graph/community/louvain_directed_multiplex_test.go - third_party/github.com/gonum/gonum - Git at Google

 // 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"
 	"reflect"
 	"sort"
 	"testing"

 	"golang.org/x/exp/rand"

 	"gonum.org/v1/gonum/floats"
 	"gonum.org/v1/gonum/floats/scalar"
 	"gonum.org/v1/gonum/graph"
 	"gonum.org/v1/gonum/graph/internal/ordered"
 	"gonum.org/v1/gonum/graph/simple"
 )

 var communityDirectedMultiplexQTests = []struct {
 	name       string
 	layers     []layer
 	structures []structure

 	wantLevels []level
 }{
 	{
 		name:   "unconnected",
 		layers: []layer{{g: unconnected, weight: 1}},
 		structures: []structure{
 			{
 				resolution: 1,
 				memberships: []intset{
 					0: linksTo(0),
 					1: linksTo(1),
 					2: linksTo(2),
 					3: linksTo(3),
 					4: linksTo(4),
 					5: linksTo(5),
 				},
 				want: math.NaN(),
 			},
 		},
 		wantLevels: []level{
 			{
 				q: math.Inf(-1), // Here math.Inf(-1) is used as a place holder for NaN to allow use of reflect.DeepEqual.
 				communities: [][]graph.Node{
 					{simple.Node(0)},
 					{simple.Node(1)},
 					{simple.Node(2)},
 					{simple.Node(3)},
 					{simple.Node(4)},
 					{simple.Node(5)},
 				},
 			},
 		},
 	},
 	{
 		name:   "simple_directed",
 		layers: []layer{{g: simpleDirected, weight: 1}},
 		// community structure and modularity calculated by C++ implementation: louvain igraph.
 		// Note that louvain igraph returns Q as an unscaled value.
 		structures: []structure{
 			{
 				resolution: 1,
 				memberships: []intset{
 					0: linksTo(0, 1),
 					1: linksTo(2, 3, 4),
 				},
 				want: 0.5714285714285716,
 				tol:  1e-10,
 			},
 		},
 		wantLevels: []level{
 			{
 				communities: [][]graph.Node{
 					{simple.Node(0), simple.Node(1)},
 					{simple.Node(2), simple.Node(3), simple.Node(4)},
 				},
 				q: 0.5714285714285716,
 			},
 			{
 				communities: [][]graph.Node{
 					{simple.Node(0)},
 					{simple.Node(1)},
 					{simple.Node(2)},
 					{simple.Node(3)},
 					{simple.Node(4)},
 				},
 				q: -1.2857142857142856,
 			},
 		},
 	},
 	{
 		name: "simple_directed_twice",
 		layers: []layer{
 			{g: simpleDirected, weight: 0.5},
 			{g: simpleDirected, weight: 0.5},
 		},
 		// community structure and modularity calculated by C++ implementation: louvain igraph.
 		// Note that louvain igraph returns Q as an unscaled value.
 		structures: []structure{
 			{
 				resolution: 1,
 				memberships: []intset{
 					0: linksTo(0, 1),
 					1: linksTo(2, 3, 4),
 				},
 				want: 0.5714285714285716,
 				tol:  1e-10,
 			},
 		},
 		wantLevels: []level{
 			{
 				q: 0.5714285714285716,
 				communities: [][]graph.Node{
 					{simple.Node(0), simple.Node(1)},
 					{simple.Node(2), simple.Node(3), simple.Node(4)},
 				},
 			},
 			{
 				q: -1.2857142857142856,
 				communities: [][]graph.Node{
 					{simple.Node(0)},
 					{simple.Node(1)},
 					{simple.Node(2)},
 					{simple.Node(3)},
 					{simple.Node(4)},
 				},
 			},
 		},
 	},
 	{
 		name: "small_dumbell",
 		layers: []layer{
 			{g: smallDumbell, edgeWeight: 1, weight: 1},
 			{g: dumbellRepulsion, edgeWeight: -1, weight: -1},
 		},
 		structures: []structure{
 			{
 				resolution: 1,
 				memberships: []intset{
 					0: linksTo(0, 1, 2),
 					1: linksTo(3, 4, 5),
 				},
 				want: 2.5714285714285716, tol: 1e-10,
 			},
 			{
 				resolution: 1,
 				memberships: []intset{
 					0: linksTo(0, 1, 2, 3, 4, 5),
 				},
 				want: 0, tol: 1e-14,
 			},
 		},
 		wantLevels: []level{
 			{
 				q: 2.5714285714285716,
 				communities: [][]graph.Node{
 					{simple.Node(0), simple.Node(1), simple.Node(2)},
 					{simple.Node(3), simple.Node(4), simple.Node(5)},
 				},
 			},
 			{
 				q: -0.857142857142857,
 				communities: [][]graph.Node{
 					{simple.Node(0)},
 					{simple.Node(1)},
 					{simple.Node(2)},
 					{simple.Node(3)},
 					{simple.Node(4)},
 					{simple.Node(5)},
 				},
 			},
 		},
 	},
 	{
 		name:   "repulsion",
 		layers: []layer{{g: repulsion, edgeWeight: -1, weight: -1}},
 		structures: []structure{
 			{
 				resolution: 1,
 				memberships: []intset{
 					0: linksTo(0, 1, 2),
 					1: linksTo(3, 4, 5),
 				},
 				want: 9.0, tol: 1e-10,
 			},
 			{
 				resolution: 1,
 				memberships: []intset{
 					0: linksTo(0),
 					1: linksTo(1),
 					2: linksTo(2),
 					3: linksTo(3),
 					4: linksTo(4),
 					5: linksTo(5),
 				},
 				want: 3, tol: 1e-14,
 			},
 		},
 		wantLevels: []level{
 			{
 				q: 9.0,
 				communities: [][]graph.Node{
 					{simple.Node(0), simple.Node(1), simple.Node(2)},
 					{simple.Node(3), simple.Node(4), simple.Node(5)},
 				},
 			},
 			{
 				q: 3.0,
 				communities: [][]graph.Node{
 					{simple.Node(0)},
 					{simple.Node(1)},
 					{simple.Node(2)},
 					{simple.Node(3)},
 					{simple.Node(4)},
 					{simple.Node(5)},
 				},
 			},
 		},
 	},
 	{
 		name: "middle_east",
 		layers: []layer{
 			{g: middleEast.friends, edgeWeight: 1, weight: 1},
 			{g: middleEast.enemies, edgeWeight: -1, weight: -1},
 		},
 		structures: []structure{
 			{
 				resolution: 1,
 				memberships: []intset{
 					0: linksTo(0, 6),
 					1: linksTo(1, 7, 9, 12),
 					2: linksTo(2, 8, 11),
 					3: linksTo(3, 4, 5, 10),
 				},
 				want: 33.818057455540355, tol: 1e-9,
 			},
 			{
 				resolution: 1,
 				memberships: []intset{
 					0: linksTo(0, 2, 3, 4, 5, 10),
 					1: linksTo(1, 7, 9, 12),
 					2: linksTo(6),
 					3: linksTo(8, 11),
 				},
 				want: 30.92749658, tol: 1e-7,
 			},
 			{
 				resolution: 1,
 				memberships: []intset{
 					0: linksTo(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12),
 				},
 				want: 0, tol: 1e-14,
 			},
 		},
 		wantLevels: []level{
 			{
 				q: 33.818057455540355,
 				communities: [][]graph.Node{
 					{simple.Node(0), simple.Node(6)},
 					{simple.Node(1), simple.Node(7), simple.Node(9), simple.Node(12)},
 					{simple.Node(2), simple.Node(8), simple.Node(11)},
 					{simple.Node(3), simple.Node(4), simple.Node(5), simple.Node(10)},
 				},
 			},
 			{
 				q: 3.8071135430916545,
 				communities: [][]graph.Node{
 					{simple.Node(0)},
 					{simple.Node(1)},
 					{simple.Node(2)},
 					{simple.Node(3)},
 					{simple.Node(4)},
 					{simple.Node(5)},
 					{simple.Node(6)},
 					{simple.Node(7)},
 					{simple.Node(8)},
 					{simple.Node(9)},
 					{simple.Node(10)},
 					{simple.Node(11)},
 					{simple.Node(12)},
 				},
 			},
 		},
 	},
 }

