VIEWS: 3 PAGES: 25 POSTED ON: 8/20/2011 Public Domain
Ising models on power-law random graphs Sander Dommers Joint work with: Cristian Giardinà Remco van der Hofstad Random Graphs and the Brain May 11, 2011 Where innovation starts Introduction 2/17 The brain is a complex network of neurons. Other examples of complex networks include social networks, information networks, technological networks. / department of mathematics and computer science Ising model 3/17 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 4/17 On a graph Gn , the ferromagnetic Ising model is given by the following Boltzmann distributions over σ ∈ {−1, +1}n , 1 µ(σ ) = exp β σi σj + B σi , Zn (β, B ) (i ,j )∈En i ∈[n] where β ≥ 0 is the inverse temperature; B is the external magnetic ﬁeld; Zn (β, B ) is a normalization factor (the partition function), i.e., Zn (β, B ) = exp β σi σj + B σi . nσ ∈{−1,1} (i ,j )∈En i ∈[n] / department of mathematics and computer science Power-law random graphs 5/17 In the conﬁguration model a graph Gn = (Vn = [n], En ) is constructed as follows. Let D have a certain distribution (the degree distribution); Assign Di half-edges to each vertex i ∈ [n], where Di 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 6/17 Start from random vertex i which has degree Di . Look at neighbors of vertex i , probability such a neighbor has degree k + 1 is approximately, (k + 1) j ∈[n] 1{Dj =k +1} j ∈[n] Dj / department of mathematics and computer science Local structure conﬁguration model for τ > 2 6/17 Start from random vertex i which has degree Di . Look at neighbors of vertex i , probability such a neighbor has degree k + 1 is approximately, (k + 1) j ∈[n] 1{Dj =k +1} /n (k + 1)P[D = k + 1] −→ , for τ > 2. j ∈[n] Dj /n E[D ] / department of mathematics and computer science Local structure conﬁguration model for τ > 2 6/17 Start from random vertex i which has degree Di . Look at neighbors of vertex i , probability such a neighbor has degree k + 1 is approximately, (k + 1) j ∈[n] 1{Dj =k +1} /n (k + 1)P[D = k + 1] −→ , for τ > 2. j ∈[n] Dj /n E[D ] Let K have distribution (the forward degree distribution), (k + 1)P[D = k + 1] P[K = k ] = . 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 ] < ∞) 7/17 Theorem (Dembo, Montanari, ’10) For a locally tree-like and uniformly sparse graph sequence {Gn }n≥1 with E[K ] < ∞, the pressure per particle, 1 ψn (β, B ) = log Zn (β, B ), n converges, for n → ∞, to E[D ] E[D ] ϕh (β, B ) ≡ log cosh(β) − E[ log(1 + tanh(β) tanh(h1 ) tanh(h2 ))] 2 2 D + E log e B 1 + tanh(β) tanh(hi ) i =1 D +e −B 1 − tanh(β) tanh(hi ) . i =1 / department of mathematics and computer science Pressure in thermodynamic limit (E[D ] < ∞) 8/17 Theorem (DGvdH, ’10) Let τ > 2. Then, in the conﬁguration model, the pressure per particle, 1 ψn (β, B ) = log Zn (β, B ), n converges almost surely, for n → ∞, to E[D ] E[D ] ϕh (β, B ) ≡ log cosh(β) − E[ log(1 + tanh(β) tanh(h1 ) tanh(h2 ))] 2 2 D + E log e B 1 + tanh(β) tanh(hi ) i =1 D +e −B 1 − tanh(β) tanh(hi ) . i =1 / department of mathematics and computer science Tree recursion 9/17 Proposition Let Kt be i.i.d. like K and B > 0. Then, the recursion Kt atanh(tanh(β) tanh(hi(t ) )), d h (t +1) = B + i =1 has a unique ﬁxed point hβ . ∗ 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 10/17 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 mj (B ) ≤ 0. ∂Bk ∂B / department of mathematics and computer science Outline of the proof 11/17 lim ψn (β, B ) n→∞ ε β ∂ ∂ = lim lim ψn (0, B ) + ψn (β , B )dβ + ψn (β , B )dβ ε↓0 n→∞ 0 ∂β ε ∂β β ∂ = ϕh (0, B ) + 0 + lim ϕ(β , B )dβ ε↓0 ε ∂β = ϕh (β, B ). / department of mathematics and computer science Outline of the proof 11/17 lim ψn (β, B ) n→∞ ε β ∂ ∂ = lim lim ψn (0, B ) + ψn (β , B )dβ + ψn (β , B )dβ ε↓0 n→∞ 0 ∂β ε ∂β β ∂ = ϕh (0, B ) + 0 + lim ϕ(β , B )dβ ε↓0 ε ∂β = ϕh (β, B ). / department of mathematics and computer science Outline of the proof 11/17 lim ψn (β, B ) n→∞ ε β ∂ ∂ = lim lim ψn (0, B ) + ψn (β , B )dβ + ψn (β , B )dβ ε↓0 n→∞ 0 ∂β ε ∂β β ∂ = ϕh (0, B ) + 0 + lim ϕ(β , B )dβ ε↓0 ε ∂β = ϕh (β, B ). / department of mathematics and computer science Outline of the proof 11/17 lim ψn (β, B ) n→∞ ε β ∂ ∂ = lim lim ψn (0, B ) + ψn (β , B )dβ + ψn (β , B )dβ ε↓0 n→∞ 0 ∂β ε ∂β β ∂ = ϕh (0, B ) + 0 + lim ϕ(β , B )dβ ε↓0 ε ∂β = ϕh (β, B ). / department of mathematics and computer science Internal energy 12/17 ∂ 1 |En | (i ,j )∈En σi σj µ ψn (β, B ) = σi σj µ = ∂β n n |En | (i ,j )∈En E[D ] −→ E σi σj µ 2 / department of mathematics and computer science Internal energy 12/17 ∂ 1 |En | (i ,j )∈En σi σj µ ψn (β, B ) = σi σj µ = ∂β n n |En | (i ,j )∈En E[D ] −→ E σi σj µ 2 E[D ] E[D ] E σi σj µ −→ E σi σj e 2 2 / department of mathematics and computer science Derivative of ϕ 13/17 ∂ E[D ] ϕhβ (β, B ) = ∗ E σi σj . ∂β 2 e E[D ] E[D ] ϕh (β, B ) = log cosh(β) − E[ log(1 + tanh(β) tanh(h1 ) tanh(h2 ))] 2 2 D D + E log e B 1 + tanh(β) tanh(hi ) + e −B 1 − tanh(β) tanh(hi ) i =1 i =1 Show that we can ignore dependence of hβ 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 hβ . ∗ / department of mathematics and computer science Thermodynamic quantities 14/17 Corollary Let τ > 2. Then, in the conﬁguration model, a.s.: The magnetization is given by n 1 ∂ m(β, B ) ≡ lim σi µ = ϕh ∗ (β, B ) = E σ0 νD +1 . n→∞ n ∂B i =1 The susceptibility is given by ∂Mn (β, B ) ∂2 χ(β, B ) ≡ lim = ϕh ∗ (β, B ). n→∞ ∂B ∂B 2 / department of mathematics and computer science Critical temperature 15/17 Deﬁne the magnetization on Gn as n 1 mn (β, B ) = σi µ . n i =1 Then, the spontaneous magnetization, = 0, β < βc ; m(β, 0+) = lim m(β, B ) B ↓0 > 0, β > βc . The critical inverse temperature βc is given by E[K ](tanh βc ) = 1. Note that, for τ ∈ (2, 3), we have E[K ] = ∞, so that βc = 0. / department of mathematics and computer science Critical exponents 16/17 Predictions by physicists (e.g. Leone, Vázquez, Vespignani, Zecchina, ’02). Critical behavior of magnetization m, and susceptibility χ . m(β, 0+ ), β ↓ βc m(βc , B ), B ↓ 0 χ (β, 0+ ), β ↓ βc τ >5 ∼ (β − βc )1/2 ∼ B 1/3 ∼ (β − βc )−1 τ ∈ (3, 5) ∼ (β − βc )1/(τ −3) ∼ B 1/(τ −2) τ ∈ (2, 3) ∼ (β − βc )1/(3−τ ) ∼ B1 ∼ (β − βc )1 / department of mathematics and computer science Critical exponents 16/17 Predictions by physicists (e.g. Leone, Vázquez, Vespignani, Zecchina, ’02). Critical behavior of magnetization m, and susceptibility χ . m(β, 0+ ), β ↓ βc m(βc , B ), B ↓ 0 χ (β, 0+ ), β ↓ βc τ >5 ∼ (β − βc )1/2 ∼ B 1/3 ∼ (β − βc )−1 τ ∈ (3, 5) ∼ (β − βc )1/(τ −3) ∼ B 1/(τ −2) τ ∈ (2, 3) ∼ (β − βc )1/(3−τ ) ∼ B1 ∼ (β − βc )1 / department of mathematics and computer science Distances in power-law random graphs 17/17 Let Hn be the graph distance between two uniformly chosen connected vertices in the conﬁguration model. Then: For τ > 3 and E[K ] > 1 (vdH, Hooghiemstra, Van Mieghem, ’05), Hn ∼ log n, For τ ∈ (2, 3) (vdH, Hooghiemstra, Znamenski, ’07), Hn ∼ log log n; / department of mathematics and computer science Distances in power-law random graphs 17/17 Let Hn be the graph distance between two uniformly chosen connected vertices in the conﬁguration model. Then: For τ > 3 and E[K ] > 1 (vdH, Hooghiemstra, Van Mieghem, ’05), Hn ∼ log n, For τ ∈ (2, 3) (vdH, Hooghiemstra, Znamenski, ’07), Hn ∼ log log n; For τ > 3 and τ ∈ (2, 3) similar results hold for the diameter of linear preferential attachment models (D, vdH, Hooghiemstra, ’10). / department of mathematics and computer science