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Asymptotic analysis of large random graphs Marion Sciauveau Joint - - PowerPoint PPT Presentation

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Asymptotic analysis of large random graphs Marion Sciauveau Joint work with J-F. Delmas and J-S. Dhersin CERMICS (ENPC) and LAGA (Paris 13) S eminaire des doctorants du CERMICS ENPC - 13 juin 2018 Marion Sciauveau Asymptotic analysis of

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Asymptotic analysis of large random graphs

Marion Sciauveau Joint work with J-F. Delmas and J-S. Dhersin

CERMICS (ENPC) and LAGA (Paris 13)

S´ eminaire des doctorants du CERMICS ENPC - 13 juin 2018

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 1 / 24

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Introduction/ Motivation

Motivation: Social networks or internet can be represented by large random graphs. Understanding their structure is therefore an important issue in mathematics. The theory of graph limits is recent and developped by Lov´ asz and Szegedy (2006) and Borgs et al. (2008).

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 2 / 24

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SLIDE 3

Introduction/ Motivation

Motivation: Social networks or internet can be represented by large random graphs. Understanding their structure is therefore an important issue in mathematics. The theory of graph limits is recent and developped by Lov´ asz and Szegedy (2006) and Borgs et al. (2008). Problems: How can we describe these large graphs ? How to characterize the convergence of sequences of graphs when the number of nodes goes to infinity ? What is the best stochastic model of random graphs to approximate these large graphs ?

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 2 / 24

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1

Introduction

2

Convergence of dense graph sequences

3

Main results

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 3 / 24

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Some notations for graphs

A finite graph G is an ordered pair (V (G), E(G)) where V (G) is the set of v(G) < +∞ vertices. E(G) is the set of e(G) edges among the collection of v(G)

2

  • unordered pairs of

vertices. G is simple if it has no self-loops and no multiple edges. G is dense when the number of edges is close to the maximal number of edges. G can be caracterized by its adjacency matrix.

1 2 3 4 5

      1 1 1 1 1 1 1 1 1 1 1 1       Figure: A graph with 5 vertices, its adjacency matrix and its pixel picture.

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 4 / 24

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Large random graphs: first example

Erd¨

  • s-R´

enyi graph Gn(p): random graph such that V (Gn(p)) = [n], edges occur independently with the same probability p, 0 < p < 1.

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 5 / 24

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Large random graphs: first example

Erd¨

  • s-R´

enyi graph Gn(p): random graph such that V (Gn(p)) = [n], edges occur independently with the same probability p, 0 < p < 1.

(a) n = 10

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 5 / 24

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Large random graphs: first example

Erd¨

  • s-R´

enyi graph Gn(p): random graph such that V (Gn(p)) = [n], edges occur independently with the same probability p, 0 < p < 1.

(a) n = 10 (b) n = 100

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 5 / 24

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Large random graphs: first example

Erd¨

  • s-R´

enyi graph Gn(p): random graph such that V (Gn(p)) = [n], edges occur independently with the same probability p, 0 < p < 1.

(a) n = 10 (b) n = 100 (c) n = 1000

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 5 / 24

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Large random graphs: first example

Erd¨

  • s-R´

enyi graph Gn(p): random graph such that V (Gn(p)) = [n], edges occur independently with the same probability p, 0 < p < 1.

(a) n = 10 (b) n = 100 (c) n = 1000 (d) W = 1

2

Figure: Erd¨

  • s-R´

enyi graph with parameter p = 1

2 ant its limit Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 5 / 24

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Large random graphs: second example

Randomly grown uniform attachment graph GUAn: random graph such that V (GUAn) = [n], Generation of the random graph: Start with a single node. Create a new node. Connect every pair of nonadjacent nodes with probability 1

k where k is the

current number of nodes.