 func TestCommunityQDirectedMultiplex(t *testing.T) {
 	for _, test := range communityDirectedMultiplexQTests {
 		g, weights, err := directedMultiplexFrom(test.layers)
 		if err != nil {
 			t.Errorf("unexpected error creating multiplex: %v", err)
 			continue
 		}

 		for _, structure := range test.structures {
 			communities := make([][]graph.Node, len(structure.memberships))
 			for i, c := range structure.memberships {
 				for n := range c {
 					communities[i] = append(communities[i], simple.Node(n))
 				}
 			}
 			q := QMultiplex(g, communities, weights, []float64{structure.resolution})
 			got := floats.Sum(q)
 			if !scalar.EqualWithinAbsOrRel(got, structure.want, structure.tol, structure.tol) && !math.IsNaN(structure.want) {
 				for _, c := range communities {
 					sort.Sort(ordered.ByID(c))
 				}
 				t.Errorf("unexpected Q value for %q %v: got: %v %.3v want: %v",
 					test.name, communities, got, q, structure.want)
 			}
 		}
 	}
 }

 func TestCommunityDeltaQDirectedMultiplex(t *testing.T) {
 tests:
 	for _, test := range communityDirectedMultiplexQTests {
 		g, weights, err := directedMultiplexFrom(test.layers)
 		if err != nil {
 			t.Errorf("unexpected error creating multiplex: %v", err)
 			continue
 		}

 		rnd := rand.New(rand.NewSource(1)).Intn
 		for _, structure := range test.structures {
 			communityOf := make(map[int64]int)
 			communities := make([][]graph.Node, len(structure.memberships))
 			for i, c := range structure.memberships {
 				for n := range c {
 					n := int64(n)
 					communityOf[n] = i
 					communities[i] = append(communities[i], simple.Node(n))
 				}
 				sort.Sort(ordered.ByID(communities[i]))
 			}
 			resolution := []float64{structure.resolution}

 			before := QMultiplex(g, communities, weights, resolution)

 			// We test exhaustively.
 			const all = true

 			l := newDirectedMultiplexLocalMover(
 				reduceDirectedMultiplex(g, nil, weights),
 				communities, weights, resolution, all)
 			if l == nil {
 				if !math.IsNaN(floats.Sum(before)) {
 					t.Errorf("unexpected nil localMover with non-NaN Q graph: Q=%.4v", before)
 				}
 				continue tests
 			}

 			// This is done to avoid run-to-run
 			// variation due to map iteration order.
 			sort.Sort(ordered.ByID(l.nodes))

 			l.shuffle(rnd)

 			for _, target := range l.nodes {
 				got, gotDst, gotSrc := l.deltaQ(target)

 				want, wantDst := math.Inf(-1), -1
 				migrated := make([][]graph.Node, len(structure.memberships))
 				for i, c := range structure.memberships {
 					for n := range c {
 						n := int64(n)
 						if n == target.ID() {
 							continue
 						}
 						migrated[i] = append(migrated[i], simple.Node(n))
 					}
 					sort.Sort(ordered.ByID(migrated[i]))
 				}

