Random Geometric Graphs Mathew D. Penrose University of Bath, UK Networks: stochastic models for populations and epidemics ICMS, Edinburgh September 2011 1 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) nπr2 for G(n, r) Often we choose p = pn or r = rn to make this expected degree Θ(1). In this case G(n, rn) resembles the following infinite system: 2 CONTINUUM PERCOLATION (see Meester and Roy 1996) Let be a homogeneous Poisson point process in Rd with intensity Pλ λ, i.e. (A) Poisson( A ) Pλ ∼ | | and (A ) are independent variables for A , A ,... disjoint. Pλ i 1 2 Gilbert Graph. Form a graph := G( ) on by connecting two Gλ Pλ Pλ Poisson points x, y iff x y 1. | − |≤ Form graph 0 := G( 0 ) similarly on 0 . Gλ Pλ ∪ { } Pλ ∪ { } Let p (λ) be the prob. that the component of 0 containing 0 has k k Gλ vertices (also depends on d). ∞ p∞(λ):=1 p (λ) be the prob. that this component is infinite. − k=0 k P 3 RANDOM GEOMETRIC GRAPHS (rescaled) (Penrose 2003) Let U1,...,Un be independently uniformly randomly scattered in a cube of volume n/λ in d-space. Form a graph on 1, 2,...,n by Gn,λ { } i j iff U U 1 ∼ | i − j|≤ Define C to be the order of the component of containing i. It | i| Gn,λ can be shown that n−1 n 1 C = k P p (λ) as n , i.e. i=1 {| i| } −→ k → ∞ for large n, P n P n−1 1 C = k p (λ) 1 {| i| }≈ k ≈ " i=1 # X That is, the proportionate number of vertices of Gn lying in components of order k, converges to pk(λ) in probability. 4 A FORMULA FOR pk(λ). k pk+1(λ)=(k + 1)λ h(x1,...,xk) Rd k Z( ) exp( λA(0, x ,...,x ))dx ...dx × − 1 k 1 k where h(x ,...,x ) is 1 if G( 0, x ,...,x ) is connected and 1 k { 1 k} 0 x x lexicographically, otherwise zero; ≺ 1 ≺···≺ k and A(0, x1,...,xk) is the volume of the union of 1-balls centred at 0, x1,...,xk. Not tractable for large k. 5 THE PHASE TRANSITION If p∞(λ) = 0, then has no infinite component, almost surely. Gλ If p∞(λ) > 0, then has a unique infinite component, almost surely. Gλ Also, p∞(λ) is nondecreasing in λ. Fundamental theorem: If d 2 then ≥ λ (d) := sup λ : p∞(λ) = 0 (0, ). c { }∈ ∞ If d = 1 then λ (d)= . From now on, assume d 2. The value of c ∞ ≥ λc(d) is not known. 6 LARGE COMPONENTS FOR THE RGG Consider again the random geometric graph (on n uniform Gn,λ random points in a cube of volume n/λ in d-space) Let L ( ) be the size of the largest component, and L ( ) the 1 Gn,λ 2 Gn,λ size of the second largest component (‘size’ measured by number of vertices). As n with λ fixed: → ∞ −1 P if λ > λ then n L ( ) p∞(λ) > 0 c 1 Gn,λ −→ if λ < λ then (log n)−1L ( ) P 1/ζ(λ) c 1 Gn,λ −→ and for the Poissonized RGG (N Poisson (n)), GNn,λ n ∼ L ( )= O(log n)d/(d−1) in probability if λ > λ 2 GNn,λ c 7 CENTRAL AND LOCAL LIMIT THEOREMS. Let K( ) be the Gn,λ number of components of . As n with λ fixed, Gn,λ → ∞ K( ) EK( ) t P Gn,λ − Gn,λ t Φ (t) := ϕ (x)dx √n ≤ → σ σ Z−∞ 2 2 2 −1/2 −x /(2σ ) where ϕσ(x) := (2πσ ) e (normal pdf); and if λ > λc, L ( ) EL ( ) P 1 Gn,λ − 1 Gn,λ t Φ (t). √n ≤ → τ Here σ and τ are positive constants, dependent on λ. Also, 1/2 z EK( n,λ sup n P [K( n,λ)= z] ϕσ − G 0. Z G − √n → z∈ 8 ISOLATED VERTICES. Suppose d = 2. Let N ( ) be the 0 Gn,λ number of isolated vertices. The expected number of isolated vertices satisfies EN ( ) n exp( πλ) 0 Gn,λ ≈ − so if we fix t and take λ(n) = (log n + t)/π, then as n , → ∞ EN ( ) e−t. 0 Gn,λ(n) → Also, N0 is approximately Poisson distributed so P [N ( )=0] exp( e−t). 