Ising models on power-law random graphs Sander Dommers Joint work - PowerPoint PPT Presentation

Ising models on power-law random graphs Sander Dommers Joint work with: Cristian Giardinà Remco van der Hofstad Prague School on Mathematical Statistical Physics August 29–September 9, 2011 Where innovation starts

Introduction 2/18 There are many complex real-world networks, e.g., ◮ Social networks (friendships, business relationships, sexual contacts, ...); ◮ Information networks (World Wide Web, citations, ...); ◮ Technological networks (Internet, airline routes, ...); ◮ Biological networks (protein interactions, neural networks,...). Sexual network Colorado Springs, USA (Potterat, et al., ’02) / department of mathematics and computer science

Introduction 2/18 There are many complex real-world networks, e.g., ◮ Social networks (friendships, business relationships, sexual contacts, ...); ◮ Information networks (World Wide Web, citations, ...); ◮ Technological networks ( Internet, airline routes, ...); ◮ Biological networks (protein interactions, neural networks,...). Small part of the Internet (http://www.fractalus.com/ steve/stuff/ipmap/) / department of mathematics and computer science

Introduction 2/18 There are many complex real-world networks, e.g., ◮ Social networks (friendships, business relationships, sexual contacts, ...); ◮ Information networks (World Wide Web, citations, ...); ◮ Technological networks (Internet, airline routes, ...); ◮ Biological networks ( protein interactions, neural networks,...). Yeast protein interaction network (Jeong, et al., ’01) / department of mathematics and computer science

Properties of complex networks 3/18 Power-law behavior Number of vertices with degree k is proportional to k − τ . Small worlds Distances in the network are small / department of mathematics and computer science

Ising model 4/18 Ising model: paradigm model in statistical physics for cooperative behavior. When studied on complex networks it can model for example opinion spreading in society. We will model complex networks with power-law random graphs. What are effects of structure of complex networks on behavior of Ising model? / department of mathematics and computer science

Definition of the Ising model 5/18 On a graph G n , the ferromagnetic Ising model is given by the following Boltzmann distributions over σ ∈ {− 1 , + 1 } n ,   1   � � Z n (β, B ) exp σ i σ j + B µ(σ) =  β σ i  , ( i , j ) ∈ E n i ∈[ n ] where ◮ β ≥ 0 is the inverse temperature; ◮ B is the external magnetic field; ◮ Z n (β, B ) is a normalization factor (the partition function ), i.e.,     � � � Z n (β, B ) = exp σ i σ j + B  β σ i  . σ ∈{− 1 , 1 } n ( i , j ) ∈ E n i ∈[ n ] / department of mathematics and computer science

Power-law random graphs 6/18 In the configuration model a graph G n = ( V n = [ n ] , E n ) is constructed as follows. ◮ Let D have a certain distribution (the degree distribution ); ◮ Assign D i half-edges to each vertex i ∈ [ n ] , where D i are i.i.d. like D (Add one half-edge to last vertex when the total number of half-edges is odd); ◮ Attach first half-edge to another half-edge uniformly at random; ◮ Continue until all half-edges are connected. Special attention to power-law degree sequences, i.e., P [ D ≥ k ] ≤ ck − (τ − 1 ) , τ > 2 . / department of mathematics and computer science

Local structure configuration model for τ > 2 7/18 Start from random vertex i which has degree D i . Look at neighbors of vertex i , probability such a neighbor has degree k + 1 is approximately, ( k + 1 ) � j ∈[ n ] 1 { D j = k + 1 } � j ∈[ n ] D j / department of mathematics and computer science

Local structure configuration model for τ > 2 7/18 Start from random vertex i which has degree D i . Look at neighbors of vertex i , probability such a neighbor has degree k + 1 is approximately, ( k + 1 ) � j ∈[ n ] 1 { D j = k + 1 } / n → ( k + 1 ) P [ D = k + 1 ] for τ > 2 . − , � j ∈[ n ] D j / n E [ D ] / department of mathematics and computer science

Local structure configuration model for τ > 2 7/18 Start from random vertex i which has degree D i . Look at neighbors of vertex i , probability such a neighbor has degree k + 1 is approximately, ( k + 1 ) � j ∈[ n ] 1 { D j = k + 1 } / n → ( k + 1 ) P [ D = k + 1 ] for τ > 2 . − , � j ∈[ n ] D j / n E [ D ] Let K have distribution (the forward degree distribution), P [ K = k ] = ( k + 1 ) P [ D = k + 1 ] . E [ D ] Locally tree-like structure: a branching process with offspring D in first generation and K in further generations. Also, uniformly sparse. / department of mathematics and computer science

Pressure in thermodynamic limit ( E [ K ] < ∞ ) 8/18 Theorem (Dembo, Montanari, ’10) For a locally tree-like and uniformly sparse graph sequence { G n } n ≥ 1 with E [ K ] < ∞ , the pressure per particle, ψ n (β, B ) = 1 n log Z n (β, B ), converges, for n → ∞ , to ϕ h (β, B ) ≡ E [ D ] log cosh (β) − E [ D ] E [ log ( 1 + tanh (β) tanh ( h 1 ) tanh ( h 2 )) ] 2 2 � � D e B � � � log 1 + tanh (β) tanh ( h i ) + E i = 1 D �� � + e − B � � 1 − tanh (β) tanh ( h i ) . i = 1 / department of mathematics and computer science