 				for i, c := range structure.memberships {
 					if i == communityOf[target.ID()] {
 						continue
 					}
 					if !(all && hasNegative(weights)) {
 						connected := false
 					search:
 						for l := 0; l < g.Depth(); l++ {
 							if weights[l] < 0 {
 								connected = true
 								break search
 							}
 							layer := g.Layer(l)
 							for n := range c {
 								if layer.HasEdgeBetween(int64(n), target.ID()) {
 									connected = true
 									break search
 								}
 							}
 						}
 						if !connected {
 							continue
 						}
 					}
 					migrated[i] = append(migrated[i], target)
 					after := QMultiplex(g, migrated, weights, resolution)
 					migrated[i] = migrated[i][:len(migrated[i])-1]
 					if delta := floats.Sum(after) - floats.Sum(before); delta > want {
 						want = delta
 						wantDst = i
 					}
 				}

 				if !scalar.EqualWithinAbsOrRel(got, want, structure.tol, structure.tol) || gotDst != wantDst {
 					t.Errorf("unexpected result moving n=%d in c=%d of %s/%.4v: got: %.4v,%d want: %.4v,%d"+
 						"\n\t%v\n\t%v",
 						target.ID(), communityOf[target.ID()], test.name, structure.resolution, got, gotDst, want, wantDst,
 						communities, migrated)
 				}
 				if gotSrc.community != communityOf[target.ID()] {
 					t.Errorf("unexpected source community index: got: %d want: %d", gotSrc, communityOf[target.ID()])
 				} else if communities[gotSrc.community][gotSrc.node].ID() != target.ID() {
 					wantNodeIdx := -1
 					for i, n := range communities[gotSrc.community] {
 						if n.ID() == target.ID() {
 							wantNodeIdx = i
 							break
 						}
 					}
 					t.Errorf("unexpected source node index: got: %d want: %d", gotSrc.node, wantNodeIdx)
 				}
 			}
 		}
 	}
 }

 func TestReduceQConsistencyDirectedMultiplex(t *testing.T) {
 tests:
 	for _, test := range communityDirectedMultiplexQTests {
 		g, weights, err := directedMultiplexFrom(test.layers)
 		if err != nil {
 			t.Errorf("unexpected error creating multiplex: %v", err)
 			continue
 		}

 		for _, structure := range test.structures {
 			if math.IsNaN(structure.want) {
 				continue tests
 			}

 			communities := make([][]graph.Node, len(structure.memberships))
 			for i, c := range structure.memberships {
 				for n := range c {
 					communities[i] = append(communities[i], simple.Node(n))
 				}
 				sort.Sort(ordered.ByID(communities[i]))
 			}

 			gQ := QMultiplex(g, communities, weights, []float64{structure.resolution})
 			gQnull := QMultiplex(g, nil, weights, nil)

 			cg0 := reduceDirectedMultiplex(g, nil, weights)
 			cg0Qnull := QMultiplex(cg0, cg0.Structure(), weights, nil)
 			if !scalar.EqualWithinAbsOrRel(floats.Sum(gQnull), floats.Sum(cg0Qnull), structure.tol, structure.tol) {
 				t.Errorf("disagreement between null Q from method: %v and function: %v", cg0Qnull, gQnull)
 			}
 			cg0Q := QMultiplex(cg0, communities, weights, []float64{structure.resolution})
 			if !scalar.EqualWithinAbsOrRel(floats.Sum(gQ), floats.Sum(cg0Q), structure.tol, structure.tol) {
 				t.Errorf("unexpected Q result after initial reduction: got: %v want :%v", cg0Q, gQ)
 			}

 			cg1 := reduceDirectedMultiplex(cg0, communities, weights)
 			cg1Q := QMultiplex(cg1, cg1.Structure(), weights, []float64{structure.resolution})
 			if !scalar.EqualWithinAbsOrRel(floats.Sum(gQ), floats.Sum(cg1Q), structure.tol, structure.tol) {
 				t.Errorf("unexpected Q result after second reduction: got: %v want :%v", cg1Q, gQ)
 			}
 		}
 	}
 }

 var localDirectedMultiplexMoveTests = []struct {
 	name       string
 	layers     []layer
 	structures []moveStructures
 }{
 	{
 		name:   "blondel",
 		layers: []layer{{g: blondel, weight: 1}, {g: blondel, weight: 0.5}},
 		structures: []moveStructures{
 			{
 				memberships: []intset{
 					0: linksTo(0, 1, 2, 4, 5),
 					1: linksTo(3, 6, 7),
 					2: linksTo(8, 9, 10, 12, 14, 15),
 					3: linksTo(11, 13),
 				},
 				targetNodes: []graph.Node{simple.Node(0)},
 				resolution:  1,
 				tol:         1e-14,
 			},
 			{
 				memberships: []intset{
 					0: linksTo(0, 1, 2, 4, 5),
 					1: linksTo(3, 6, 7),
 					2: linksTo(8, 9, 10, 12, 14, 15),
 					3: linksTo(11, 13),
 				},
 				targetNodes: []graph.Node{simple.Node(3)},
 				resolution:  1,
 				tol:         1e-14,
 			},
 			{
 				memberships: []intset{
 					0: linksTo(0, 1, 2, 4, 5),
 					1: linksTo(3, 6, 7),
 					2: linksTo(8, 9, 10, 12, 14, 15),
 					3: linksTo(11, 13),
 				},
 				// Case to demonstrate when A_aa != k_a^𝛼.
 				targetNodes: []graph.Node{simple.Node(3), simple.Node(2)},
 				resolution:  1,
 				tol:         1e-14,
 			},
 		},
 	},
 }

 func TestMoveLocalDirectedMultiplex(t *testing.T) {
 	for _, test := range localDirectedMultiplexMoveTests {
 		g, weights, err := directedMultiplexFrom(test.layers)
 		if err != nil {
 			t.Errorf("unexpected error creating multiplex: %v", err)
 			continue
 		}

 		for _, structure := range test.structures {
 			communities := make([][]graph.Node, len(structure.memberships))
 			for i, c := range structure.memberships {
 				for n := range c {
 					communities[i] = append(communities[i], simple.Node(n))
 				}
 				sort.Sort(ordered.ByID(communities[i]))
 			}

