Phase Transition of Inhomogeneous Random Graphs lie de Panafieu - PowerPoint PPT Presentation

Phase Transition of Inhomogeneous Random Graphs Élie de Panafieu Liafa, Université Paris-Diderot May 28, 2013 Joint work with Vlady Ravelomanana Élie de Panafieu Phase Transition of Inhomogeneous Random Graphs

Introduction The inhomogeneous graphs model encodes several tractable SAT and CSP problems [Söderberg 02] [Bollobàs Janson Riordan 07] Élie de Panafieu Phase Transition of Inhomogeneous Random Graphs

Introduction The inhomogeneous graphs model encodes several tractable SAT and CSP problems [Söderberg 02] [Bollobàs Janson Riordan 07] bipartite graphs and satisfiable quantified 2-XOR-SAT formulas Élie de Panafieu Phase Transition of Inhomogeneous Random Graphs

Introduction The inhomogeneous graphs model encodes several tractable SAT and CSP problems [Söderberg 02] [Bollobàs Janson Riordan 07] bipartite graphs and satisfiable quantified 2-XOR-SAT formulas Asymptotic analysis of the number of inhomogeneous graphs (some differences with the original model) following the approach of the giant paper [Janson Knuth Łuczak Pittel 93] Élie de Panafieu Phase Transition of Inhomogeneous Random Graphs

Introduction The inhomogeneous graphs model encodes several tractable SAT and CSP problems [Söderberg 02] [Bollobàs Janson Riordan 07] bipartite graphs and satisfiable quantified 2-XOR-SAT formulas Asymptotic analysis of the number of inhomogeneous graphs (some differences with the original model) following the approach of the giant paper [Janson Knuth Łuczak Pittel 93] Phase transition of the modeled problems Élie de Panafieu Phase Transition of Inhomogeneous Random Graphs

Introduction The inhomogeneous graphs model encodes several tractable SAT and CSP problems [Söderberg 02] [Bollobàs Janson Riordan 07] bipartite graphs and satisfiable quantified 2-XOR-SAT formulas Asymptotic analysis of the number of inhomogeneous graphs (some differences with the original model) following the approach of the giant paper [Janson Knuth Łuczak Pittel 93] Phase transition of the modeled problems probability for a graph to be bipartite [Pittel Yeum 10], probability of satisfiability of a quantified 2-XOR-SAT formula [Creignou Daudé Egly 07] Élie de Panafieu Phase Transition of Inhomogeneous Random Graphs

Bipartite Graphs [Pittel Yeum 10] Élie de Panafieu Phase Transition of Inhomogeneous Random Graphs

Bipartite Graphs [Pittel Yeum 10] Each vertex v receives a color c ( v ) , Élie de Panafieu Phase Transition of Inhomogeneous Random Graphs

Bipartite Graphs [Pittel Yeum 10] Each vertex v receives a color c ( v ) , the edges are weighted according to the color of their ends using R = ( 0 1 1 0 ) , weight 1 2 for each connected component. � cc ( G ) � 1 � weight ( c ( G )) : = R c ( a ) , c ( b ) 2 ( a , b ) ∈ E ( G ) weight = 0 Élie de Panafieu Phase Transition of Inhomogeneous Random Graphs

Bipartite Graphs [Pittel Yeum 10] Each vertex v receives a color c ( v ) , the edges are weighted according to the color of their ends using R = ( 0 1 1 0 ) , weight 1 2 for each connected component. � cc ( G ) � 1 � weight ( c ( G )) : = R c ( a ) , c ( b ) 2 ( a , b ) ∈ E ( G ) weight = 1 4 Élie de Panafieu Phase Transition of Inhomogeneous Random Graphs

