The scaling limit of critical random graphs

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The scaling limit of critical random graphs

L. Addario–Berry

¹ N. Broutin²

C. Goldschmidt

³

1Department of Mathematics and Statistics McGill University

2Projet Algorithms INRIA Paris-Rocquencourt

3Department of Statistics University of Warwick

April 5, 2011

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Plan

1

Random graphs

2

Exploration and branching processes

3

Large random trees

4

Comparing trees and Gromov–Hausdorff distance

5

Continuum random tree

6

Depth-first search and metric structure of large graphs

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Erd˝os–R´enyi random graphs

nlabelled vertices{1,2, . . . ,n}

G_n,p: each edge is present with probabilityp∈[0,1]

1 2

3

4

5

6 7

8

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Evolution of random graphs

Sⁿ₁≥Sⁿ₂≥Sⁿ₃. . .: sizes of connected components

Different regimes for sizes asn→ ∞:

np=1−:Sⁿ₁=O(logn) np=1:∀k≥1 fixedSⁿ

k= Θ(n^2/3) np=1+:Sⁿ₁= Θ(n),Sⁿ₂=O(logn)

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Evolution of random graphs

np=1−:Sⁿ

1=O(logn)

np=1:∀k≥1 fixedSⁿ

k= Θ(n^2/3) np=1+:Sⁿ₁= Θ(n),Sⁿ₂=O(logn)

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Evolution of random graphs

np=1−:Sⁿ

1=O(logn) np=1:∀k≥1 fixedSⁿ_k= Θ(n^2/3)

np=1+:Sⁿ₁= Θ(n),Sⁿ₂=O(logn)

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Evolution of random graphs

np=1−:Sⁿ

1=O(logn) np=1:∀k≥1 fixedSⁿ_k= Θ(n^2/3) np=1+:Sⁿ₁= Θ(n),Sⁿ₂=O(logn)

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Evolution of random graphs

np=1−:Sⁿ

1=O(logn) np=1:∀k≥1 fixedSⁿ_k= Θ(n^2/3) np=1+:Sⁿ₁= Θ(n),Sⁿ₂=O(logn)

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Component sizes and component structure

Important observation:

Given its sizem, a connected component ofG(n,p)is distributed asG(m,p) conditioned on being connected.

⇒Study separately:

sizes of connected components their structure given the size

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What does a component look like?

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Inside the critical window: np = 1 + λn

^−1/3

for λ ∈ R

Theorem (Aldous, 1997)

(Sⁿ_i;Nⁿ_i)the size and surplus of thei-th largest component ofG_n,p

((n^−2/3Sⁿ_i;N_iⁿ) :i≥1)−−−→^d

n→∞ ((Si;N_i) :i≥1)

as a sequence in`²_&={x= (x₁,x₂, . . .) :x₁≥x₂≥ · · · ≥0,P

i≥1x²_i <∞}.

t W^λ(t)

W^λ(t) =W(t) +tλ−^t₂²

t B^λ(t)

B^λ(t) =W^λ(t)−inf0<s<tW^λ(s)

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Graph exploration and a branching process/random walk

Exposure vertices one after another:

Initialization:

one active vertexA0=1 At each time step:

Ai+1=Ai−1+Bin(n−i−Ai,p)

The walkAiand sizes of components Fork≥0, setξk:=inf{i:Ai=1−k}

ξk−ξk−1size of thek-th explored component n−i−Ai

unexplored vertices

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Asymptotics for A

_i

and phases in random graphs

Random walk with step distribution

Ai+1−Ai=Bin(n−i−Ai,p)−1

The regimes whennp=c

E[A_i+1−A_i] =EBin(n−i−A_i,p)−1≈c−1 c<1:A_i→ −∞

c=1: approx recurrent c>1:Ai→ ∞

Branching process point of view

New vertices:EBin(n−i−A_i,p)≈c c<1: subcritical

c=1: critical c>1: supercritical

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The critical window

Parametrization of the probability for the critical window p=1/n+λn^−4/3withλ∈R

Ai+1−Ai=Bin

n−i−Ai,1

n +λn^−4/3

−1

So fori∼tn^2/3,

E[Ai+1−Ai]∼(λ−t)n^−1/3 and Var[Ai+1−Ai]∼1

Central Limit Theorem for martingales:

A_tn2/3

n^1/3

t≥0

→

λt−t²

2 +W(t)

t≥0

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Strategy to describe random graphs

1

Decompose into connected components

2

Extract a tree from each connected component

3

Describe the trees

4

Describe how to put back surplus edges

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Real trees

x y

z t

x y

z t

The tree in a continuous excursion

continuous excursionf(0) =f(1) =0,f(s)>0,s∈(0,1) metricdf(x,y) =f(x) +f(y)−2 inf_s∈[x,y]f(s).

x∼f yiffdf(x,y) =0

the metric space([0,1]/∼f,df)has a tree structure

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Comparing subsets of a metric space: Hausdorff distance

For two pointsx,y∈(M,d)

d(x,y) =inf{ >0:y∈B(x, )andx∈B(y, )}

x y

For two subsetsP,Qof a metric space(M,d):Hausdorffdistance

dH(P,Q) =inf







>0:P⊆ [

x∈Q

B(x, )andQ⊆[

y∈P

B(y, )





 .

