## The scaling limit of critical random graphs

### L. Addario–Berry

^{1}N. Broutin

^{2}

### C. Goldschmidt

^{3}

1Department of Mathematics and Statistics McGill University

2Projet Algorithms INRIA Paris-Rocquencourt

3Department of Statistics University of Warwick

### April 5, 2011

## 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

## 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

## Evolution of random graphs

S^{n}_{1}≥S^{n}_{2}≥S^{n}_{3}. . .: sizes of connected components

Different regimes for sizes asn→ ∞:

np=1−:S^{n}_{1}=O(logn)
np=1:∀k≥1 fixedS^{n}

k= Θ(n^{2/3})
np=1+:S^{n}_{1}= Θ(n),S^{n}_{2}=O(logn)

## Evolution of random graphs

S^{n}_{1}≥S^{n}_{2}≥S^{n}_{3}. . .: sizes of connected components

Different regimes for sizes asn→ ∞:

np=1−:S^{n}

1=O(logn)

np=1:∀k≥1 fixedS^{n}

k= Θ(n^{2/3})
np=1+:S^{n}_{1}= Θ(n),S^{n}_{2}=O(logn)

## Evolution of random graphs

S^{n}_{1}≥S^{n}_{2}≥S^{n}_{3}. . .: sizes of connected components

Different regimes for sizes asn→ ∞:

np=1−:S^{n}

1=O(logn)
np=1:∀k≥1 fixedS^{n}_{k}= Θ(n^{2/3})

np=1+:S^{n}_{1}= Θ(n),S^{n}_{2}=O(logn)

## Evolution of random graphs

S^{n}_{1}≥S^{n}_{2}≥S^{n}_{3}. . .: sizes of connected components

Different regimes for sizes asn→ ∞:

np=1−:S^{n}

1=O(logn)
np=1:∀k≥1 fixedS^{n}_{k}= Θ(n^{2/3})
np=1+:S^{n}_{1}= Θ(n),S^{n}_{2}=O(logn)

## Evolution of random graphs

S^{n}_{1}≥S^{n}_{2}≥S^{n}_{3}. . .: sizes of connected components

Different regimes for sizes asn→ ∞:

np=1−:S^{n}

1=O(logn)
np=1:∀k≥1 fixedS^{n}_{k}= Θ(n^{2/3})
np=1+:S^{n}_{1}= Θ(n),S^{n}_{2}=O(logn)

## 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

## What does a component look like?

## Inside the critical window: np = 1 + λn

^{−1/3}

## for λ ∈ R

Theorem (Aldous, 1997)

(S^{n}_{i};N^{n}_{i})the size and surplus of thei-th largest component ofG_{n,p}

((n^{−2/3}S^{n}_{i};N_{i}^{n}) :i≥1)−−−→^{d}

n→∞ ((Si;N_{i}) :i≥1)

as a sequence in`^{2}_{&}={x= (x_{1},x_{2}, . . .) :x_{1}≥x_{2}≥ · · · ≥0,P

i≥1x^{2}_{i} <∞}.

t
W^{λ}(t)

W^{λ}(t) =W(t) +tλ−^{t}_{2}^{2}

t
B^{λ}(t)

B^{λ}(t) =W^{λ}(t)−inf0<s<tW^{λ}(s)

## 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

## 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

## The critical window

Parametrization of the probability for the critical window
p=1/n+λn^{−4/3}withλ∈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_{tn}2/3

n^{1/3}

t≥0

→

λt−t^{2}

2 +W(t)

t≥0

## 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

## 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

## 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

## 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

## Comparing metric spaces: Gromov–Hausdorff distance

Using isometries

Embed both spacesisometricallyinto the same bigger spaceM:

d_{GH}((M1,d_{1}); (M2,d_{2})) =infd_{H}(M1,M_{2}),

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

(x_{1},x2),(y1,y2)∈R

{|d1(x1,y1)−d2(x2,y2)|}

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

R dist(R)

## 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^{−1/2}Tn

−−−→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.

## 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

## 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(·))

## Understanding Depth-first search in graphs

X(0) =0andX(i+1)−X(i) = #children ofiminus 1

Question:

what can we change without changing the tree we obtain?

Answer:

Can add any of thea(T)edges between two verticesactive at the same time

## Understanding Depth-first search in graphs

X(0) =0andX(i+1)−X(i) = #children ofiminus 1

Question:

what can we change without changing the tree we obtain?

Answer:

Can add any of thea(T)edges between two verticesactive at the same time

## Understanding Depth-first search in graphs

X(0) =0andX(i+1)−X(i) = #children ofiminus 1

Question:

what can we change without changing the tree we obtain?

Answer:

Can add any of thea(T)edges between two verticesactive at the same time

## Understanding Depth-first search in graphs

X(0) =0andX(i+1)−X(i) = #children ofiminus 1

Question:

what can we change without changing the tree we obtain?

Answer:

Can add any of thea(T)edges between two verticesactive at the same time

## Understanding Depth-first search in graphs

X(0) =0andX(i+1)−X(i) = #children ofiminus 1

Question:

what can we change without changing the tree we obtain?