 			r := reduceDirectedMultiplex(reduceDirectedMultiplex(g, nil, weights), communities, weights)

 			l := newDirectedMultiplexLocalMover(r, r.communities, weights, []float64{structure.resolution}, true)
 			for _, n := range structure.targetNodes {
 				dQ, dst, src := l.deltaQ(n)
 				if dQ > 0 {
 					before := floats.Sum(QMultiplex(r, l.communities, weights, []float64{structure.resolution}))
 					l.move(dst, src)
 					after := floats.Sum(QMultiplex(r, l.communities, weights, []float64{structure.resolution}))
 					want := after - before
 					if !scalar.EqualWithinAbsOrRel(dQ, want, structure.tol, structure.tol) {
 						t.Errorf("unexpected deltaQ: got: %v want: %v", dQ, want)
 					}
 				}
 			}
 		}
 	}
 }

 func TestLouvainDirectedMultiplex(t *testing.T) {
 	const louvainIterations = 20

 	for _, test := range communityDirectedMultiplexQTests {
 		g, weights, err := directedMultiplexFrom(test.layers)
 		if err != nil {
 			t.Errorf("unexpected error creating multiplex: %v", err)
 			continue
 		}

 		if test.structures[0].resolution != 1 {
 			panic("bad test: expect resolution=1")
 		}
 		want := make([][]graph.Node, len(test.structures[0].memberships))
 		for i, c := range test.structures[0].memberships {
 			for n := range c {
 				want[i] = append(want[i], simple.Node(n))
 			}
 			sort.Sort(ordered.ByID(want[i]))
 		}
 		sort.Sort(ordered.BySliceIDs(want))

 		var (
 			got   *ReducedDirectedMultiplex
 			bestQ = math.Inf(-1)
 		)
 		// Modularize is randomised so we do this to
 		// ensure the level tests are consistent.
 		src := rand.New(rand.NewSource(1))
 		for i := 0; i < louvainIterations; i++ {
 			r := ModularizeMultiplex(g, weights, nil, true, src).(*ReducedDirectedMultiplex)
 			if q := floats.Sum(QMultiplex(r, nil, weights, nil)); q > bestQ || math.IsNaN(q) {
 				bestQ = q
 				got = r

 				if math.IsNaN(q) {
 					// Don't try again for non-connected case.
 					break
 				}
 			}

 			var qs []float64
 			for p := r; p != nil; p = p.Expanded().(*ReducedDirectedMultiplex) {
 				qs = append(qs, floats.Sum(QMultiplex(p, nil, weights, nil)))
 			}

 			// Recovery of Q values is reversed.
 			if reverse(qs); !sort.Float64sAreSorted(qs) {
 				t.Errorf("Q values not monotonically increasing: %.5v", qs)
 			}
 		}

 		gotCommunities := got.Communities()
 		for _, c := range gotCommunities {
 			sort.Sort(ordered.ByID(c))
 		}
 		sort.Sort(ordered.BySliceIDs(gotCommunities))
 		if !reflect.DeepEqual(gotCommunities, want) {
 			t.Errorf("unexpected community membership for %s Q=%.4v:\n\tgot: %v\n\twant:%v",
 				test.name, bestQ, gotCommunities, want)
 			continue
 		}

 		var levels []level
 		for p := got; p != nil; p = p.Expanded().(*ReducedDirectedMultiplex) {
 			var communities [][]graph.Node
 			if p.parent != nil {
 				communities = p.parent.Communities()
 				for _, c := range communities {
 					sort.Sort(ordered.ByID(c))
 				}
 				sort.Sort(ordered.BySliceIDs(communities))
 			} else {
 				communities = reduceDirectedMultiplex(g, nil, weights).Communities()
 			}
 			q := floats.Sum(QMultiplex(p, nil, weights, nil))
 			if math.IsNaN(q) {
 				// Use an equalable flag value in place of NaN.
 				q = math.Inf(-1)
 			}
 			levels = append(levels, level{q: q, communities: communities})
 		}
 		if !reflect.DeepEqual(levels, test.wantLevels) {
 			t.Errorf("unexpected level structure:\n\tgot: %v\n\twant:%v", levels, test.wantLevels)
 		}
 	}
 }

 func TestNonContiguousDirectedMultiplex(t *testing.T) {
 	g := simple.NewDirectedGraph()
 	for _, e := range []simple.Edge{
 		{F: simple.Node(0), T: simple.Node(1)},
 		{F: simple.Node(4), T: simple.Node(5)},
 	} {
 		g.SetEdge(e)
 	}

 	func() {
 		defer func() {
 			r := recover()
 			if r != nil {
 				t.Error("unexpected panic with non-contiguous ID range")
 			}
 		}()
 		ModularizeMultiplex(DirectedLayers{g}, nil, nil, true, nil)
 	}()
 }

 func TestNonContiguousWeightedDirectedMultiplex(t *testing.T) {
 	g := simple.NewWeightedDirectedGraph(0, 0)
 	for _, e := range []simple.WeightedEdge{
 		{F: simple.Node(0), T: simple.Node(1), W: 1},
 		{F: simple.Node(4), T: simple.Node(5), W: 1},
 	} {
 		g.SetWeightedEdge(e)
 	}

 	func() {
 		defer func() {
 			r := recover()
 			if r != nil {
 				t.Error("unexpected panic with non-contiguous ID range")
 			}
 		}()
 		ModularizeMultiplex(DirectedLayers{g}, nil, nil, true, nil)
 	}()
 }

 func BenchmarkLouvainDirectedMultiplex(b *testing.B) {
 	src := rand.New(rand.NewSource(1))
 	for i := 0; i < b.N; i++ {
 		ModularizeMultiplex(DirectedLayers{dupGraphDirected}, nil, nil, true, src)
 	}
 }