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 6 / 24

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Large random graphs: second example

Randomly grown uniform attachment graph GUAn: random graph such that V (GUAn) = [n], Generation of the random graph: Start with a single node. Create a new node. Connect every pair of nonadjacent nodes with probability 1

k where k is the

current number of nodes. (a) n = 10

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 6 / 24

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Large random graphs: second example

Randomly grown uniform attachment graph GUAn: random graph such that V (GUAn) = [n], Generation of the random graph: Start with a single node. Create a new node. Connect every pair of nonadjacent nodes with probability 1

k where k is the

current number of nodes. (a) n = 10 (b) n = 100

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 6 / 24

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SLIDE 14

Large random graphs: second example

Randomly grown uniform attachment graph GUAn: random graph such that V (GUAn) = [n], Generation of the random graph: Start with a single node. Create a new node. Connect every pair of nonadjacent nodes with probability 1

k where k is the

current number of nodes. (a) n = 10 (b) n = 100 (c) n = 1000

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 6 / 24

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SLIDE 15

Large random graphs: second example

Randomly grown uniform attachment graph GUAn: random graph such that V (GUAn) = [n], Generation of the random graph: Start with a single node. Create a new node. Connect every pair of nonadjacent nodes with probability 1

k where k is the

current number of nodes. (a) n = 10 (b) n = 100 (c) n = 1000 (d) W(x, y) =

1 − max(x, y)

Figure: Randomly grown uniform attachment graph ant its limit

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 6 / 24

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From graphs to graphons

Graph F Graphon W Vertex set V (F ) σ-finite measure space ([0, 1], B([0, 1]), λ) Adjacency matrix A : V (F ) × V (F ) → {0, 1} Symmetric, measurable function W : [0, 1]2 → [0, 1]

Figure: Exponential graphon: W (x, y) =

ex+y 1+ex+y

We denote by W the space of all graphons.

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 7 / 24

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1

Introduction

2

Convergence of dense graph sequences

3

Main results

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 8 / 24

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Motivation

We consider a sequence of dense finite graphs (Gn : n ∈ N∗) i.e. such that limn→+∞

2e(Gn) v(Gn)(v(Gn)−1) > 0.

Questions: When can we say that this sequence is convergent ? How can we caracterize the convergence ? What is the limit ? How can we generate a sequence of graphs whose limit is a given graphon ?

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 9 / 24

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A key object: homomorphism densities

Let F and G be two simple graphs and let ϕ be a map from V (F) to V (G). We define several types of homomorphism:

Set F Definition Density Hom(F, G) ϕ such that {i, j} ∈ E(F ) ⇒ {ϕ(i), ϕ(j)} ∈ E(G) t(F, G) = |Hom(F,G)|

v(G)v(F )

Inj(F, G) ϕ ∈ Hom(F, G) injective tinj(F, G) = |Inj(F,G)|

Av(F ) v(G)

Ind(F, G) ϕ injective such that {i, j} ∈ E(F ) ⇔ {ϕ(i), ϕ(j)} ∈ E(G) tind(F, G) = |Ind(F,G)|

Av(F ) v(G)

F G

ϕ ϕ ϕ

Figure: ϕ is as injective homomorphism but not an induced homomorphism.

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 10 / 24

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How to charaterize the convergence of sequence of dense graphs ?

Theorem [Lov´ asz, Szegedy (2006)] A sequence of simple graphs (Gn : n ∈ N∗) is called convergent if the sequence (tinj(F, Gn) : n ∈ N∗) has a limit for every simple graph F. The limit can be represented as a graphon.

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 11 / 24

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How to charaterize the convergence of sequence of dense graphs ?

Theorem [Lov´ asz, Szegedy (2006)] A sequence of simple graphs (Gn : n ∈ N∗) is called convergent if the sequence (tinj(F, Gn) : n ∈ N∗) has a limit for every simple graph F. The limit can be represented as a graphon. Theorem [Lov´ asz, Szegedy (2006)] A sequence of graphs (Gn : n ∈ N∗) is said to converge to a graphon W if for every finite simple graph F, we have lim

n→+∞ tinj(F, Gn) = t(F, W),

where t(F, W) = tinj(F, W) =

  • [0,1]v(F )
  • {i,j}∈E(F )

W(xi, xj)

  • k∈V (F )

dxk.

