Random Geometric Graphs

Transcription

1 Random Geometric Graphs Mathew D. Penrose University of Bath, UK Networks: stochastic models for populations and epidemics ICMS, Edinburgh September

2 MODELS of RANDOM GRAPHS Erdos-Renyi G(n, p): start with the complete n-graph, retain each edge with probability p (independently). Random geometric graph G(n, r). Place n points uniformly at random in [0, 1] 2. Connect any two points distance at most r apart. Consider n large and p or r small. Expected degree of a typical vertex is approximately: np for G(n, p) nπr 2 for G(n, r) Often we choose p = p n or r = r n to make this expected degree Θ(1). In this case G(n, r n ) resembles the following infinite system: 2

3 CONTINUUM PERCOLATION (see Meester and Roy 1996) Let P λ be a homogeneous Poisson point process in R d with intensity λ, i.e. P λ (A) Poisson( A ) and P λ (A i ) are independent variables for A 1, A 2,... disjoint. Gilbert Graph. Form a graph G λ := G(P λ ) on P λ by connecting two Poisson points x, y iff x y 1. Form graph G 0 λ := G(P λ {0}) similarly on P λ {0}. Let p k (λ) be the prob. that the component of Gλ 0 vertices (also depends on d). containing 0 has k p (λ) := 1 k=0 p k(λ) be the prob. that this component is infinite. 3

4 RANDOM GEOMETRIC GRAPHS (rescaled) (Penrose 2003) Let U 1,...,U n be independently uniformly randomly scattered in a cube of volume n/λ in d-space. Form a graph G n,λ on {1, 2,...,n} by i j iff U i U j 1 Define C i to be the order of the component of G n,λ containing i. It can be shown that n 1 n i=1 1{ C i = k} P p k (λ) as n, i.e. for large n, [ ] n P n 1 1{ C i = k} p k (λ) 1 i=1 That is, the proportionate number of vertices of G n lying in components of order k, converges to p k (λ) in probability. 4

5 A FORMULA FOR p k (λ). p k+1 (λ) = (k + 1)λ k (R d ) k h(x 1,...,x k ) exp( λa(0, x 1,...,x k ))dx 1...dx k where h(x 1,...,x k ) is 1 if G({0, x 1,...,x k }) is connected and 0 x 1 x k lexicographically, otherwise zero; and A(0, x 1,...,x k ) is the volume of the union of 1-balls centred at 0, x 1,...,x k. Not tractable for large k. 5

6 THE PHASE TRANSITION If p (λ) = 0, then G λ has no infinite component, almost surely. If p (λ) > 0, then G λ has a unique infinite component, almost surely. Also, p (λ) is nondecreasing in λ. Fundamental theorem: If d 2 then λ c (d) := sup{λ : p (λ) = 0} (0, ). If d = 1 then λ c (d) =. From now on, assume d 2. The value of λ c (d) is not known. 6

7 LARGE COMPONENTS FOR THE RGG Consider again the random geometric graph G n,λ (on n uniform random points in a cube of volume n/λ in d-space) Let L 1 (G n,λ ) be the size of the largest component, and L 2 (G n,λ ) the size of the second largest component ( size measured by number of vertices). As n with λ fixed: if λ > λ c then n 1 L 1 (G n,λ ) if λ < λ c then (log n) 1 L 1 (G n,λ ) P p (λ) > 0 P 1/ζ(λ) and for the Poissonized RGG G Nn,λ (N n Poisson (n)), L 2 (G Nn,λ) = O(log n) d/(d 1) in probability if λ > λ c 7

8 CENTRAL AND LOCAL LIMIT THEOREMS. Let K(G n,λ ) be the number of components of G n,λ. As n with λ fixed, [ ] K(Gn,λ ) EK(G n,λ ) t P t Φ σ (t) := ϕ σ (x)dx n where ϕ σ (x) := (2πσ 2 ) 1/2 e x2 /(2σ 2) (normal pdf); and if λ > λ c, [ ] L1 (G n,λ ) EL 1 (G n,λ ) P t Φ τ (t). n sup z Z Here σ and τ are positive constants, dependent on λ. Also, { ( )} z n 1/2 EK(Gn,λ P[K(G n,λ ) = z] ϕ σ 0. n 8

9 ISOLATED VERTICES. Suppose d = 2. Let N 0 (G n,λ ) be the number of isolated vertices. The expected number of isolated vertices satisfies EN 0 (G n,λ ) n exp( πλ) so if we fix t and take λ(n) = (log n + t)/π, then as n, EN 0 (G n,λ(n) ) e t. Also, N 0 is approximately Poisson distributed so P[N 0 (G n,λ(n) ) = 0] exp( e t ). 9

