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Gibbs conditioning, atypical consensus and splitting Gibbs measures on random regular graphs

I-Hsun Chen, Ivan Lee, Kavita Ramanan, Sarath Yasodharan

Abstract

Given n independent Bernoulli(p) random variables X_i, i = 1, ..., n, representing the opinions of individuals connected by an underlying random k-regular graph G_n on {1, ..., n}, we show that when conditioned on an atypical empirical consensus, which is the normalized sum of X_i X_j over neighboring vertices i, j, the joint distribution of the random variables converges, as n goes to infinity, to an Ising measure on the infinite k-regular tree T^k with a specific external field that depends only on the bias parameter p, and a temperature that depends on both p and the atypical consensus value. In particular, we show that conditional on the empirical consensus being smaller (respy, larger) than typical, the limit is a translation-invariant splitting (TIS) antiferromagnetic (respy, ferromagnetic) Ising measure on T^k. Moreover, if the bias is zero, then there is a phase transition: when the consensus exceeds k/(k-1), the conditional limits could be either the plus or minus boundary condition Ising measures. Furthermore, when X_i, i = 1, ..., n, are i.i.d. on a finite space, we show that when conditioned on an atypical value of the scaled sum of h(X_i, X_j) over neighboring vertices i and j, for any symmetric edge potential h, the limiting joint distribution of {X_i} lies in the set of (possibly degenerate) TIS Gibbs measures on T^k. The proofs leverage a tractable form of the large deviation rate function for component empirical measures of random regular graphs with i.i.d. marks and Gibbs conditioning principles, and entail careful analyses of associated non-convex constrained optimization problems. As a by-product of our results, we also obtain an (asymptotic) analog of the maximum entropy principle for Gibbs measures on random regular graphs.

Gibbs conditioning, atypical consensus and splitting Gibbs measures on random regular graphs

Abstract

Given n independent Bernoulli(p) random variables X_i, i = 1, ..., n, representing the opinions of individuals connected by an underlying random k-regular graph G_n on {1, ..., n}, we show that when conditioned on an atypical empirical consensus, which is the normalized sum of X_i X_j over neighboring vertices i, j, the joint distribution of the random variables converges, as n goes to infinity, to an Ising measure on the infinite k-regular tree T^k with a specific external field that depends only on the bias parameter p, and a temperature that depends on both p and the atypical consensus value. In particular, we show that conditional on the empirical consensus being smaller (respy, larger) than typical, the limit is a translation-invariant splitting (TIS) antiferromagnetic (respy, ferromagnetic) Ising measure on T^k. Moreover, if the bias is zero, then there is a phase transition: when the consensus exceeds k/(k-1), the conditional limits could be either the plus or minus boundary condition Ising measures. Furthermore, when X_i, i = 1, ..., n, are i.i.d. on a finite space, we show that when conditioned on an atypical value of the scaled sum of h(X_i, X_j) over neighboring vertices i and j, for any symmetric edge potential h, the limiting joint distribution of {X_i} lies in the set of (possibly degenerate) TIS Gibbs measures on T^k. The proofs leverage a tractable form of the large deviation rate function for component empirical measures of random regular graphs with i.i.d. marks and Gibbs conditioning principles, and entail careful analyses of associated non-convex constrained optimization problems. As a by-product of our results, we also obtain an (asymptotic) analog of the maximum entropy principle for Gibbs measures on random regular graphs.
Paper Structure (56 sections, 21 theorems, 160 equations, 3 figures)

This paper contains 56 sections, 21 theorems, 160 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background and discussion of results
  3. A new maximum "entropy" principle
  4. Outline.
  5. Common notation and terminology
  6. Measure-theoretic notation
  7. Graph notation:
  8. Marked Graph notation
  9. Main results: How atypical consensus occurs
  10. Random graph model
  11. Conditional limits and phase transitions in two-spin systems
  12. Ising measure on infinite regular trees
  13. Ising measures on infinite regular trees
  14. The conditional limit of the component empirical measure for two-spin systems
  15. Connecting back to the conditional behavior of the joint mark distribution
  16. ...and 41 more sections

Key Result

Lemma 2.6

Fix $\kappa\geq 2$ and $B\in \mathbb{R}$. In the ferromagnetic regime $\beta>0$, there exists a critical threshold $\beta_*=\beta_*(\kappa,B)>0$ such that the map $\Gamma = \Gamma(\kappa,\beta,B)$ has a unique fixed point $\theta^*$ if and only if $(\beta, B)$ lie in the uniqueness regime Moreover, the following properties hold in the ferromagnetic regime: Furthermore, $\beta_*(2,B)=\infty$ for

Figures (3)

  • Figure 1: All $\mathsf{TIS}$ Ising measures in different parameter regimes $(\beta,B)\in \mathbb{R}\times \mathbb{R}$.
  • Figure 2: Two-spin with consensus functional for $\kappa=5$. Top: non-uniform mark distribution $\nu$ (no phase transition). Bottom: uniform mark distribution $\nu$ (phase transition).
  • Figure 3: $J^\nu_5(s,t)$ with non-uniform mark distribution $\nu (1) = 1/3, \nu(-1) = 2/3$

Theorems & Definitions (80)

  • Definition 1.1: Depth-$r$ marginals
  • Remark 1.2
  • Remark 1.3
  • Remark 2.1: Typical Asymptotic Behavior and the True Law
  • Definition 2.2: Ising measures on the infinite regular tree
  • Remark 2.3
  • Definition 2.4: Ising Cavity Map
  • Remark 2.5
  • Lemma 2.6: Known Properties of the Ising Cavity Map
  • proof
  • ...and 70 more