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 11 / 24

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How to understand t(F, W) ?

Let F be a finite simple graph with p vertices. let G be a finite simple graph with n vertices then we have the density of F in G: t(F, G) = 1 np

  • β∈Sn,p
  • {i,j}∈E(F )

1{{βi,βj}∈E(G)}. let W be a graphon then we have the density of F in W: t(F, W) = 1 λ([0, 1])p

  • [0,1]p
  • k∈V (F )
  • {i,j}∈E(F )

W(xi, xj).

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 12 / 24

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Generating dense graphs with a given number of vertices from a graphon

Given a graphon, W , we construct a W -random generated Gn(W ), as follows:

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 13 / 24

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Generating dense graphs with a given number of vertices from a graphon

Given a graphon, W , we construct a W -random generated Gn(W ), as follows: vertex set [n]

1 2 3 4

Figure: G4(W )

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 13 / 24

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Generating dense graphs with a given number of vertices from a graphon

Given a graphon, W , we construct a W -random generated Gn(W ), as follows: vertex set [n] X = (Xi : i ∈ N∗) i.i.d r.v. uniform on [0, 1] 1 X1 X2 X3 X4

1 2 3 4

Figure: G4(W )

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 13 / 24

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Generating dense graphs with a given number of vertices from a graphon

Given a graphon, W , we construct a W -random generated Gn(W ), as follows: vertex set [n] X = (Xi : i ∈ N∗) i.i.d r.v. uniform on [0, 1] 1 X1 X2 X3 X4 {i, j} is an edge in Gn(W ) with probability W (Xi, Xj).

1 2 3 4

Figure: G4(W )

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 13 / 24

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SLIDE 27

Generating dense graphs with a given number of vertices from a graphon

Given a graphon, W , we construct a W -random generated Gn(W ), as follows: vertex set [n] X = (Xi : i ∈ N∗) i.i.d r.v. uniform on [0, 1] 1 X1 X2 X3 X4 {i, j} is an edge in Gn(W ) with probability W (Xi, Xj).

1 2 3 4

Figure: G4(W )

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 13 / 24

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Generating dense graphs with a given number of vertices from a graphon

Given a graphon, W , we construct a W -random generated Gn(W ), as follows: vertex set [n] X = (Xi : i ∈ N∗) i.i.d r.v. uniform on [0, 1] 1 X1 X2 X3 X4 {i, j} is an edge in Gn(W ) with probability W (Xi, Xj).

1 2 3 4

Figure: G4(W )

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 13 / 24

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SLIDE 29

Generating dense graphs with a given number of vertices from a graphon

Given a graphon, W , we construct a W -random generated Gn(W ), as follows: vertex set [n] X = (Xi : i ∈ N∗) i.i.d r.v. uniform on [0, 1] 1 X1 X2 X3 X4 {i, j} is an edge in Gn(W ) with probability W (Xi, Xj).

1 2 3 4

Figure: G4(W )

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 13 / 24

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SLIDE 30

Generating dense graphs with a given number of vertices from a graphon

Given a graphon, W , we construct a W -random generated Gn(W ), as follows: vertex set [n] X = (Xi : i ∈ N∗) i.i.d r.v. uniform on [0, 1] 1 X1 X2 X3 X4 {i, j} is an edge in Gn(W ) with probability W (Xi, Xj).

1 2 3 4

Figure: G4(W )

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 13 / 24

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SLIDE 31

Generating dense graphs with a given number of vertices from a graphon

Given a graphon, W , we construct a W -random generated Gn(W ), as follows: vertex set [n] X = (Xi : i ∈ N∗) i.i.d r.v. uniform on [0, 1] 1 X1 X2 X3 X4 {i, j} is an edge in Gn(W ) with probability W (Xi, Xj).

