source rnd which should return
// an integer in the range [0,n).
func (l *undirectedLocalMover) shuffle(rnd func(n int) int) {
l.moved = false
for i := range l.nodes[:len(l.nodes)-1] {
j := i + rnd(len(l.nodes)-i)
l.nodes[i], l.nodes[j] = l.nodes[j], l.nodes[i]
}
}
// move moves the node at src to the community at dst.
func (l *undirectedLocalMover) move(dst int, src commIdx) {
l.moved = true
l.changed = true
srcComm := l.communities[src.community]
n := srcComm[src.node]
l.memberships[n.ID()] = dst
l.communities[dst] = append(l.communities[dst], n)
srcComm[src.node], srcComm[len(srcComm)-1] = srcComm[len(srcComm)-1], nil
l.communities[src.community] = srcComm[:len(srcComm)-1]
}
// deltaQ returns the highest gain in modularity attainable by moving
// n from its current community to another connected community and
// the index of the chosen destination. The index into the
// undirectedLocalMover's communities field is returned in src if n
// is in communities.
func (l *undirectedLocalMover) deltaQ(n graph.Node) (deltaQ float64, dst int, src commIdx) {
id := n.ID()
a_aa := l.weight(id, id)
k_a := l.edgeWeightOf[id]
m2 := l.m2
gamma := l.resolution
// Find communities connected to n.
connected := make(set.Ints)
// The following for loop is equivalent to:
//
// for _, v := range l.g.From(n) {
// connected.Add(l.memberships[v.ID()])
// }
//
// This is done to avoid an allocation.
for _, vid := range l.g.edges[id] {
connected.Add(l.memberships[vid])
}
// Insert the node's own community.
connected.Add(l.memberships[id])
candidates := make([]int, 0, len(connected))
for i := range connected {
candidates = append(candidates, i)
}
sort.Ints(candidates)
// Calculate the highest modularity gain
// from moving into another community and
// keep the index of that community.
var dQremove float64
dQadd, dst, src := math.Inf(-1), -1, commIdx{-1, -1}
for _, i := range candidates {
c := l.communities[i]
var k_aC, sigma_totC float64 // C is a substitution for ^𝛼 or ^𝛽.
var removal bool
for j, u := range c {
uid := u.ID()
if uid == id {
if src.community != -1 {
panic("community: multiple sources")
}
src = commIdx{i, j}
removal = true
}
k_aC += l.weight(id, uid)
// sigma_totC could be kept for each community
// and updated for moves, changing the calculation
// of sigma_totC here from O(n_c) to O(1), but
// in practice the time savings do not appear
// to be compelling and do not make up for the
// increase in code complexity and space required.
sigma_totC += l.edgeWeightOf[uid]
}
// See louvain.tex for a derivation of these equations.
switch {
case removal:
// The community c was the current community,
// so calculate the change due to removal.
dQremove = k_aC /*^𝛼*/ - a_aa - gamma*k_a*(sigma_totC /*^𝛼*/ -k_a)/m2
default:
// Otherwise calculate the change due to an addition
// to c and retain if it is the current best.
dQ := k_aC /*^𝛽*/ - gamma*k_a*sigma_totC /*^𝛽*/ /m2
if dQ > dQadd {
dQadd = dQ
dst = i
}
}
}
return 2 * (dQadd - dQremove) / m2, dst, src
}