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

lim P [ρn,k = ρn(N

The limit distribution of ρn(N

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. The clique and chromatic number.

In some cases, non-uniform distributions of the vertices V1,...,Vn have been considered.

18 CONTINUITY OF p∞(λ)

Clearly p∞(λ) = 0 for λ < λc

Also p∞(λ) is increasing in λ on λ > λc.

Less trivially, it is known that p∞(λ) is continuous in λ on λ (λ , ) and is right continuous at λ = λ , i.e. ∈ c ∞ c

p∞(λc)= lim p∞(λ). λ↓λc

So p∞( ) is continuous on (0, ) iﬀ · ∞

p∞(λc) = 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 LARGE-k ASYMPTOTICS FOR pk(λ)

Suppose λ < λc. Then there exists ζ(λ) > 0 such that

−1 ζ(λ)= lim n log pn(λ) n→∞ − or more informally, p (λ) (e−ζ(λ))n. n ∼ n Proof uses subadditivity. With x1 := (x1,...,xn), recall

n n n −λA(0,x1 ) n pn+1(λ)=(n + 1)λ h(x1 )e dx1 . Z Setting q := p /(n + 1), can show q q q − , so n n+1 n m ≤ n+m 1 log q /(n 1) inf ≥ ( log q /(n 1)) := ζ as n . − n − → n 1 − n − → ∞ That ζ(λ) > 0 is a deeper result.

20 LARGE-k ASYMPTOTICS: THE SUPERCRITICAL CASE

Suppose λ > λc. Then

−(d−1)/d lim sup n log pn(λ) < 0 n→∞ −(d−1)/d lim inf n log pn(λ) > n→∞ −∞ Loosely speaking, this says that in the supercritical case pn(λ) decays exponentially in n1−1/d, whereas in the subcritical case it decays exponentially in n.

21 HIGH DIMENSIONAL ASYMPTOTICS (Penrose 1996)

d Let θd denote the volume of the unit ball in R . Suppose λ = λ(d) is chosen so that λ(d)θd = µ (this is the expected degree of the origin 0 in 0). Gλ Asymptotically as d with µ ﬁxed, the structure of C converges → ∞ 0 to that of a branching process (Z0,Z1,...) with Poisson (µ) oﬀspring distribution (Z0 = 1,Zn is nth generation size). So (Penrose 1996): p (λ(d)) p˜ (µ) := P [ ∞ Z = k]; k → k n=0 n ∞ p∞(λ(d)) ψ(µ) := P [P Zn = ]; → n=0 ∞ θdλc(d) 1. P →

22 DETAILS OF THE BRANCHING PROCESS THEORY:

∞ kk−2 p˜ (µ)= P [ Z = k]= µk−1e−kµ. k n (k 1)! n=0 X − 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 OTHER MODELS OF CONTINUUM PERCOLATION The Boolean model. Let Ξ be the union of balls of random • radius centred at the points of (with some speciﬁed radius Pλ distribution). Look at the connected components of Ξ. The random connection model. Given , let each pair (x, y) • Pλ of points of be connected with probability f( x y ) with f some Pλ | − | speciﬁed function satisfying

f( x )dx < Rd | | ∞ Z Both of these are generalizations of the model we have been considering.

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

25 HIGH-INTENSITY ASYMPTOTICS Let θ denote the volume of the unit ball in Rd. As λ , → ∞ p∞(λ) 1 and in fact (Penrose 1991) →

1 p∞(λ) p (λ) = exp( λθ) − ∼ 1 − This says that for large λ, given the unlikely event that the component of 0 containing 0 is ﬁnite, it is likely to consist of an Gλ isolated vertex. Also, for k 1, ≥ λ log p (λ)= λθ +(d 1)k log + O(1) − k+1 − k as λ with k ﬁxed (Alexander 1991). → ∞

26 References

[1] Bollob´as, 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.