@MISC{Goldberg_higherorder, author = {A. Goldberg}, title = {HIGHER ORDER MARKOV RANDOM FIELDS FOR INDEPENDENT SETS}, year = {} }

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Abstract

It is well-known that if one samples from the independent sets of a large regular graph of large girth using a pairwise Markov random eld (i.e. hardcore model) in the uniqueness regime, each excluded node has a binomially distributed number of included neighbors in the limit. In this paper, motivated by applications to the design of communication networks, we pose the question of how to sample from the independent sets of such a graph so that the number of included neighbors of each excluded node has a dierent distribution of our choosing. We observe that higher order Markov random elds are well-suited to this task, and investigate the properties of these models. For the family of so-called reverse ultra log-concave distributions, which in-cludes the truncated Poisson and geometric, we give necessary and sucient conditions for the natural higher order Markov random eld which induces the desired distribution to be in the uniqueness regime, in terms of the set of solutions to a certain system of equations. We also show that these Markov random elds undergo a phase transi-tion, and give explicit bounds on the associated critical activity, which we prove to exhibit a certain robustness. For distributions which are small perturbations around the binomial distribution realized by the hardcore model at critical activity, we give a description of the cor-responding uniqueness regime in terms of a simple polyhedral cone. Our analysis reveals an interesting non-monotonicity with regards to biasing towards excluded nodes with no included neighbors. We con-clude with a broader discussion of the potential use of higher order Markov random elds for analyzing independent sets in graphs. 1. Introduction. Recently