Pressure in thermodynamic limit ( E [ D ] < ∞ ) 9/18 Theorem (DGvdH, ’10) Let τ > 2 . Then, in the configuration model, the pressure per particle, ψ n (β, B ) = 1 n log Z n (β, B ), converges almost surely, for n → ∞ , to ϕ h (β, B ) ≡ E [ D ] log cosh (β) − E [ D ] E [ log ( 1 + tanh (β) tanh ( h 1 ) tanh ( h 2 )) ] 2 2 � � D � e B � � log 1 + tanh (β) tanh ( h i ) + E i = 1 D �� � + e − B � � 1 − tanh (β) tanh ( h i ) . i = 1 / department of mathematics and computer science

Tree recursion 10/18 Proposition Let K t be i.i.d. like K and B > 0 . Then, the recursion K t h ( t + 1 ) d � atanh ( tanh (β) tanh ( h ( t ) = B + i )), i = 1 has a unique fixed point h ∗ β . Interpretation: the effective field of a vertex in a tree expressed in that of its neighbors. Uniqueness shown by showing that effect of boundary conditions on generation t vanishes for t → ∞ . / department of mathematics and computer science

Correlation inequalities 11/18 Lemma (Griffiths, ’67, Kelly, Sherman, ’68) For a ferromagnet with positive external field, the magnetization at a vertex will not decrease, when ◮ The number of edges increases; ◮ The external magnetic field increases; ◮ The temperature decreases. Lemma (Griffiths, Hurst, Sherman, ’70) For a ferromagnet with positive external field, the magnetization is concave in the external fields, i.e., ∂ 2 m j ( B ) ≤ 0 . ∂ B k ∂ B ℓ / department of mathematics and computer science

Outline of the proof 12/18 n →∞ ψ n (β, B ) lim � ε � β ∂ ∂ � ∂β ′ ψ n (β ′ , B ) d β ′ + ∂β ′ ψ n (β ′ , B ) d β ′ � = lim ε ↓ 0 lim ψ n ( 0 , B ) + n →∞ 0 ε � β ∂ ∂β ′ ϕ(β ′ , B ) d β ′ = ϕ h ( 0 , B ) + 0 + lim ε ↓ 0 ε = ϕ h (β, B ). / department of mathematics and computer science

Outline of the proof 12/18 n →∞ ψ n (β, B ) lim � ε � β ∂ ∂ � ∂β ′ ψ n (β ′ , B ) d β ′ + ∂β ′ ψ n (β ′ , B ) d β ′ � = lim ε ↓ 0 lim ψ n ( 0 , B ) + n →∞ 0 ε � β ∂ ∂β ′ ϕ(β ′ , B ) d β ′ = ϕ h ( 0 , B ) + 0 + lim ε ↓ 0 ε = ϕ h (β, B ). / department of mathematics and computer science

Outline of the proof 12/18 n →∞ ψ n (β, B ) lim � ε � β ∂ ∂ � ∂β ′ ψ n (β ′ , B ) d β ′ + ∂β ′ ψ n (β ′ , B ) d β ′ � = lim ε ↓ 0 lim ψ n ( 0 , B ) + n →∞ 0 ε � β ∂ ∂β ′ ϕ(β ′ , B ) d β ′ = ϕ h ( 0 , B ) + 0 + lim ε ↓ 0 ε = ϕ h (β, B ). / department of mathematics and computer science

Outline of the proof 12/18 n →∞ ψ n (β, B ) lim � ε � β ∂ ∂ � ∂β ′ ψ n (β ′ , B ) d β ′ + ∂β ′ ψ n (β ′ , B ) d β ′ � = lim ε ↓ 0 lim ψ n ( 0 , B ) + n →∞ 0 ε � β ∂ ∂β ′ ϕ(β ′ , B ) d β ′ = ϕ h ( 0 , B ) + 0 + lim ε ↓ 0 ε = ϕ h (β, B ). / department of mathematics and computer science

Internal energy 13/18 � � � σ i σ j ∂β ψ n (β, B ) = 1 µ = | E n | ∂ ( i , j ) ∈ E n � µ � � σ i σ j n n | E n | ( i , j ) ∈ E n → E [ D ] �� � � − E σ i σ j µ 2 / department of mathematics and computer science

Internal energy 13/18 � � � σ i σ j ∂β ψ n (β, B ) = 1 µ = | E n | ∂ ( i , j ) ∈ E n � µ � � σ i σ j n n | E n | ( i , j ) ∈ E n → E [ D ] �� � � − E σ i σ j µ 2 E [ D ] → E [ D ] �� � � �� � � E σ i σ j − E σ i σ j 2 µ 2 e / department of mathematics and computer science

Derivative of ϕ 14/18 ∂ β (β, B ) = E [ D ] �� � � ∂β ϕ h ∗ E σ i σ j . e 2 ϕ h (β, B ) = E [ D ] log cosh (β) − E [ D ] E [ log ( 1 + tanh (β) tanh ( h 1 ) tanh ( h 2 )) ] 2 2 � � D D �� e B � + e − B � � � � � log 1 + tanh (β) tanh ( h i ) 1 − tanh (β) tanh ( h i ) + E i = 1 i = 1 ◮ Show that we can ignore dependence of h ∗ β on β ; ( Interpolation techniques. Split analysis into two parts, one for small degrees and one for large degrees ) ◮ Compute the derivative with assuming β fixed in h ∗ β . / department of mathematics and computer science