 func directedMultiplexFrom(raw []layer) (DirectedLayers, []float64, error) {
 	var layers []graph.Directed
 	var weights []float64
 	for _, l := range raw {
 		g := simple.NewWeightedDirectedGraph(0, 0)
 		for u, e := range l.g {
 			// Add nodes that are not defined by an edge.
 			if g.Node(int64(u)) == nil {
 				g.AddNode(simple.Node(u))
 			}
 			for v := range e {
 				w := 1.0
 				if l.edgeWeight != 0 {
 					w = l.edgeWeight
 				}
 				g.SetWeightedEdge(simple.WeightedEdge{F: simple.Node(u), T: simple.Node(v), W: w})
 			}
 		}
 		layers = append(layers, g)
 		weights = append(weights, l.weight)
 	}
 	g, err := NewDirectedLayers(layers...)
 	if err != nil {
 		return nil, nil, err
 	}
 	return g, weights, nil
 }
	// 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"
	"reflect"
	"sort"
	"testing"

	"golang.org/x/exp/rand"

	"gonum.org/v1/gonum/floats"
	"gonum.org/v1/gonum/floats/scalar"
	"gonum.org/v1/gonum/graph"
	"gonum.org/v1/gonum/graph/internal/ordered"
	"gonum.org/v1/gonum/graph/simple"
	)

	var communityDirectedMultiplexQTests = []struct {
	name string
	layers []layer
	structures []structure

	wantLevels []level
	}{
	{
	name: "unconnected",
	layers: []layer{{g: unconnected, weight: 1}},
	structures: []structure{
	{
	resolution: 1,
	memberships: []intset{
	0: linksTo(0),
	1: linksTo(1),
	2: linksTo(2),
	3: linksTo(3),
	4: linksTo(4),
	5: linksTo(5),
	},
	want: math.NaN(),
	},
	},
	wantLevels: []level{
	{
	q: math.Inf(-1), // Here math.Inf(-1) is used as a place holder for NaN to allow use of reflect.DeepEqual.
	communities: [][]graph.Node{
	{simple.Node(0)},
	{simple.Node(1)},
	{simple.Node(2)},
	{simple.Node(3)},
	{simple.Node(4)},
	{simple.Node(5)},
	},
	},
	},
	},
	{
	name: "simple_directed",
	layers: []layer{{g: simpleDirected, weight: 1}},
	// community structure and modularity calculated by C++ implementation: louvain igraph.
	// Note that louvain igraph returns Q as an unscaled value.
	structures: []structure{
	{
	resolution: 1,
	memberships: []intset{
	0: linksTo(0, 1),
	1: linksTo(2, 3, 4),
	},
	want: 0.5714285714285716,
	tol: 1e-10,
	},
	},
	wantLevels: []level{
	{
	communities: [][]graph.Node{
	{simple.Node(0), simple.Node(1)},
	{simple.Node(2), simple.Node(3), simple.Node(4)},
	},
	q: 0.5714285714285716,
	},
	{
	communities: [][]graph.Node{
	{simple.Node(0)},
	{simple.Node(1)},
	{simple.Node(2)},
	{simple.Node(3)},
	{simple.Node(4)},
	},
	q: -1.2857142857142856,
	},
	},
	},
	{
	name: "simple_directed_twice",
	layers: []layer{
	{g: simpleDirected, weight: 0.5},
	{g: simpleDirected, weight: 0.5},
	},
	// community structure and modularity calculated by C++ implementation: louvain igraph.
	// Note that louvain igraph returns Q as an unscaled value.
	structures: []structure{
	{
	resolution: 1,
	memberships: []intset{
	0: linksTo(0, 1),
	1: linksTo(2, 3, 4),
	},
	want: 0.5714285714285716,
	tol: 1e-10,
	},
	},
	wantLevels: []level{
	{
	q: 0.5714285714285716,
	communities: [][]graph.Node{
	{simple.Node(0), simple.Node(1)},
	{simple.Node(2), simple.Node(3), simple.Node(4)},
	},
	},
	{
	q: -1.2857142857142856,
	communities: [][]graph.Node{
	{simple.Node(0)},
	{simple.Node(1)},
	{simple.Node(2)},
	{simple.Node(3)},
	{simple.Node(4)},
	},
	},
	},
	},
	{
	name: "small_dumbell",
	layers: []layer{
	{g: smallDumbell, edgeWeight: 1, weight: 1},
	{g: dumbellRepulsion, edgeWeight: -1, weight: -1},
	},
	structures: []structure{
	{
	resolution: 1,
	memberships: []intset{
	0: linksTo(0, 1, 2),
	1: linksTo(3, 4, 5),
	},
	want: 2.5714285714285716, tol: 1e-10,
	},
	{
	resolution: 1,
	memberships: []intset{
	0: linksTo(0, 1, 2, 3, 4, 5),
	},
	want: 0, tol: 1e-14,
	},
	},
	wantLevels: []level{
	{
	q: 2.5714285714285716,
	communities: [][]graph.Node{
	{simple.Node(0), simple.Node(1), simple.Node(2)},
	{simple.Node(3), simple.Node(4), simple.Node(5)},
	},
	},
	{
	q: -0.857142857142857,
	communities: [][]graph.Node{
	{simple.Node(0)},
	{simple.Node(1)},
	{simple.Node(2)},
	{simple.Node(3)},
	{simple.Node(4)},
	{simple.Node(5)},
	},
	},
	},
	},
	{
	name: "repulsion",
	layers: []layer{{g: repulsion, edgeWeight: -1, weight: -1}},
	structures: []structure{
	{
	resolution: 1,
	memberships: []intset{
	0: linksTo(0, 1, 2),
	1: linksTo(3, 4, 5),
	},
	want: 9.0, tol: 1e-10,
	},
	{
	resolution: 1,
	memberships: []intset{
	0: linksTo(0),
	1: linksTo(1),
	2: linksTo(2),
	3: linksTo(3),
	4: linksTo(4),
	5: linksTo(5),
	},
	want: 3, tol: 1e-14,
	},
	},
	wantLevels: []level{
	{
	q: 9.0,
	communities: [][]graph.Node{
	{simple.Node(0), simple.Node(1), simple.Node(2)},
	{simple.Node(3), simple.Node(4), simple.Node(5)},
	},
	},
	{
	q: 3.0,
	communities: [][]graph.Node{
	{simple.Node(0)},
	{simple.Node(1)},
	{simple.Node(2)},
	{simple.Node(3)},
	{simple.Node(4)},
	{simple.Node(5)},
	},
	},
	},
	},
	{
	name: "middle_east",
	layers: []layer{
	{g: middleEast.friends, edgeWeight: 1, weight: 1},
	{g: middleEast.enemies, edgeWeight: -1, weight: -1},
	},
	structures: []structure{
	{
	resolution: 1,
	memberships: []intset{
	0: linksTo(0, 6),
	1: linksTo(1, 7, 9, 12),
	2: linksTo(2, 8, 11),
	3: linksTo(3, 4, 5, 10),
	},
	want: 33.818057455540355, tol: 1e-9,
	},
	{
	resolution: 1,
	memberships: []intset{
	0: linksTo(0, 2, 3, 4, 5, 10),
	1: linksTo(1, 7, 9, 12),
	2: linksTo(6),
	3: linksTo(8, 11),
	},
	want: 30.92749658, tol: 1e-7,
	},
	{
	resolution: 1,
	memberships: []intset{
	0: linksTo(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12),
	},
	want: 0, tol: 1e-14,
	},
	},
	wantLevels: []level{
	{
	q: 33.818057455540355,
	communities: [][]graph.Node{
	{simple.Node(0), simple.Node(6)},
	{simple.Node(1), simple.Node(7), simple.Node(9), simple.Node(12)},
	{simple.Node(2), simple.Node(8), simple.Node(11)},
	{simple.Node(3), simple.Node(4), simple.Node(5), simple.Node(10)},
	},
	},
	{
	q: 3.8071135430916545,
	communities: [][]graph.Node{
	{simple.Node(0)},
	{simple.Node(1)},
	{simple.Node(2)},
	{simple.Node(3)},
	{simple.Node(4)},
	{simple.Node(5)},
	{simple.Node(6)},
	{simple.Node(7)},
	{simple.Node(8)},
	{simple.Node(9)},
	{simple.Node(10)},
	{simple.Node(11)},
	{simple.Node(12)},
	},
	},
	},
	},
	}