0 Gn,λ(n) → − 9 CONNECTIVITY. Note is connected iff K( )=1. Gn,λ Gn,λ Clearly P [K( )=1] P [N ( ) = 0]. Again taking Gn,λ ≤ 0 Gn,λ(n) λ(n) = (log n + t)/π with t fixed, it turns out (Penrose 1997) that −t lim P [K( n,λ(n))=1]= lim P [N0( n,λ(n)) = 0] = exp( e ) n→∞ G n→∞ G − or in other words, lim (P [K( n,λ(n)) > 1] P [N0( n,λ(n)) > 0]) = 0. n→∞ G − G 10 THE CONNECTIVITY THRESHOLD Let V1,...,Vn be independently uniformly randomly scattered in [0, 1]d. Form a graph r on 1, 2,...,n by Gn { } i j iff V V r. ∼ | i − j |≤ Given the values of V1,...,Vn, define the connectivity threshold ρn(K = 1), and the no-isolated-vertex threshold ρn(N0 = 0), by ρ (K = 1) = min r : K( r ) = 1 ; n { Gn } ρ (N = 0) = min r : N ( r ) = 1 . n 0 { 0 Gn } 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 Taking λ(n) = (log n + t)/π with t fixed, we have √λ(n)/n P [K( )=1]= P [K( n )=1] Gn,λ(n) G = P ρ (K = 1) (log n + t)/(πn) n ≤ h i so the earlier result p −t lim P [K( n,λ(n))=1]= lim P [N0( n,λ(n)) = 0] = exp( e ) n→∞ G n→∞ G − implies 2 −t lim P [nπ(ρn(K = 1)) log n t] = exp(e ) n→∞ − ≤ and likewise for ρn(N0) = 0. In fact we have a stronger result: lim P [ρn(K =1)= ρn(N0 = 0)] = 1. n→∞ 12 MULTIPLE CONNECTIVITY. Given k N, a graph is ∈ k-connected if for any two distinct vertices there are k disjoint paths connecting them. r Let ρn,k be the smallest r such that Gn is k-connected. Let ρ (N = 0) be the smallest r such that r has no vertex of n <k Gn degree less than k (a random variable determined by V1,...,Vn). Then lim P [ρn,k = ρn(N<k = 0)] = 1. n→∞ The limit distribution of ρn(N<k = 0) can be determined via Poisson approximation, as with ρn(N0 = 0). 13 HAMILTONIAN PATHS Let ρ (Ham) be the smallest r such that r has a Hamiltonian path n Gn (i.e. a self-avoiding tour through all the vertices). Clearly ρ (Ham) ρ (N = 0). n ≥ n <2 It has recently been established (Balogh, Bollobas, Walters, Krivelevich, M¨uller 2009) that lim P [ρn(Ham) = ρn(N<2 = 0)] = 1. n→∞ 14 SMALL SUBGRAPHS Let Γ be a fixed connected graph of order k. Assume Γ is feasible, i.e. P [ Γ] > 0 for some λ. Gk,λ ∼ Let J ( ) be the number of components isomorphic to Γ. Γ Gn,λ Let H ( ) be the number of induced subgraphs isomomorphic to Γ Gn,λ Γ (e.g. number of triangles). As n with λ fixed, → ∞ n−1J ( ) p , n−1H ( ) µ a.s. Γ Gn,λ → Γ Γ Gn,λ → Γ where pΓ(λ) and µΓ(λ) are given by formulae similar to pn(λ). In terms of 0 (the Poisson Gilbert graph), p is the probability 0 is Gλ Γ in a component isomorphic to Γ, and µΓ is the expected number of induced Γ-subgraphs incident to 0. cf. random simplicial complexes 15 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 , → ∞ H ( ) EH ( ) P Γ Gn,λ − Γ Gn,λ t Φ (t). √n ≤ → σ 1/2 z EHΓ( n,λ) sup n P [HΓ( n,λ)= z] ϕσ − G 0, Z G − √n → z∈ and J ( ) EJ ( ) P Γ Gn,λ − Γ Gn,λ t Φ (t), √n ≤ → τ 1/2 z EHΓ( n,λ) sup n P [HΓ( n,λ)= z] ϕτ − G 0. Z G − √n → z∈ 16 k d(k−1) POISSON LIMITS. Consider G(n, r ) with n rn c. n → (Equivalent to with λ n−1/(k−1)). Then Gn,λn n ∝ n EH (G(n, r )) rd(k−1)µ (1) c′ Γ n ∼ k n Γ → In fact, for n large both HΓ(G(n, rn)) and JΓ(G(n, rn)) are approximately Poisson (c′). Now suppose n3r2d c and let N be the number of vertices of n → 2 degree 2. Then N2 is approximately compound Poisson. Reason: N2 = N∧ + 3N△, and N∧ and N△ converge to independent Poissons. 17 OTHER PROPERTIES OF RGGS Asymptotic behaviour of other quantities arising from of random geometric graph have been considered, including The largest and smallest degree.