10 CONNECTIVITY. Note G n,λ is connected iff K(G n,λ ) = 1. Clearly P[K(G n,λ ) = 1] P[N 0 (G n,λ(n) ) = 0]. Again taking λ(n) = (log n + t)/π with t fixed, it turns out (Penrose 1997) that lim P[K(G n,λ(n)) = 1] = lim P[N 0(G n,λ(n) ) = 0] = exp( e t ) n n or in other words, lim (P[K(G n,λ(n)) > 1] P[N 0 (G n,λ(n) ) > 0]) = 0. n 10

11 THE CONNECTIVITY THRESHOLD Let V 1,...,V n be independently uniformly randomly scattered in [0, 1] d. Form a graph G r n on {1, 2,...,n} by i j iff V i V j r. Given the values of V 1,...,V n, define the connectivity threshold ρ n (K = 1), and the no-isolated-vertex threshold ρ n (N 0 = 0), by ρ n (K = 1) = min{r : K(G r n) = 1}; ρ n (N 0 = 0) = min{r : N 0 (G r n) = 1}. The preceding result can be interpreted as giving the limiting distributions of these thresholds (suitably scaled and centred) as n : they have the same limiting behaviour. 11

12 Taking λ(n) = (log n + t)/π with t fixed, we have so the earlier result λ(n)/n P[K(G n,λ(n) ) = 1] = P[K(Gn ) = 1] [ = P ρ n (K = 1) ] (log n + t)/(πn) lim P[K(G n,λ(n)) = 1] = lim P[N 0(G n,λ(n) ) = 0] = exp( e t ) n n implies lim P[nπ(ρ n(k = 1)) 2 log n t] = exp(e t ) n and likewise for ρ n (N 0 ) = 0. In fact we have a stronger result: lim P[ρ n(k = 1) = ρ n (N 0 = 0)] = 1. n 12

13 MULTIPLE CONNECTIVITY. Given k N, a graph is k-connected if for any two distinct vertices there are k disjoint paths connecting them. Let ρ n,k be the smallest r such that G r n is k-connected. Let ρ n (N <k = 0) be the smallest r such that G r n has no vertex of degree less than k (a random variable determined by V 1,...,V n ). Then lim P[ρ n,k = ρ n (N <k = 0)] = 1. n The limit distribution of ρ n (N <k = 0) can be determined via Poisson approximation, as with ρ n (N 0 = 0). 13

14 HAMILTONIAN PATHS Let ρ n (Ham) be the smallest r such that G r n has a Hamiltonian path (i.e. a self-avoiding tour through all the vertices). Clearly ρ n (Ham) ρ n (N <2 = 0). It has recently been established (Balogh, Bollobas, Walters, Krivelevich, Müller 2009) that lim P[ρ n(ham) = ρ n (N <2 = 0)] = 1. n 14

15 SMALL SUBGRAPHS Let Γ be a fixed connected graph of order k. Assume Γ is feasible, i.e. P[G k,λ Γ] > 0 for some λ. Let J Γ (G n,λ ) be the number of components isomorphic to Γ. Let H Γ (G n,λ ) be the number of induced subgraphs isomomorphic to Γ (e.g. number of triangles). As n with λ fixed, n 1 J Γ (G n,λ ) p Γ, n 1 H Γ (G n,λ ) µ Γ a.s. where p Γ (λ) and µ Γ (λ) are given by formulae similar to p n (λ). In terms of G 0 λ (the Poisson Gilbert graph), p Γ is the probability 0 is in a component isomorphic to Γ, and µ Γ is the expected number of induced Γ-subgraphs incident to 0. cf. random simplicial complexes 15

16 CLT AND LLT FOR SMALL SUBGRAPHS. There are constants σ(γ, λ) and τ(γ, λ) (this time we have integral formulae) such that as n, P [ HΓ (G n,λ ) EH Γ (G n,λ ) n ] t Φ σ (t). { ( )} z sup n 1/2 EHΓ (G n,λ ) P[H Γ (G n,λ ) = z] ϕ σ 0, z Z n and P [ JΓ (G n,λ ) EJ Γ (G n,λ ) n ] t Φ τ (t), { ( )} z sup n 1/2 EHΓ (G n,λ ) P[H Γ (G n,λ ) = z] ϕ τ 0. z Z n 16

17 POISSON LIMITS. Consider G(n, r n ) with n k r d(k 1) n c. (Equivalent to G n,λn with λ n n 1/(k 1) ). Then ( ) n EH Γ (G(n, r n )) rn d(k 1) µ Γ (1) c k In fact, for n large both H Γ (G(n, r n )) and J Γ (G(n, r n )) are approximately Poisson (c ). Now suppose n 3 r 2d n c and let N 2 be the number of vertices of degree 2. Then N 2 is approximately compound Poisson. Reason: N 2 = N + 3N, and N and N converge to independent Poissons. 17