P

Q

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Comparing subsets of a metric space: Hausdorff distance

For two pointsx,y∈(M,d)

d(x,y) =inf{ >0:y∈B(x, )andx∈B(y, )}

x y

For two subsetsP,Qof a metric space(M,d):Hausdorffdistance

dH(P,Q) =inf







>0:P⊆ [

x∈Q

B(x, )andQ⊆[

y∈P

B(y, )





 .

P

Q

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Comparing metric spaces: Gromov–Hausdorff distance

Using isometries

Embed both spacesisometricallyinto the same bigger spaceM:

d_GH((M1,d₁); (M2,d₂)) =infd_H(M1,M₂),

where the infimum is over all metric spacesMcontaining both(M1,d1)and (M2,d2).

Using distortions of correspondences

Map each space inside the other using acorrespondenceR ⊆M1×M2

dist(R) = sup

(x₁,x2),(y1,y2)∈R

{|d1(x1,y1)−d2(x2,y2)|}

dGH((M1,d1); (M2,d2)) = 1 2inf

R dist(R)

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The Brownian continuum random tree (CRT)

LetT(2e)be the real tree encoded bytwicea standard Brownian excursione Theorem (Aldous)

LetTnbe a Cayley tree of sizen, seen as a metric space with the graph distance. Then,

n^−1/2Tn

−−−→d n→∞ T(2e) with the Gromov–Hausdorff distance.

Some remarks:

extends to all Galton–Watson trees with finite variance progeny the trees in random maps

if infinite variance: L´evy trees, stable trees.

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Representation of trees

A canonical order for the nodes:

sort children by increasing label

Depth-first order 1

3 5 8

9 6 7

2 4

1

2 6 7

3 8 9

4 5

Three different encodings of trees as nonnegative paths:

Contour processC Height processH Depth-first walkX

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Walks associated with large random trees

Aldous, Le Gall, Marckert–Mokkadem

LetTnbe a Cayley tree of sizen.

Theorem (Marckert–Mokkadem)

Lete= (e(t),0≤t≤1)be a standard Brownian excursion. Then, X(bn· c)

√n ;H(bn· c)

√n ;C(b2n· c)

√n

d

−−−→n→∞ (e(·);2e(·);2e(·))

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Understanding Depth-first search in graphs

Partition the all the graphsGaccording to their canonical treeT(G)

Gn= [

T∈Tⁿ

{G:T(G) =T}

Each graphGhas a unique canonical treeT(G)

Each canoninal treeTyields2^a(T)graphs:#{G:T(G) =T}=2^a(T)

Uniform connected component ofG_n,pwithmvertices PickT˜_m=Twith probability∝(1−p)^−a(T) Add each allowed edge with probabilityp.

Uniform connected connected graph withmvertices andm−1+sedges PickT˜_m=Twith probability∝ ^a(T)_s

Addsrandom allowed edges.

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Limit of canonical trees

In the critical regime:p∼1/nandm∼n^2/3:

(1−p)^−a(T^m⁾∼exp(m^−3/2a(Tm))∼exp(R1 0 e(s)ds)

Definition (Tilted excursion)

Letebe a standard Brownian excursion:

P(˜e∈ B) = Eh

1[e∈ B]exp(R1

0 e(s)ds)i Eh

exp(R1

0 e(s)ds)i .

LetT˜_mbe picked such thatP T˜_m=T

∝(1−m^−3/2)^−a(T)

Theorem X˜ⁿ(bn· c)

√n ;C˜ⁿ(b2n· c)

√n ;H˜ⁿ(bn· c)

√n

!

−−−→d

n→∞ (˜e(·);2˜e(·);2˜e(·)).

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The limit of tilted trees

LetT(˜e^(σ))the real tree encoded by a tilted excursion˜e^(σ)of lengthσ.

Theorem

Letp∼1/nandm∼σn^2/3,σ >0. LetG^p_mbe a uniform connected component ofGn,ponmvertices.

T(G^p_m) n^1/3

−−−→d

n→∞ T(2˜e^(σ)), with the Gromov–Hausdorff distance.

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Connected graphs as marked excursions

Bijection between:

Excursion of lengthmwith some points underX:(X,P).

Connected graphs onmverticesG^X(X,P)

X(s)

s

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Need bothXandH Xprovides abijection

Hprovides themetric structure

i

i i

G^X G^H

Limit using the “Depth-First construction”G^X

m^−1/2X˜^m(bm· c)→˜eand marks converge to a Poisson point processP

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Limit of connected components: construction

•ξ

Metric characterization using the “height construction”G^H

T(2˜e)where each point(ξx, ξy)ofPidentifies the leafξxwith the point at distanceξyfrom the root on the pathJ0, ξxK.

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The limit of critical random graphs

M_iⁿthei-th largest connected component ofGn,pandSⁿ_i its size.

Mⁿ= (Mⁿ₁,M₂ⁿ, . . .)as a sequence of metric spaces, with distance

d(A,B) =



 X

i≥1

dGH(Ai,Bi)⁴





1/4

n^−2/3Sⁿ= (n^−2/3Sⁿ₁, . . .)in

`²_&={(x1,x2, . . .) :x1≥0, . . . ,P

i≥1x²_i <∞}.

Theorem

(n^−1/3Mⁿ,n^−2/3Sⁿ)−−−→^d

n→∞ (M,S) where Sis the ordered sequence of excursion lengths ofB^λ

GivenS= (S1,S2, . . .),(M1,M2, . . .)are independentg(˜e^(Sⁱ⁾,Pi).