	func TestCommunityQDirectedMultiplex(t *testing.T) {
	for _, test := range communityDirectedMultiplexQTests {
	g, weights, err := directedMultiplexFrom(test.layers)
	if err != nil {
	t.Errorf("unexpected error creating multiplex: %v", err)
	continue
	}

	for _, structure := range test.structures {
	communities := make([][]graph.Node, len(structure.memberships))
	for i, c := range structure.memberships {
	for n := range c {
	communities[i] = append(communities[i], simple.Node(n))
	}
	}
	q := QMultiplex(g, communities, weights, []float64{structure.resolution})
	got := floats.Sum(q)
	if !scalar.EqualWithinAbsOrRel(got, structure.want, structure.tol, structure.tol) && !math.IsNaN(structure.want) {
	for _, c := range communities {
	sort.Sort(ordered.ByID(c))
	}
	t.Errorf("unexpected Q value for %q %v: got: %v %.3v want: %v",
	test.name, communities, got, q, structure.want)
	}
	}
	}
	}

	func TestCommunityDeltaQDirectedMultiplex(t *testing.T) {
	tests:
	for _, test := range communityDirectedMultiplexQTests {
	g, weights, err := directedMultiplexFrom(test.layers)
	if err != nil {
	t.Errorf("unexpected error creating multiplex: %v", err)
	continue
	}

	rnd := rand.New(rand.NewSource(1)).Intn
	for _, structure := range test.structures {
	communityOf := make(map[int64]int)
	communities := make([][]graph.Node, len(structure.memberships))
	for i, c := range structure.memberships {
	for n := range c {
	n := int64(n)
	communityOf[n] = i
	communities[i] = append(communities[i], simple.Node(n))
	}
	sort.Sort(ordered.ByID(communities[i]))
	}
	resolution := []float64{structure.resolution}

	before := QMultiplex(g, communities, weights, resolution)

	// We test exhaustively.
	const all = true

	l := newDirectedMultiplexLocalMover(
	reduceDirectedMultiplex(g, nil, weights),
	communities, weights, resolution, all)
	if l == nil {
	if !math.IsNaN(floats.Sum(before)) {
	t.Errorf("unexpected nil localMover with non-NaN Q graph: Q=%.4v", before)
	}
	continue tests
	}

	// This is done to avoid run-to-run
	// variation due to map iteration order.
	sort.Sort(ordered.ByID(l.nodes))

	l.shuffle(rnd)

	for _, target := range l.nodes {
	got, gotDst, gotSrc := l.deltaQ(target)

	want, wantDst := math.Inf(-1), -1
	migrated := make([][]graph.Node, len(structure.memberships))
	for i, c := range structure.memberships {
	for n := range c {
	n := int64(n)
	if n == target.ID() {
	continue
	}
	migrated[i] = append(migrated[i], simple.Node(n))
	}
	sort.Sort(ordered.ByID(migrated[i]))
	}

	for i, c := range structure.memberships {
	if i == communityOf[target.ID()] {
	continue
	}
	if !(all && hasNegative(weights)) {
	connected := false
	search:
	for l := 0; l < g.Depth(); l++ {
	if weights[l] < 0 {
	connected = true
	break search
	}
	layer := g.Layer(l)
	for n := range c {
	if layer.HasEdgeBetween(int64(n), target.ID()) {
	connected = true
	break search
	}
	}
	}
	if !connected {
	continue
	}
	}
	migrated[i] = append(migrated[i], target)
	after := QMultiplex(g, migrated, weights, resolution)
	migrated[i] = migrated[i][:len(migrated[i])-1]
	if delta := floats.Sum(after) - floats.Sum(before); delta > want {
	want = delta
	wantDst = i
	}
	}