1 2 3 4

Figure: G4(W )

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 13 / 24

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SLIDE 32

Generating dense graphs with a given number of vertices from a graphon

Given a graphon, W , we construct a W -random generated Gn(W ), as follows: vertex set [n] X = (Xi : i ∈ N∗) i.i.d r.v. uniform on [0, 1] 1 X1 X2 X3 X4 {i, j} is an edge in Gn(W ) with probability W (Xi, Xj).

1 2 3 4

Figure: G4(W )

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 13 / 24

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SLIDE 33

1

Introduction

2

Convergence of dense graph sequences

3

Main results

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 14 / 24

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SLIDE 34

A preliminary result

Let W ∈ W be a graphon (Gn(W) : n ∈ N∗) be the associated W-random graphs.

Proposition

For all finite simple graph F and n ≥ v(F), we have E [tinj(F, Gn(W))] = t(F, W).

Proof: E

  • tinj(F, Gn(W))
  • =

1 Ap

n

  • β∈Sn,p

E  

  • {i,j}∈E(F )

1{{βi,βj}∈E(Gn(W ))}   = 1 Ap

n

  • β∈Sn,p

E  E  

  • {i,j}∈E(F )

1{{βi,βj}∈E(Gn(W ))}

  • X

    = 1 Ap

n

  • β∈Sn,p

E  

  • {i,j}∈E(F )

W(Xi, Xj)   = t(F, W).

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 15 / 24

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An overview of existing results

Let W ∈ W and (Gn(W) : n ∈ N∗) the associated sequence of W-random graphs.

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 16 / 24

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An overview of existing results

Let W ∈ W and (Gn(W) : n ∈ N∗) the associated sequence of W-random graphs. Questions: What is the limit of this sequence ?

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 16 / 24

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An overview of existing results

Let W ∈ W and (Gn(W) : n ∈ N∗) the associated sequence of W-random graphs. Questions: What is the limit of this sequence ? Theorem [Borgs, Chayes, Lov´ asz, S´

  • s and Vesztergombi (2008)]

We have almost surely, for every finite simple graph F, lim

n→+∞ tinj(F, Gn(W)) = t(F, W).

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 16 / 24

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SLIDE 38

An overview of existing results

Let W ∈ W and (Gn(W) : n ∈ N∗) the associated sequence of W-random graphs. Questions: What is the limit of this sequence ? Theorem [Borgs, Chayes, Lov´ asz, S´

  • s and Vesztergombi (2008)]

We have almost surely, for every finite simple graph F, lim

n→+∞ tinj(F, Gn(W)) = t(F, W).

What are the fluctuations associated to this convergence ?

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 16 / 24

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SLIDE 39

An overview of existing results

Let W ∈ W and (Gn(W) : n ∈ N∗) the associated sequence of W-random graphs. Questions: What is the limit of this sequence ? Theorem [Borgs, Chayes, Lov´ asz, S´

  • s and Vesztergombi (2008)]

We have almost surely, for every finite simple graph F, lim

n→+∞ tinj(F, Gn(W)) = t(F, W).

What are the fluctuations associated to this convergence ? Theorem [F´ eray, M´ eliot and Nikeghbali (2017)] We have the following convergence in distribution: √n (tinj(F, Gn(W)) − t(F, W))

L

− →

n→∞ N

  • 0, σ(F)2

.

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 16 / 24

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What happens when W ≡ p ?

F is a finite simple graphs with p vertices and e edges W ≡ p with 0 < p < 1 We have: t(F, W) =

  • [0,1]v(F )
  • {i,j}∈E(F )

W(xi, xj)

  • =p
  • k∈V (F )

dxk = pe. and σ(F)2 = 0.

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 17 / 24

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What happens when W ≡ p ?