18 OTHER PROPERTIES OF RGGS Asymptotic behaviour of other quantities arising from of random geometric graph have been considered, including The largest and smallest degree. The clique and chromatic number. In some cases, non-uniform distributions of the vertices V 1,...,V n have been considered. 18

19 CONTINUITY OF p (λ) Clearly p (λ) = 0 for λ < λ c Also p (λ) is increasing in λ on λ > λ c. Less trivially, it is known that p (λ) is continuous in λ on λ (λ c, ) and is right continuous at λ = λ c, i.e. p (λ c ) = lim λ λc p (λ). So p ( ) is continuous on (0, ) iff p (λ c ) = 0. This is known to hold for d = 2 (Alexander 1996) and for large d (Tanemura 1996). It is conjectured to hold for all d. 19

20 LARGE-k ASYMPTOTICS FOR p k (λ) Suppose λ < λ c. Then there exists ζ(λ) > 0 such that ( ζ(λ) = lim n 1 log p n (λ) ) n or more informally, p n (λ) (e ζ(λ) ) n. Proof uses subadditivity. With x n 1 := (x 1,...,x n ), recall p n+1 (λ) = (n + 1)λ n h(x n 1)e λa(0,xn 1 ) dx n 1. Setting q n := p n+1 /(n + 1), can show q n q m q n+m 1, so log q n /(n 1) inf n 1 ( log q n /(n 1)) := ζ as n. That ζ(λ) > 0 is a deeper result. 20

21 LARGE-k ASYMPTOTICS: THE SUPERCRITICAL CASE Suppose λ > λ c. Then lim sup n lim inf n ( ) n (d 1)/d log p n (λ) ( n (d 1)/d log p n (λ) ) < 0 > Loosely speaking, this says that in the supercritical case p n (λ) decays exponentially in n 1 1/d, whereas in the subcritical case it decays exponentially in n. 21

22 HIGH DIMENSIONAL ASYMPTOTICS (Penrose 1996) Let θ d denote the volume of the unit ball in R d. Suppose λ = λ(d) is chosen so that λ(d)θ d = µ (this is the expected degree of the origin 0 in G 0 λ ). Asymptotically as d with µ fixed, the structure of C 0 converges to that of a branching process (Z 0, Z 1,...) with Poisson (µ) offspring distribution (Z 0 = 1, Z n is nth generation size). So (Penrose 1996): p k (λ(d)) p k (µ) := P[ n=0 Z n = k]; p (λ(d)) ψ(µ) := P [ n=0 Z n = ]; θ d λ c (d) 1. 22

23 DETAILS OF THE BRANCHING PROCESS THEORY: p k (µ) = P[ Z n = k] = n=0 kk 2 (k 1)! µk 1 e kµ. t = 1 ψ(µ) is the smallest positive solution to t = exp(µ(t 1)). In particular, ψ(µ) = 0, if µ 1 ψ(µ) > 0, if µ > 1 23

24 OTHER MODELS OF CONTINUUM PERCOLATION The Boolean model. Let Ξ be the union of balls of random radius centred at the points of P λ (with some specified radius distribution). Look at the connected components of Ξ. The random connection model. Given P λ, let each pair (x, y) of points of P λ be connected with probability f( x y ) with f some specified function satisfying R d f( x )dx < Both of these are generalizations of the model we have been considering. 24

25 Nearest-neighbour percolation. Fix k N. Join each point of P λ by an undirected edge to its k nearest neighbours in P λ. Look at components of resulting graph. Lilypond model. At time zero, start growing a ball of unit rate outwards from each point of P λ, stopping as soon as it hits another ball. This yields a maximal system of non-overlapping balls on P. Voronoi cell percolation. Let each vertex of P λ be coloured red (with probability p), otherwise blue. Look at connectivity properties of the union of red Voronoi cells. 25

26 HIGH-INTENSITY ASYMPTOTICS Let θ denote the volume of the unit ball in R d. As λ, p (λ) 1 and in fact (Penrose 1991) 1 p (λ) p 1 (λ) = exp( λθ) This says that for large λ, given the unlikely event that the component of Gλ 0 containing 0 is finite, it is likely to consist of an isolated vertex. Also, for k 1, log p k+1 (λ) = λθ + (d 1)k log λ k + O(1) as λ with k fixed (Alexander 1991). 26

27 References [1] Bollobás, B. and Riordan, O. (2006) Percolation. Cambridge University Press. [2] Franceschetti, M. and Meester, R. (2008) Random Networks in Communiction. Cambridge University Press. [3] Meester, R. and Roy, R. (1996) Continuum Percolation. Cambridge University Press. [4] Penrose, M.D. (2003). Random Geometric Graphs. Oxford University Press. 27