Bipartite Graphs [Pittel Yeum 10] Each vertex v receives a color c ( v ) , the edges are weighted according to the color of their ends using R = ( 0 1 1 0 ) , weight 1 2 for each connected component. � cc ( G ) � 1 � weight ( c ( G )) : = R c ( a ) , c ( b ) 2 ( a , b ) ∈ E ( G ) weight = 1 4 + 1 4 Élie de Panafieu Phase Transition of Inhomogeneous Random Graphs

Bipartite Graphs [Pittel Yeum 10] Each vertex v receives a color c ( v ) , the edges are weighted according to the color of their ends using R = ( 0 1 1 0 ) , weight 1 2 for each connected component. � cc ( G ) � 1 � weight ( c ( G )) : = R c ( a ) , c ( b ) 2 ( a , b ) ∈ E ( G ) weight = 1 4 + 1 4 + 1 4 Élie de Panafieu Phase Transition of Inhomogeneous Random Graphs

Bipartite Graphs [Pittel Yeum 10] Each vertex v receives a color c ( v ) , the edges are weighted according to the color of their ends using R = ( 0 1 1 0 ) , weight 1 2 for each connected component. � cc ( G ) � 1 � weight ( c ( G )) : = R c ( a ) , c ( b ) 2 ( a , b ) ∈ E ( G ) weight = 1 4 + 1 4 + 1 4 + 1 4 Élie de Panafieu Phase Transition of Inhomogeneous Random Graphs

Bipartite Graphs [Pittel Yeum 10] Each vertex v receives a color c ( v ) , the edges are weighted according to the color of their ends using R = ( 0 1 1 0 ) , weight 1 2 for each connected component. � cc ( G ) � 1 � weight ( c ( G )) : = R c ( a ) , c ( b ) 2 ( a , b ) ∈ E ( G ) � 1 if G is bipartite, � weight ( c ( G )) = 0 otherwise. c weight = 1 4 + 1 4 + 1 4 + 1 4 Élie de Panafieu Phase Transition of Inhomogeneous Random Graphs

Bipartite Graphs [Pittel Yeum 10] Each vertex v receives a color c ( v ) , the edges are weighted according to the color of their ends using R = ( 0 1 1 0 ) , weight 1 2 for each connected component. � cc ( G ) � 1 � weight ( c ( G )) : = R c ( a ) , c ( b ) 2 ( a , b ) ∈ E ( G ) � 1 if G is bipartite, � weight ( c ( G )) = 0 otherwise. c The number of ( n , m ) -bipartite graphs is weight = 1 4 + 1 4 + 1 4 + 1 4 � � weight ( c ( G )) . 2 ( n , m ) : = g ( 0 1 1 0 ) , 1 ( n , m ) -graph G c Élie de Panafieu Phase Transition of Inhomogeneous Random Graphs

Inhomogeneous Graphs [Söderberg 02] [Bollobàs Janson Riordan 07] R ∈ Sym q × q ( R ≥ 0 ) and σ > 0. A ( R , σ ) -graph is: a vertex colored graph c ( G ) , with weight R c ( s ) , c ( t ) on each edge ( s , t ) , and weight σ for each connected component. weight ( c ( G )) : = σ cc ( G ) � R c ( a ) , c ( b ) , ( a , b ) ∈ E ( G ) � � weight ( c ( G )) . g R , σ ( n , m ) : = ( n , m ) -graph G c Élie de Panafieu Phase Transition of Inhomogeneous Random Graphs

Inhomogeneous Graphs [Söderberg 02] [Bollobàs Janson Riordan 07] R ∈ Sym q × q ( R ≥ 0 ) and σ > 0. A ( R , σ ) -graph is: a vertex colored graph c ( G ) , with weight R c ( s ) , c ( t ) on each edge ( s , t ) , and weight σ for each connected component. weight ( c ( G )) : = σ cc ( G ) � R c ( a ) , c ( b ) , ( a , b ) ∈ E ( G ) � � weight ( c ( G )) . g R , σ ( n , m ) : = ( n , m ) -graph G c weight = σ 2 R 1 , 1 R 3 1 , 2 R 2 1 , 3 R 2 2 , 3 Élie de Panafieu Phase Transition of Inhomogeneous Random Graphs

