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

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

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

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

Introduction 1 Convergence of dense graph sequences 2 Main results 3 Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 3 / 24

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. � v ( G ) � E ( G ) is the set of e ( G ) edges among the collection of unordered pairs of 2 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   0 1 0 0 1 1 0 0 0 1     0 0 0 1 1 5     0 0 1 0 1   1 1 1 1 0 4 3 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

Large random graphs: first example Erd¨ os-R´ enyi graph G n ( p ) : random graph such that V ( G n ( 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

Large random graphs: first example Erd¨ os-R´ enyi graph G n ( p ) : random graph such that V ( G n ( 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

Large random graphs: first example Erd¨ os-R´ enyi graph G n ( p ) : random graph such that V ( G n ( 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

Large random graphs: first example Erd¨ os-R´ enyi graph G n ( p ) : random graph such that V ( G n ( 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

Large random graphs: first example Erd¨ os-R´ enyi graph G n ( p ) : random graph such that V ( G n ( 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¨ enyi graph with parameter p = 1 os-R´ 2 ant its limit Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 5 / 24

Large random graphs: second example Randomly grown uniform attachment graph GUA n : random graph such that V ( GUA n ) = [ 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

Large random graphs: second example Randomly grown uniform attachment graph GUA n : random graph such that V ( GUA n ) = [ 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

Large random graphs: second example Randomly grown uniform attachment graph GUA n : random graph such that V ( GUA n ) = [ 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

Large random graphs: second example Randomly grown uniform attachment graph GUA n : random graph such that V ( GUA n ) = [ 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

Large random graphs: second example Randomly grown uniform attachment graph GUA n : random graph such that V ( GUA n ) = [ 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

From graphs to graphons Graph F Graphon W σ -finite measure space Vertex set V ( F ) ([0 , 1] , B ([0 , 1]) , λ ) Adjacency matrix Symmetric, measurable function W : [0 , 1] 2 → [0 , 1] A : V ( F ) × V ( F ) → { 0 , 1 } ex + y Figure: Exponential graphon: W ( x, 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

Introduction 1 Convergence of dense graph sequences 2 Main results 3 Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 8 / 24

Motivation We consider a sequence of dense finite graphs ( G n : n ∈ N ∗ ) i.e. such that 2 e ( G n ) lim n → + ∞ v ( G n )( v ( G n ) − 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

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 ϕ such that t ( F, G ) = | Hom ( F,G ) | Hom ( F, G ) { i, j } ∈ E ( F ) ⇒ { ϕ ( i ) , ϕ ( j ) } ∈ E ( G ) v ( G ) v ( F ) t inj ( F, G ) = | Inj ( F,G ) | Inj ( F, G ) ϕ ∈ Hom ( F, G ) injective Av ( F ) v ( G ) t ind ( F, G ) = | Ind ( F,G ) | ϕ injective such that Ind ( F, G ) Av ( F ) { i, j } ∈ E ( F ) ⇔ { ϕ ( i ) , ϕ ( j ) } ∈ E ( G ) v ( G ) ϕ ϕ ϕ F G Figure: ϕ is as injective homomorphism but not an induced homomorphism. Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 10 / 24

How to charaterize the convergence of sequence of dense graphs ? Theorem [Lov´ asz, Szegedy (2006)] A sequence of simple graphs ( G n : n ∈ N ∗ ) is called convergent if the sequence ( t inj ( F, G n ) : 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

How to charaterize the convergence of sequence of dense graphs ? Theorem [Lov´ asz, Szegedy (2006)] A sequence of simple graphs ( G n : n ∈ N ∗ ) is called convergent if the sequence ( t inj ( F, G n ) : 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 ( G n : n ∈ N ∗ ) is said to converge to a graphon W if for every finite simple graph F , we have n → + ∞ t inj ( F, G n ) = t ( F, W ) , lim where � � � t ( F, W ) = t inj ( F, W ) = W ( x i , x j ) dx k . [0 , 1] v ( F ) { i,j }∈ E ( F ) k ∈ V ( F ) Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 11 / 24

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 � � 1 {{ β i ,β j }∈ E ( G ) } . n p β ∈S n,p { i,j }∈ E ( F ) let W be a graphon then we have the density of F in W : � 1 � � t ( F, W ) = W ( x i , x j ) . λ ([0 , 1]) p [0 , 1] p k ∈ V ( F ) { i,j }∈ E ( F ) Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 12 / 24

Generating dense graphs with a given number of vertices from a graphon Given a graphon, W , we construct a W -random generated G n ( W ) , as follows: Marion Sciauveau Asymptotic analysis of large random graphs JPS - 18 mai 13 / 24