Abstract:
The hardcore model on a graph G with parameter \lambda > 0 is a probability measure on the collection of all independent sets of G , that assigns to each indep...Show MoreMetadata
Abstract:
The hardcore model on a graph G with parameter \lambda > 0 is a probability measure on the collection of all independent sets of G , that assigns to each independent set I a probability proportional to \lambda ^{|I|} . In this paper we consider the problem of estimating the parameter \lambda given a single sample from the hardcore model on a graph G . To bypass the computational intractability of the maximum likelihood method, we use the maximum pseudo-likelihood (MPL) estimator, which for the hardcore model has a surprisingly simple closed form expression. We show that for any sequence of graphs \{G_{N}\}_{N \geq 1} , where G_{N} is a graph on N vertices, the MPL estimate of \lambda is \sqrt {N} -consistent (that is, it converges to the true parameter at rate 1/\sqrt {N} ), whenever the graph sequence has uniformly bounded average degree. We then extend our methods to obtain estimates for the vector of activity parameters in general H -coloring models, in which restrictions between adjacent colors are encoded by a constraint graph H . These constitute an important class of Markov random fields that includes all hard-constraint models, which arise in a broad array of fields including combinatorics, statistical physics, and communication networks. Given a single sample from an H -coloring model, we derive sufficient conditions under which the MPL estimate is \sqrt {N} -consistent. Moreover, we verify the sufficient conditions for H -coloring models for which there is at least one ‘unconstrained’ color (that is, there exists at least one vertex in the constraint graph H that is connected to all vertices), as long as the graph sequence has uniformly bounded average degree. This applies to many H -coloring examples such as the Widom-Rowlinson and multi-state hard-core models. On the other hand, for the q -coloring model, which falls outside this class, we show that the condition can fail and consis...
Published in: IEEE Transactions on Information Theory ( Volume: 67, Issue: 10, October 2021)