Factor models on locally tree-like graphs

Abstract

We consider homogeneous factor models on uniformly sparse graph sequences converging locally to a (unimodular) random tree $T$, and study the existence of the free energy density $\phi$, the limit of the log-partition function divided by the number of vertices $n$ as $n$ tends to infinity. We provide a new interpolation scheme and use it to prove existence of, and to explicitly compute, the quantity $\phi$ subject to uniqueness of a relevant Gibbs measure for the factor model on $T$. By way of example we compute $\phi$ for the independent set (or hard-core) model at low fugacity, for the ferromagnetic Ising model at all parameter values, and for the ferromagnetic Potts model with both weak enough and strong enough interactions. Even beyond uniqueness regimes our interpolation provides useful explicit bounds on $\phi$. In the regimes in which we establish existence of the limit, we show that it coincides with the Bethe free energy functional evaluated at a suitable fixed point of the belief propagation (Bethe) recursions on $T$. In the special case that $T$ has a Galton-Watson law, this formula coincides with the nonrigorous "Bethe prediction" obtained by statistical physicists using the "replica" or "cavity" methods. Thus our work is a rigorous generalization of these heuristic calculations to the broader class of sparse graph sequences converging locally to trees. We also provide a variational characterization for the Bethe prediction in this general setting, which is of independent interest.Comment: Published in at http://dx.doi.org/10.1214/12-AOP828 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Universality of the mean-field for the Potts model

“…For the complete graph it has been long known that the mean-field prediction is indeed tight for both Ising and Potts measure (see [28,29,30]). However, for locally tree-like graphs (see [23,Definition 1.1]) this is not the case. Indeed, in [20] it is shown that the Bethe prediction is the correct answer for Ising measures on such graphs when the limiting tree is a Galton-Watson tree whose off-spring distribution have a finite variance.…”

“…In [26] it was extended for power law distribution, and finally in [23] it was extended to full generality. Moreover the same was shown be true for the Potts model on regular graphs in [23,24].…”

Convergent sequences of sparse graphs: A large deviations approach

“…Proof. The existence of the limit (19) follows immediately by monotonicity: note that hom(G, H) is monotonically non-decreasing in every element of the weight matrix A.…”

Harnessing the Bethe free energy

“…According to the cavity method, is determined by the “limiting local structure” of the random factor graph . To formalise this concept, we adapt the concept of local weak convergence of graph sequences to our current setup, thereby generalising the approach taken in . Definition A ‐ template consists of a ‐model , a connected factor graph and a root r H , which is a variable or factor node.…”

“…Suppose that we fix , , , as in Definition 3.1 and that M = (M n ) n is a sequence of ( , , , )-models such that M n = (V n , F n , d n , t n , (ψ a ) a∈Fn ) has size n. Let us write G = G(M n ) for the sake of brevity. According to the cavity method, lim n→∞ 1 n E[ln Z(G)] is determined by the "limiting local structure" of the random factor graph G. To formalise this concept, we adapt the concept of local weak convergence of graph sequences [8,35] to our current setup, thereby generalising the approach taken in [23]. A ( , , , )-template consists of a ( , , , )-model M, a connected factor graph H ∈ G(M) and a root r H , which is a variable or factor node.…”