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## The scaling limit of critical random graphs

1 N. Broutin2

### C. Goldschmidt

3

1Department of Mathematics and Statistics McGill University

2Projet Algorithms INRIA Paris-Rocquencourt

3Department of Statistics University of Warwick

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## Erd˝os–R´enyi random graphs

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

Gn,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

Sn1Sn2Sn3. . .: sizes of connected components

Different regimes for sizes asn→ ∞:

np=1:Sn1=O(logn) np=1:∀k1 fixedSn

k= Θ(n2/3) np=1+:Sn1= Θ(n),Sn2=O(logn)

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## Evolution of random graphs

Sn1Sn2Sn3. . .: sizes of connected components

Different regimes for sizes asn→ ∞:

np=1:Sn

1=O(logn)

np=1:∀k1 fixedSn

k= Θ(n2/3) np=1+:Sn1= Θ(n),Sn2=O(logn)

(6)

## Evolution of random graphs

Sn1Sn2Sn3. . .: sizes of connected components

Different regimes for sizes asn→ ∞:

np=1:Sn

1=O(logn) np=1:∀k1 fixedSnk= Θ(n2/3)

np=1+:Sn1= Θ(n),Sn2=O(logn)

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## Evolution of random graphs

Sn1Sn2Sn3. . .: sizes of connected components

Different regimes for sizes asn→ ∞:

np=1:Sn

1=O(logn) np=1:∀k1 fixedSnk= Θ(n2/3) np=1+:Sn1= Θ(n),Sn2=O(logn)

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## Evolution of random graphs

Sn1Sn2Sn3. . .: sizes of connected components

Different regimes for sizes asn→ ∞:

np=1:Sn

1=O(logn) np=1:∀k1 fixedSnk= Θ(n2/3) 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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−1/3

## for λ ∈ R

Theorem (Aldous, 1997)

(Sni;Nni)the size and surplus of thei-th largest component ofGn,p

((n−2/3Sni;Nin) :i≥1)−−−→d

n→∞ ((Si;Ni) :i≥1)

as a sequence in`2&={x= (x1,x2, . . .) :x1x2≥ · · · ≥0,P

i≥1x2i <∞}.

t Wλ(t)

Wλ(t) =W(t) +t22

t Bλ(t)

Bλ(t) =Wλ(t)inf0<s<tWλ(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

(13)

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

c=1: approx recurrent c>1:Ai→ ∞

Branching process point of view

New vertices:EBin(niAi,p)c c<1: subcritical

c=1: critical c>1: supercritical

(14)

## 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∼tn2/3,

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

Central Limit Theorem for martingales:

Atn2/3

n1/3

t≥0

λt−t2

2 +W(t)

t≥0

(15)

1

2

3

4

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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 infs∈[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

(19)

## Comparing metric spaces: Gromov–Hausdorff distance

Using isometries

Embed both spacesisometricallyinto the same bigger spaceM:

dGH((M1,d1); (M2,d2)) =infdH(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.

(21)

## 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

LetTnbe a Cayley tree of sizen.

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(·))

(23)

## 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?

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

(24)

## 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?

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

(25)

## 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?

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

(26)

## 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?

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

(27)

## 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?

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

(28)

## 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?

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

(29)

## 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?

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

(30)

## 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?

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

(31)

## 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?

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

(32)

## 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?

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

(33)

## 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?

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

(34)

## 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?

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

(35)

## 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?

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

(36)

## 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?

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

(37)

## 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?

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

(38)

## 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?

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

(39)

## 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?

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

(40)

## 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?

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

(41)

## Canonical tree and sampling random 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 treeTyields2a(T)graphs:#{G:T(G) =T}=2a(T)

Uniform connected component ofGn,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

(42)

## Limit of canonical trees

In the critical regime:p∼1/nandm∼n2/3:

(1−p)−a(Tm)∼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(·)).

(43)

## The limit of tilted trees

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

Theorem

Letp∼1/nandm∼σn2/3,σ >0. LetGpmbe a uniform connected component ofGn,ponmvertices.

T(Gpm) n1/3

−−−→d

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

(44)

## Connected graphs as marked excursions

Bijection between:

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

Connected graphs onmverticesGX(X,P)

X(s)

s

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## Connected graphs as marked excursions

Bijection between:

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

Connected graphs onmverticesGX(X,P)

X(s)

s

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## Connected graphs as marked excursions

Bijection between:

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

Connected graphs onmverticesGX(X,P)

i

i X(s)

s

(47)

## Connected graphs as marked excursions

Bijection between:

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

Connected graphs onmverticesGX(X,P)

i

i X(s)

s

(48)

## Connected graphs as marked excursions

Bijection between:

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

Connected graphs onmverticesGX(X,P)

i

i X(s)

s

(49)

## The limit of connected components: convergence

Need bothXandH Xprovides abijection

Hprovides themetric structure

i

i i

GX GH

Limit using the “Depth-First construction”GX

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

(50)

## Limit of connected components: construction

•ξ

Metric characterization using the “height construction”GH

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

(51)

## The limit of critical random graphs

Minthei-th largest connected component ofGn,pandSni 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≥1x2i <∞}.

Theorem

(n−1/3Mn,n−2/3Sn)−−−→d

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

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

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