Phase transitions in the edge-triangle exponential random graph model
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1 Phase transitions in the edge-triangle exponential random graph model Mei Yin Department of Mathematics, Brown University April 28, 2014
2 Probability space: The set G n of all simple graphs G n on n vertices. Probability mass function: ( P β n(g n = exp n 2 (β 1 e(g n + β 2 t(g n ψn β. e(g n = 2 E(Gn and t(g n 2 n = densities of G n. Normalization constant ψn β : ψ β n = 1 n 2 log G n G n exp 6 T (Gn n 3 are the edge and triangle ( n 2 (β 1 e(g n + β 2 t(g n. Survey: Fienberg, Introduction to papers on the modeling and analysis of network data I & II. arxiv: &
3 (Razborov and others
4 β 2 = 0: ( P β n(g n = exp n 2 (β 1 e(g n ψn β ( = exp 2β 1 E(G n n 2 ψn β. Erdős-Rényi graph G(n, ρ, P ρ n(g n = ρ E(Gn (1 ρ( n 2 E(G n. Include edges independently with parameter ρ = e 2β 1 /(1 + e 2β 1. exp(n 2 ψn β = ( ( n 1 2 exp (2β 1 E(G n =. 1 ρ G n G n
5 What happens when β 2 0? Problem: Graphs with different numbers of vertices belong to different probability spaces! Solution: Theory of graph limits (graphons! (Lovász and coauthors; earlier work of Aldous and Hoover Graphon space W is the space of all symmetric measurable functions h(x, y from [0, 1] 2 into [0, 1]. The interval [0, 1] represents a continuum of vertices, and h(x, y denotes the probability of putting an edge between x and y.
6 What happens when β 2 0? Problem: Graphs with different numbers of vertices belong to different probability spaces! Solution: Theory of graph limits (graphons! (Lovász and coauthors; earlier work of Aldous and Hoover Graphon space W is the space of all symmetric measurable functions h(x, y from [0, 1] 2 into [0, 1]. The interval [0, 1] represents a continuum of vertices, and h(x, y denotes the probability of putting an edge between x and y.
7 Example: Erdős-Rényi graph G(n, ρ, h(x, y = ρ. Example: Any G n G n, { 1, if ( nx, ny is an edge in Gn ; h(x, y = 0, otherwise.
8 Large deviation: ( lim n ψβ n = max β 1 e(h + β 2 t(h I (hdxdy, h W [0,1] 2 where: e(h = [0,1] h(x, ydxdy, t(h = h(x, yh(y, zh(z, xdxdydz, [0,1] 3 and I : [0, 1] R is the function I (u = 1 2 u log u + 1 (1 u log(1 u. 2 Let F be the set of maximizers. G n lies close to F with high probability for large n. (Chatterjee and Varadhan; Chatterjee and Diaconis
9 β 2 0: Let n. G n behaves like the Erdős-Rényi graph G(n, u, where u [0, 1] maximizes l(u = β 1 u + β 2 u u log u 1 (1 u log(1 u. 2 (Chatterjee and Diaconis; Häggström and Jonasson; Bhamidi, Bresler, and Sly Critical point is ( 1 2 log 2 3 4, (Radin and Y
10 Fix β 1. Let n and then let β 2. G n looks like a 1 complete bipartite graph with fraction of edges randomly 1+e 2β 1 deleted. (Chatterjee and Diaconis Take β 1 = aβ 2 + b. Fix a and b. Let n and then let β 2. G n exhibits quantized behavior, jumping from one complete multipartite graph to another, and the jumps happen precisely at the normal lines of an infinite polytope. Idea: Minimize ae + t. (Y, Rinaldo, and Fadnavis; related work in Handcock; Rinaldo, Fienberg, and Zhou
11 What happens if we interchange the limits in n and β 2? Discontinuous transition along critical directions. Idea: Study the induced probability distribution on S n = {(e(g n, t(g n, G n G n } [0, 1] β = 60, β = 110, n = β 1 = 50, β 2 = 36, n = β 1 = 80, β 2 = 40, n = 30
12 Fix e. What happens if we only consider graphs whose edge density is close to e, say e(g n e < α? (conditional Probability mass function: ( P e,β n,α(g n = exp n 2 (βt(g n ψn,α e,β. 6 T (Gn t(g n = is the triangle density of G n 3 n. (conditional Normalization constant ψ e,β ψ e,β n,α = 1 n 2 log G n G n: e(g n e <α n,α: exp ( n 2 βt(g n.
13 Large deviation: where: lim lim α 0 n ψe,β n,α = max h W:e(h=e e(h = [0,1] ( βt(h I (hdxdy, [0,1] 2 h(x, ydxdy, t(h = h(x, yh(y, zh(z, xdxdydz, [0,1] 3 and I : [0, 1] R is the function I (u = 1 2 u log u + 1 (1 u log(1 u. 2 Let F be the set of maximizers. G n lies close to F with high (conditional probability for large n. (Kenyon and Y
14 Fix e and t. Consider graphons with e(h = e and t(h = t that minimize [0,1] I (hdxdy. Why are we interested? 2 They are useful for studying exponential models without constraints: ( = max e,t max h W β 1 e(h + β 2 t(h I (hdxdy [0,1] 2 ( max β 1 e + β 2 t I (hdxdy [0,1] 2, h W:e(h=e,t(h=t and with constraints: = max t max h W:e(h=e max h W:e(h=e,t(h=t ( βt(h I (hdxdy [0,1] 2 ( βt I (hdxdy. [0,1] 2
15 Special strip: Fix e = 1 2 and t e3. Graphon h(x, y that minimizes [0,1] 2 I (hdxdy is given by { 1 h(x, y = 2 + ɛ, if x < 1 2 < y or x > 1 2 > y; 1 2 ɛ, if x, y < 1 2 or x, y > 1 2, where 0 ɛ = (e 3 t (Radin and Sadun Then ( max βt(h I (hdxdy h W:e(h=e [0,1] 2 = max (β(e 3 ɛ 3 I ( 12 + ɛ. ɛ [0, 1 2 ] As β decreases from 0 to, G n jumps from being Erdős-Rényi to almost complete bipartite (with ɛ > 0.47 at β 2.7, skipping a large portion of the e = 1 2 line. (Kenyon and Y
16 Fix β 1 and β 2. (macro Euler-Lagrange equation for the maximizer: β 1 + 3β 2 h(x, zh(y, zdz = 1 2 log h(x, y 1 h(x, y. [0,1] (Chatterjee and Diaconis Fix e and t. (micro Euler-Lagrange equation for the maximizer: [0,1] h(x, zh(y, zdz = λ + µ log h(x, y 1 h(x, y. Plus: Stronger! Holds for the entire (e, t-space. Minus: The relationship between e, t and λ, µ is not as explicit. (Kenyon and Y Thank You!
17 Fix β 1 and β 2. (macro Euler-Lagrange equation for the maximizer: β 1 + 3β 2 h(x, zh(y, zdz = 1 2 log h(x, y 1 h(x, y. [0,1] (Chatterjee and Diaconis Fix e and t. (micro Euler-Lagrange equation for the maximizer: [0,1] h(x, zh(y, zdz = λ + µ log h(x, y 1 h(x, y. Plus: Stronger! Holds for the entire (e, t-space. Minus: The relationship between e, t and λ, µ is not as explicit. (Kenyon and Y Thank You!
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