F is a finite simple graphs with p vertices and e edges W ≡ p with 0 < p < 1 We have: t(F, W) =

  • [0,1]v(F )
  • {i,j}∈E(F )

W(xi, xj)

  • =p
  • k∈V (F )

dxk = pe. and σ(F)2 = 0. Theorem [Nowicki (1989) or Nowicki and Janson (1991)] We have the following convergence in distribution: √n (tinj(F, Gn(W)) − pe)

L

− →

n→∞ N (0, 0) .

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 17 / 24

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SLIDE 42

What happens when W ≡ p ?

F is a finite simple graphs with p vertices and e edges W ≡ p with 0 < p < 1 We have: t(F, W) =

  • [0,1]v(F )
  • {i,j}∈E(F )

W(xi, xj)

  • =p
  • k∈V (F )

dxk = pe. and σ(F)2 = 0. Theorem [Nowicki (1989) or Nowicki and Janson (1991)] We have the following convergence in distribution: n (tinj(F, Gn(W)) − pe)

L

− →

n→∞ N

  • 0, ˜

σ(F)2 .

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 17 / 24

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Generalization for partially labeled graphs

F and G are two simple graphs and ϕ is a map from V (F) to V (G) k ∈ [v(F)] ℓ = (ℓ1, . . . , ℓk) ∈ Sv(F ),k s.t. ℓi ∈ [v(F)] and ℓ1, . . . , ℓk are all distinct α = (α1, . . . , αk) ∈ Sv(G),k s.t. αi ∈ [v(G)] and α1, . . . , αk are all distinct.

Set F Definition Density Inj(F ℓ, Gα) ϕ ∈ Inj(F, G) such that ϕ(ℓi) = αi, for all i ∈ [k] tInj(F ℓ, Gα) =

  • Inj(F ℓ,Gα)
  • Av(F )−k

v(G)−k

F ℓ Gα

ℓ1 ℓ2 α2 α1

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 18 / 24

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Our results

W ∈ W and (Gn(W) : n ∈ N∗) the associated sequence of W-random graphs d ≥ 1 and F = (Fm : 1 ≤ m ≤ d) a sequence of finite simple graphs with the same number of nodes p k ∈ [p] and ℓ ∈ Sn,p We define the random probability measure ΓF,ℓ

n

by, ΓF ,ℓ

n

= tinj(F ℓ, Gα

n(W))

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 19 / 24

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SLIDE 45

Our results

W ∈ W and (Gn(W) : n ∈ N∗) the associated sequence of W-random graphs d ≥ 1 and F = (Fm : 1 ≤ m ≤ d) a sequence of finite simple graphs with the same number of nodes p k ∈ [p] and ℓ ∈ Sn,p We define the random probability measure ΓF,ℓ

n

by, for all g ∈ B+([0, 1]d): ΓF ,ℓ

n (g) =

g

  • tinj(F ℓ, Gα

n(W))

  • Marion Sciauveau

Asymptotic analysis of large random graphs JPS - 18 mai 19 / 24

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SLIDE 46

Our results

W ∈ W and (Gn(W) : n ∈ N∗) the associated sequence of W-random graphs d ≥ 1 and F = (Fm : 1 ≤ m ≤ d) a sequence of finite simple graphs with the same number of nodes p k ∈ [p] and ℓ ∈ Sn,p We define the random probability measure ΓF,ℓ

n

by, for all g ∈ B+([0, 1]d): ΓF ,ℓ

n (g) =

1 Ak

n

  • α∈Sn,k

g

  • tinj(F ℓ, Gα

n(W))

  • Marion Sciauveau

Asymptotic analysis of large random graphs JPS - 18 mai 19 / 24

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SLIDE 47

Our results

W ∈ W and (Gn(W) : n ∈ N∗) the associated sequence of W-random graphs d ≥ 1 and F = (Fm : 1 ≤ m ≤ d) a sequence of finite simple graphs with the same number of nodes p k ∈ [p] and ℓ ∈ Sn,p We define the random probability measure ΓF,ℓ

n

by, for all g ∈ B+([0, 1]d): ΓF ,ℓ

n (g) =

1 Ak

n

  • α∈Sn,k

g

  • tinj(F ℓ, Gα

n(W))

