// NavigableSmallWorld constructs an N-dimensional grid with guaranteed local connectivity
// and random long-range connectivity in the destination, dst. The dims parameters specifies
// the length of each of the N dimensions, p defines the Manhattan distance between local
// nodes, and q defines the number of out-going long-range connections from each node. Long-
// range connections are made with a probability proportional to |d(u,v)|^-r where d is the
// Manhattan distance between non-local nodes.
//
// The algorithm is essentially as described on p4 of http://www.cs.cornell.edu/home/kleinber/swn.pdf.
func NavigableSmallWorld(dst GraphBuilder, dims []int, p, q int, r float64, src *rand.Rand) (err error) {
if p < 1 {
return fmt.Errorf("gen: bad local distance: p=%v", p)
}
if q < 0 {
return fmt.Errorf("gen: bad distant link count: q=%v", q)
}
if r < 0 {
return fmt.Errorf("gen: bad decay constant: r=%v", r)
}
n := 1
for _, d := range dims {
n *= d
}
for i := 0; i < n; i++ {
if !dst.Has(simple.Node(i)) {
dst.AddNode(simple.Node(i))
}
}
hasEdge := dst.HasEdgeBetween
d, isDirected := dst.(graph.Directed)
if isDirected {
hasEdge = d.HasEdgeFromTo
}
locality := make([]int, len(dims))
for i := range locality {
locality[i] = p*2 + 1
}
iterateOver(dims, func(u []int) {
uid := idFrom(u, dims)
iterateOver(locality, func(delta []int) {
d := manhattanDelta(u, delta, dims, -p)
if d == 0 || d > p {
return
}
vid := idFromDelta(u, delta, dims, -p)
e := simple.Edge{F: simple.Node(uid), T: simple.Node(vid), W: 1}
if uid > vid {
e.F, e.T = e.T, e.F
}
if !hasEdge(e.From(), e.To()) {
dst.SetEdge(e)
}
if !isDirected {
return
}
e.F, e.T = e.T, e.F
if !hasEdge(e.From(), e.To()) {
dst.SetEdge(e)
}
})
})
defer func() {
r := recover()
if r != nil {
if r != "depleted distribution" {
panic(r)
}
err = errors.New("depleted distribution")
}
}()
w := make([]float64, n)
ws := sample.NewWeighted(w, src)
iterateOver(dims, func(u []int) {
uid := idFrom(u, dims)
iterateOver(dims, func(v []int) {
d := manhattanBetween(u, v)
if d <= p {
return
}
w[idFrom(v, dims)] = math.Pow(float64(d), -r)
})
ws.ReweightAll(w)
for i := 0; i < q; i++ {
vid, ok := ws.Take()
if !ok {
panic("depleted distribution")
}
e := simple.Edge{F: simple.Node(uid), T: simple.Node(vid), W: 1}
if !isDirected && uid > vid {
e.F, e.T = e.T, e.F
}
if !hasEdge(e.From(), e.To()) {
dst.SetEdge(e)
}
}
for i := range w {
w[i] = 0
}
})
return nil
}
// SmallWorldsBB constructs a small worlds graph of order n in the destination, dst.
// Node degree is specified by d and edge replacement by the probability, p.
// If src is not nil it is used as the random source, otherwise rand.Float64 is used.
// The graph is constructed in O(nd) time.
//
// The algorithm used is described in http://algo.uni-konstanz.de/publications/bb-eglrn-05.pdf
func SmallWorldsBB(dst GraphBuilder, n, d int, p float64, src *rand.Rand) error {
if d < 1 || d > (n-1)/2 {
return fmt.Errorf("gen: bad degree: d=%d", d)
}
if p == 0 {
return nil
}
if p < 0 || p >= 1 {
return fmt.Errorf("gen: bad replacement: p=%v", p)
}
var (
rnd func() float64
rndN func(int) int
)
if src == nil {
rnd = rand.Float64
rndN = rand.Intn
} else {
rnd = src.Float64
rndN = src.Intn
}
hasEdge := dst.HasEdgeBetween
dg, isDirected := dst.(graph.Directed)
if isDirected {
hasEdge = dg.HasEdgeFromTo
}
for i := 0; i < n; i++ {
if !dst.Has(simple.Node(i)) {
dst.AddNode(simple.Node(i))
}
}
nChoose2 := (n - 1) * n / 2
lp := math.Log(1 - p)
// Add forward edges for all graphs.
k := int(math.Log(1-rnd()) / lp)
m := 0
replace := make(map[int]int)
for v := 0; v < n; v++ {
for i := 1; i <= d; i++ {
if k > 0 {
j := v*(v-1)/2 + (v+i)%n
ej := simple.Edge{W: 1}
ej.T, ej.F = edgeNodesFor(j)
if !hasEdge(ej.From(), ej.To()) {
dst.SetEdge(ej)
}
k--
m++
em := simple.Edge{W: 1}
em.T, em.F = edgeNodesFor(m)
if !hasEdge(em.From(), em.To()) {
replace[j] = m
} else {
replace[j] = replace[m]
}
} else {
k = int(math.Log(1-rnd()) / lp)
}
}
}
for i := m + 1; i <= n*d && i < nChoose2; i++ {
r := rndN(nChoose2-i) + i
er := simple.Edge{W: 1}
er.T, er.F = edgeNodesFor(r)
if !hasEdge(er.From(), er.To()) {
dst.SetEdge(er)
} else {
er.T, er.F = edgeNodesFor(replace[r])
if !hasEdge(er.From(), er.To()) {
dst.SetEdge(er)
}
}
ei := simple.Edge{W: 1}
ei.T, ei.F = edgeNodesFor(i)
if !hasEdge(ei.From(), ei.To()) {
replace[r] = i
} else {
replace[r] = replace[i]
}
}
// Add backward edges for directed graphs.
if !isDirected {
return nil
}
k = int(math.Log(1-rnd()) / lp)
m = 0
replace = make(map[int]int)
for v := 0; v < n; v++ {
for i := 1; i <= d; i++ {
if k > 0 {
j := v*(v-1)/2 + (v+i)%n
ej := simple.Edge{W: 1}
ej.F, ej.T = edgeNodesFor(j)
if !hasEdge(ej.From(), ej.To()) {
dst.SetEdge(ej)
}
k--
m++
if !hasEdge(edgeNodesFor(m)) {
replace[j] = m
} else {
replace[j] = replace[m]
}
} else {
k = int(math.Log(1-rnd()) / lp)
}
}
}
for i := m + 1; i <= n*d && i < nChoose2; i++ {
r := rndN(nChoose2-i) + i
er := simple.Edge{W: 1}
er.F, er.T = edgeNodesFor(r)
if !hasEdge(er.From(), er.To()) {
dst.SetEdge(er)
} else {
er.F, er.T = edgeNodesFor(replace[r])
if !hasEdge(er.From(), er.To()) {
dst.SetEdge(er)
}
}
if !hasEdge(edgeNodesFor(i)) {
replace[r] = i
} else {
replace[r] = replace[i]
}
}
return nil
}