Answer:

Can add any of thea(T)edges between two verticesactive at the same time

## Understanding Depth-first search in graphs

X(0) =0andX(i+1)−X(i) = #children ofiminus 1

Question:

what can we change without changing the tree we obtain?

Answer:

Can add any of thea(T)edges between two verticesactive at the same time

## Understanding Depth-first search in graphs

X(0) =0andX(i+1)−X(i) = #children ofiminus 1

Question:

what can we change without changing the tree we obtain?

Answer:

Can add any of thea(T)edges between two verticesactive at the same time

## Understanding Depth-first search in graphs

X(0) =0andX(i+1)−X(i) = #children ofiminus 1

Question:

what can we change without changing the tree we obtain?

Answer:

Can add any of thea(T)edges between two verticesactive at the same time

## Understanding Depth-first search in graphs

X(0) =0andX(i+1)−X(i) = #children ofiminus 1

Question:

what can we change without changing the tree we obtain?

Answer:

Can add any of thea(T)edges between two verticesactive at the same time

## Understanding Depth-first search in graphs

X(0) =0andX(i+1)−X(i) = #children ofiminus 1

Question:

what can we change without changing the tree we obtain?

Answer:

Can add any of thea(T)edges between two verticesactive at the same time

## Understanding Depth-first search in graphs

X(0) =0andX(i+1)−X(i) = #children ofiminus 1

Question:

what can we change without changing the tree we obtain?

Answer:

Can add any of thea(T)edges between two verticesactive at the same time

## Understanding Depth-first search in graphs

X(0) =0andX(i+1)−X(i) = #children ofiminus 1

Question:

what can we change without changing the tree we obtain?

Answer:

Can add any of thea(T)edges between two verticesactive at the same time

## Understanding Depth-first search in graphs

X(0) =0andX(i+1)−X(i) = #children ofiminus 1

Question:

what can we change without changing the tree we obtain?

Answer:

Can add any of thea(T)edges between two verticesactive at the same time

## Understanding Depth-first search in graphs

X(0) =0andX(i+1)−X(i) = #children ofiminus 1

Question:

what can we change without changing the tree we obtain?

Answer:

Can add any of thea(T)edges between two verticesactive at the same time

## Understanding Depth-first search in graphs

X(0) =0andX(i+1)−X(i) = #children ofiminus 1

Question:

what can we change without changing the tree we obtain?

Answer:

Can add any of thea(T)edges between two verticesactive at the same time

## Understanding Depth-first search in graphs

X(0) =0andX(i+1)−X(i) = #children ofiminus 1

Question:

what can we change without changing the tree we obtain?

Answer:

Can add any of thea(T)edges between two verticesactive at the same time

## Understanding Depth-first search in graphs

X(0) =0andX(i+1)−X(i) = #children ofiminus 1

Question:

what can we change without changing the tree we obtain?

Answer:

Can add any of thea(T)edges between two verticesactive at the same time

## Understanding Depth-first search in graphs

X(0) =0andX(i+1)−X(i) = #children ofiminus 1

Question:

what can we change without changing the tree we obtain?

Answer:

Can add any of thea(T)edges between two verticesactive at the same time

## Canonical tree and sampling random graphs

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

Gn= [

T∈T^{n}

{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,p}withmvertices
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.

## Limit of canonical trees

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

(1−p)^{−a(T}^{m}^{)}∼exp(m^{−3/2}a(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˜_{m}be 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(·)).

## 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}_{m}be a uniform connected
component ofGn,ponmvertices.

T(G^{p}_{m})
n^{1/3}

−−−→d

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

## Connected graphs as marked excursions

Bijection between:

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

Connected graphs onmverticesG^{X}(X,P)

X(s)

s

## Connected graphs as marked excursions

Bijection between:

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

Connected graphs onmverticesG^{X}(X,P)

X(s)

s

## Connected graphs as marked excursions

Bijection between:

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

Connected graphs onmverticesG^{X}(X,P)

i

i X(s)

s

## Connected graphs as marked excursions

Bijection between:

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

Connected graphs onmverticesG^{X}(X,P)

i

i X(s)

s

## Connected graphs as marked excursions

Bijection between:

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

Connected graphs onmverticesG^{X}(X,P)

i

i X(s)

s

## The limit of connected components: convergence

Need bothXandH Xprovides abijection

Hprovides themetric structure

i

i i

G^{X} G^{H}

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

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

## 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.

## The limit of critical random graphs

M_{i}^{n}thei-th largest connected component ofGn,pandS^{n}_{i} its size.

M^{n}= (M^{n}_{1},M_{2}^{n}, . . .)as a sequence of metric spaces, with distance

d(A,B) =

X

i≥1

dGH(Ai,Bi)^{4}

1/4

n^{−2/3}S^{n}= (n^{−2/3}S^{n}_{1}, . . .)in

`^{2}_{&}={(x1,x2, . . .) :x1≥0, . . . ,P

i≥1x^{2}_{i} <∞}.

Theorem

(n^{−1/3}M^{n},n^{−2/3}S^{n})−−−→^{d}

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

GivenS= (S1,S2, . . .),(M1,M2, . . .)are independentg(˜e^{(S}^{i}^{)},Pi).