  • Theorem

Invariance principle: (ΓF,ℓ

n

: n ∈ N∗) converges a.s. for the weak topology towards a deterministic probability measure ΓF,ℓ

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 19 / 24

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SLIDE 48

Our results

W ∈ W and (Gn(W) : n ∈ N∗) the associated sequence of W-random graphs d ≥ 1 and F = (Fm : 1 ≤ m ≤ d) a sequence of finite simple graphs with the same number of nodes p k ∈ [p] and ℓ ∈ Sn,p We define the random probability measure ΓF,ℓ

n

by, for all g ∈ B+([0, 1]d): ΓF ,ℓ

n (g) =

1 Ak

n

  • α∈Sn,k

g

  • tinj(F ℓ, Gα

n(W))

  • Theorem

Invariance principle: (ΓF,ℓ

n

: n ∈ N∗) converges a.s. for the weak topology towards a deterministic probability measure ΓF,ℓ and its fluctuations: for all g ∈ C2([0, 1]d), we have: √n

  • ΓF,ℓ

n (g) − ΓF,ℓ(g)

  • L

− →

n→∞ N

  • 0, σF,ℓ(g)2

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 19 / 24

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SLIDE 49

A limiting gaussian process

Let F be the set of all simple finite graphs.

Theorem

We have the following convergence of finite-dimensional distributions: √n (tinj(F, Gn) − t(F, W)) : F ∈ F

  • (fdd)

− − − − − →

n→+∞

Θ, where Θ = (ΘF : F ∈ F) is a centered Gaussian process.

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 20 / 24

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SLIDE 50

Central limit theorem for other homomorphism densities

Let F be a finite simple graph with p = v(F) and W ∈ W a graphon.

1

Because |tinj(F, G) − t(F, G)| ≤ 1 n

  • p

2

  • ,

we have the following convergence in distribution:

Theorem

√n (t(F, Gn) − t(F, W))

L

− →

n→∞ N

  • 0, σ(F)2

,

2

Because tind(F, G) =

  • F ′:F ≤F ′

(−1)e(F ′)−e(F )tinj(F ′, G), we have the following convergence in distribution:

Theorem

√n (tind(F, Gn) − tind(F, W))

L

− →

n→∞ N

  • 0, σind(F)2

.

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 21 / 24

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SLIDE 51

Asymptotics for the empirical degrees CDF (1)

Let W be a graphon. We define its normalized degree function D by: D(x) =

  • [0,1]

W(x, y)dy, ∀x ∈ [0, 1]. Let y ∈ (0, 1). Recall ΓF ,ℓ

n (g) =

1 Ak

n

  • α∈Sn,k

g

  • tinj(F ℓ, Gα

n(W))

  • Then with F = K2, ℓ ∈ {1, 2}, k = 1 and g(x) = 1[0,D(y)](x), we get that

ΓF ,ℓ

n (g) = 1

n

n

  • i=1

1

D(n)

i

≤D(y) ,

where D(n)

i

= tinj(K•

2, Gi n(W)), for all i ∈ [n].

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 22 / 24

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SLIDE 52

Asymptotics for the empirical degrees CDF (2)

For y ∈ (0, 1), the empirical CDF of the degrees of the graph Gn is: Λn(y) = 1 n

n

  • i=1

1

D(n)

i

≤D(y) .

Theorem

Under regularity conditions on W, we have the following convergence of finite-dimensional distributions: √n (Λn(y) − y) : y ∈ (0, 1)

  • (fdd)

− − − − − →

n→+∞

χ, where χ = (χ(y) : y ∈ (0, 1)) is a centered Gaussian process.

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 23 / 24

slide-53
SLIDE 53

Thank you for your attention !

Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 24 / 24