Quantified 2-Xor-Sat Formulas [Creignou Daudé Egly 07] ∀ x , y , ∃ a , b ,..., h , a ⊕ b = x , a ⊕ h = y , a ⊕ c = x , b ⊕ e = x , d ⊕ f = x , d ⊕ g = y , e ⊕ h = y Élie de Panafieu Phase Transition of Inhomogeneous Random Graphs

Quantified 2-Xor-Sat Formulas [Creignou Daudé Egly 07] ∀ x , y , ∃ a , b ,..., h , a ⊕ b = x , a ⊕ h = y , a ⊕ c = x , b ⊕ e = x , d ⊕ f = x , d ⊕ g = y , e ⊕ h = y Élie de Panafieu Phase Transition of Inhomogeneous Random Graphs

Quantified 2-Xor-Sat Formulas [Creignou Daudé Egly 07] ∀ x , y , ∃ a , b ,..., h , a ⊕ b = x , a ⊕ h = y , a ⊕ c = x , b ⊕ e = x , d ⊕ f = x , d ⊕ g = y , e ⊕ h = y satisfiable iff each cycle contains an even number of x and y . Élie de Panafieu Phase Transition of Inhomogeneous Random Graphs

Quantified 2-Xor-Sat Formulas [Creignou Daudé Egly 07] ∀ x , y , ∃ a , b ,..., h , a ⊕ b = x , a ⊕ h = y , a ⊕ c = x , b ⊕ e = x , d ⊕ f = x , d ⊕ g = y , e ⊕ h = y satisfiable iff each cycle contains an even number of x and y . 00 10 01 11 00 x y � 0 1 1 0 , σ = 1 � 1 0 0 1 , R = 10 x y 1 0 0 1 4 0 1 1 0 01 y x 11 y x Élie de Panafieu Phase Transition of Inhomogeneous Random Graphs

Quantified 2-Xor-Sat Formulas [Creignou Daudé Egly 07] ∀ x , y , ∃ a , b ,..., h , a ⊕ b = x , a ⊕ h = y , a ⊕ c = x , b ⊕ e = x , d ⊕ f = x , d ⊕ g = y , e ⊕ h = y satisfiable iff each cycle contains an even number of x and y . 00 10 01 11 00 x y � 0 1 1 0 , σ = 1 � 1 0 0 1 , R = 10 x y 1 0 0 1 4 0 1 1 0 01 y x 11 y x The number of satisfiable quantified 2-Xor-Sat formulas with n existantial variables and m clauses is g R , σ ( n , m ) . Élie de Panafieu Phase Transition of Inhomogeneous Random Graphs

Sub-Critical Density of Edges When m n < c ( 1 − ǫ ) and n → ∞ , with high probability a ( n , m ) - ( R , σ ) -graph consists of trees and unicycle components. Élie de Panafieu Phase Transition of Inhomogeneous Random Graphs

Sub-Critical Density of Edges When m n < c ( 1 − ǫ ) and n → ∞ , with high probability a ( n , m ) - ( R , σ ) -graph consists of trees and unicycle components. ← → rooted tree T i ( z ) = z exp ( T ( z )) R i Symbolic method → → T ) R ) − 1 → T = ( I − diag ( z ∂ T Élie de Panafieu Phase Transition of Inhomogeneous Random Graphs

Sub-Critical Density of Edges When m n < c ( 1 − ǫ ) and n → ∞ , with high probability a ( n , m ) - ( R , σ ) -graph consists of trees and unicycle components. ← → → → → � 1 − z rooted tree T i ( z ) = z exp ( T ( z )) R i T ∼ t 0 − t 1 ρ + ... Drmota-Lalley-Wood Theorem Élie de Panafieu Phase Transition of Inhomogeneous Random Graphs