	if !scalar.EqualWithinAbsOrRel(got, want, structure.tol, structure.tol) \|\| gotDst != wantDst {
	t.Errorf("unexpected result moving n=%d in c=%d of %s/%.4v: got: %.4v,%d want: %.4v,%d"+
	"\n\t%v\n\t%v",
	target.ID(), communityOf[target.ID()], test.name, structure.resolution, got, gotDst, want, wantDst,
	communities, migrated)
	}
	if gotSrc.community != communityOf[target.ID()] {
	t.Errorf("unexpected source community index: got: %d want: %d", gotSrc, communityOf[target.ID()])
	} else if communities[gotSrc.community][gotSrc.node].ID() != target.ID() {
	wantNodeIdx := -1
	for i, n := range communities[gotSrc.community] {
	if n.ID() == target.ID() {
	wantNodeIdx = i
	break
	}
	}
	t.Errorf("unexpected source node index: got: %d want: %d", gotSrc.node, wantNodeIdx)
	}
	}
	}
	}
	}

	func TestReduceQConsistencyDirectedMultiplex(t *testing.T) {
	tests:
	for _, test := range communityDirectedMultiplexQTests {
	g, weights, err := directedMultiplexFrom(test.layers)
	if err != nil {
	t.Errorf("unexpected error creating multiplex: %v", err)
	continue
	}

	for _, structure := range test.structures {
	if math.IsNaN(structure.want) {
	continue tests
	}

	communities := make([][]graph.Node, len(structure.memberships))
	for i, c := range structure.memberships {
	for n := range c {
	communities[i] = append(communities[i], simple.Node(n))
	}
	sort.Sort(ordered.ByID(communities[i]))
	}

	gQ := QMultiplex(g, communities, weights, []float64{structure.resolution})
	gQnull := QMultiplex(g, nil, weights, nil)

	cg0 := reduceDirectedMultiplex(g, nil, weights)
	cg0Qnull := QMultiplex(cg0, cg0.Structure(), weights, nil)
	if !scalar.EqualWithinAbsOrRel(floats.Sum(gQnull), floats.Sum(cg0Qnull), structure.tol, structure.tol) {
	t.Errorf("disagreement between null Q from method: %v and function: %v", cg0Qnull, gQnull)
	}
	cg0Q := QMultiplex(cg0, communities, weights, []float64{structure.resolution})
	if !scalar.EqualWithinAbsOrRel(floats.Sum(gQ), floats.Sum(cg0Q), structure.tol, structure.tol) {
	t.Errorf("unexpected Q result after initial reduction: got: %v want :%v", cg0Q, gQ)
	}

	cg1 := reduceDirectedMultiplex(cg0, communities, weights)
	cg1Q := QMultiplex(cg1, cg1.Structure(), weights, []float64{structure.resolution})
	if !scalar.EqualWithinAbsOrRel(floats.Sum(gQ), floats.Sum(cg1Q), structure.tol, structure.tol) {
	t.Errorf("unexpected Q result after second reduction: got: %v want :%v", cg1Q, gQ)
	}
	}
	}
	}

	var localDirectedMultiplexMoveTests = []struct {
	name string
	layers []layer
	structures []moveStructures
	}{
	{
	name: "blondel",
	layers: []layer{{g: blondel, weight: 1}, {g: blondel, weight: 0.5}},
	structures: []moveStructures{
	{
	memberships: []intset{
	0: linksTo(0, 1, 2, 4, 5),
	1: linksTo(3, 6, 7),
	2: linksTo(8, 9, 10, 12, 14, 15),
	3: linksTo(11, 13),
	},
	targetNodes: []graph.Node{simple.Node(0)},
	resolution: 1,
	tol: 1e-14,
	},
	{
	memberships: []intset{
	0: linksTo(0, 1, 2, 4, 5),
	1: linksTo(3, 6, 7),
	2: linksTo(8, 9, 10, 12, 14, 15),
	3: linksTo(11, 13),
	},
	targetNodes: []graph.Node{simple.Node(3)},
	resolution: 1,
	tol: 1e-14,
	},
	{
	memberships: []intset{
	0: linksTo(0, 1, 2, 4, 5),
	1: linksTo(3, 6, 7),
	2: linksTo(8, 9, 10, 12, 14, 15),
	3: linksTo(11, 13),
	},
	// Case to demonstrate when A_aa != k_a^𝛼.
	targetNodes: []graph.Node{simple.Node(3), simple.Node(2)},
	resolution: 1,
	tol: 1e-14,
	},
	},
	},
	}

	func TestMoveLocalDirectedMultiplex(t *testing.T) {
	for _, test := range localDirectedMultiplexMoveTests {
	g, weights, err := directedMultiplexFrom(test.layers)
	if err != nil {
	t.Errorf("unexpected error creating multiplex: %v", err)
	continue
	}

	for _, structure := range test.structures {
	communities := make([][]graph.Node, len(structure.memberships))
	for i, c := range structure.memberships {
	for n := range c {
	communities[i] = append(communities[i], simple.Node(n))
	}
	sort.Sort(ordered.ByID(communities[i]))
	}

	r := reduceDirectedMultiplex(reduceDirectedMultiplex(g, nil, weights), communities, weights)

