Document Sample

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

DOCUMENT INFO

Shared By:

Categories:

Tags:
Ising models, Ising model, random graphs, partition function, square lattice, quasiperiodic tilings, boundary conditions, free energy, J. Phys, Penrose tiling

Stats:

views: | 3 |

posted: | 8/20/2011 |

language: | English |

pages: | 25 |

OTHER DOCS BY liuhongmei

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.