	l := newDirectedMultiplexLocalMover(r, r.communities, weights, []float64{structure.resolution}, true)
	for _, n := range structure.targetNodes {
	dQ, dst, src := l.deltaQ(n)
	if dQ > 0 {
	before := floats.Sum(QMultiplex(r, l.communities, weights, []float64{structure.resolution}))
	l.move(dst, src)
	after := floats.Sum(QMultiplex(r, l.communities, weights, []float64{structure.resolution}))
	want := after - before
	if !scalar.EqualWithinAbsOrRel(dQ, want, structure.tol, structure.tol) {
	t.Errorf("unexpected deltaQ: got: %v want: %v", dQ, want)
	}
	}
	}
	}
	}
	}

	func TestLouvainDirectedMultiplex(t *testing.T) {
	const louvainIterations = 20

	for _, test := range communityDirectedMultiplexQTests {
	g, weights, err := directedMultiplexFrom(test.layers)
	if err != nil {
	t.Errorf("unexpected error creating multiplex: %v", err)
	continue
	}

	if test.structures[0].resolution != 1 {
	panic("bad test: expect resolution=1")
	}
	want := make([][]graph.Node, len(test.structures[0].memberships))
	for i, c := range test.structures[0].memberships {
	for n := range c {
	want[i] = append(want[i], simple.Node(n))
	}
	sort.Sort(ordered.ByID(want[i]))
	}
	sort.Sort(ordered.BySliceIDs(want))

	var (
	got *ReducedDirectedMultiplex
	bestQ = math.Inf(-1)
	)
	// Modularize is randomised so we do this to
	// ensure the level tests are consistent.
	src := rand.New(rand.NewSource(1))
	for i := 0; i < louvainIterations; i++ {
	r := ModularizeMultiplex(g, weights, nil, true, src).(*ReducedDirectedMultiplex)
	if q := floats.Sum(QMultiplex(r, nil, weights, nil)); q > bestQ \|\| math.IsNaN(q) {
	bestQ = q
	got = r

	if math.IsNaN(q) {
	// Don't try again for non-connected case.
	break
	}
	}

	var qs []float64
	for p := r; p != nil; p = p.Expanded().(*ReducedDirectedMultiplex) {
	qs = append(qs, floats.Sum(QMultiplex(p, nil, weights, nil)))
	}

	// Recovery of Q values is reversed.
	if reverse(qs); !sort.Float64sAreSorted(qs) {
	t.Errorf("Q values not monotonically increasing: %.5v", qs)
	}
	}

	gotCommunities := got.Communities()
	for _, c := range gotCommunities {
	sort.Sort(ordered.ByID(c))
	}
	sort.Sort(ordered.BySliceIDs(gotCommunities))
	if !reflect.DeepEqual(gotCommunities, want) {
	t.Errorf("unexpected community membership for %s Q=%.4v:\n\tgot: %v\n\twant:%v",
	test.name, bestQ, gotCommunities, want)
	continue
	}

	var levels []level
	for p := got; p != nil; p = p.Expanded().(*ReducedDirectedMultiplex) {
	var communities [][]graph.Node
	if p.parent != nil {
	communities = p.parent.Communities()
	for _, c := range communities {
	sort.Sort(ordered.ByID(c))
	}
	sort.Sort(ordered.BySliceIDs(communities))
	} else {
	communities = reduceDirectedMultiplex(g, nil, weights).Communities()
	}
	q := floats.Sum(QMultiplex(p, nil, weights, nil))
	if math.IsNaN(q) {
	// Use an equalable flag value in place of NaN.
	q = math.Inf(-1)
	}
	levels = append(levels, level{q: q, communities: communities})
	}
	if !reflect.DeepEqual(levels, test.wantLevels) {
	t.Errorf("unexpected level structure:\n\tgot: %v\n\twant:%v", levels, test.wantLevels)
	}
	}
	}

	func TestNonContiguousDirectedMultiplex(t *testing.T) {
	g := simple.NewDirectedGraph()
	for _, e := range []simple.Edge{
	{F: simple.Node(0), T: simple.Node(1)},
	{F: simple.Node(4), T: simple.Node(5)},
	} {
	g.SetEdge(e)
	}

	func() {
	defer func() {
	r := recover()
	if r != nil {
	t.Error("unexpected panic with non-contiguous ID range")
	}
	}()
	ModularizeMultiplex(DirectedLayers{g}, nil, nil, true, nil)
	}()
	}

	func TestNonContiguousWeightedDirectedMultiplex(t *testing.T) {
	g := simple.NewWeightedDirectedGraph(0, 0)
	for _, e := range []simple.WeightedEdge{
	{F: simple.Node(0), T: simple.Node(1), W: 1},
	{F: simple.Node(4), T: simple.Node(5), W: 1},
	} {
	g.SetWeightedEdge(e)
	}

	func() {
	defer func() {
	r := recover()
	if r != nil {
	t.Error("unexpected panic with non-contiguous ID range")
	}
	}()
	ModularizeMultiplex(DirectedLayers{g}, nil, nil, true, nil)
	}()
	}

	func BenchmarkLouvainDirectedMultiplex(b *testing.B) {
	src := rand.New(rand.NewSource(1))
	for i := 0; i < b.N; i++ {
	ModularizeMultiplex(DirectedLayers{dupGraphDirected}, nil, nil, true, src)
	}
	}

	func directedMultiplexFrom(raw []layer) (DirectedLayers, []float64, error) {
	var layers []graph.Directed
	var weights []float64
	for _, l := range raw {
	g := simple.NewWeightedDirectedGraph(0, 0)
	for u, e := range l.g {
	// Add nodes that are not defined by an edge.
	if g.Node(int64(u)) == nil {
	g.AddNode(simple.Node(u))
	}
	for v := range e {
	w := 1.0
	if l.edgeWeight != 0 {
	w = l.edgeWeight
	}
	g.SetWeightedEdge(simple.WeightedEdge{F: simple.Node(u), T: simple.Node(v), W: w})
	}
	}
	layers = append(layers, g)
	weights = append(weights, l.weight)
	}
	g, err := NewDirectedLayers(layers...)
	if err != nil {
	return nil, nil, err
	}
	return g, weights, nil
	}