Antar Bandyopadhyay

Counting without sampling: new algorithms for enumeration problems using statistical physics

We propose a new type of approximate counting algorithms for the problems of enumerating the number of independent sets and proper colorings in low degree graphs with large girth. Our algorithms are not based on a commonly used Markov chain technique, but rather are inspired by recent developments in statistical physics in connection with correlation decay properties of Gibbs measures and its implications to uniqueness of Gibbs measures on infinite trees, reconstruction problems and local weak convergence methods. On a negative side, our algorithms provide ∈-approximations only to the logarithms of the size of a feasible set (also known as free energy in statistical physics). But on the positive side, unlike Markov chain based algorithms, our approach provides deterministic as opposed to probabilistic guarantee on approximations. Moreover, for some regular graphs we obtain explicit values for the counting problem. For example, we show that every 4-regular n-node graph with large girth has asymptotically (1.494 ...)n independent sets, and in every r-regular graph with n nodes and large girth the number of q ≥ r + 1-proper colorings is asymptotically (q(1-1/q)r/2)n for large n. In statistical physics terminology, we compute explicitly the partition function (free energy) in these cases. We extend our results to random regular graphs graphs also. The explicit results obtained in this paper would be hard to derive via Markov chain sampling technique.

Endogeny for the Logistic Recursive Distributional Equation

In this article we prove the endogeny and bivariate uniqueness property for a particular “max-type” recursive distributional equation (RDE). The RDE we consider is the so called logistic RDE, which appears in the proof of the ζ(2)-limit of the random assignment problem using the local weak convergence method proved by D. Aldous [Probab. Theory Related Fields 93 (1992)(4), 507 – 534]. This article provides a non-trivial application of the general theory developed by D. Aldous and A. Bandyopadhyay [Ann. Appl. Probab. 15 (2005)(2), 1047 – 1110]. The proofs involves analytic arguments, which illustrate the need to develop more analytic tools for studying such max-type RDEs.

Pólya urn schemes with infinitely many colors

In this work we introduce a new urn model with infinite but countably many colors indexed by an appropriate infinite set. We mainly focus on d-dimensional integer lattice and replacement matrix associated with bounded increment random walks on it. We prove central and local limit theorems for the expected configuration of the urn and show that irrespective of the null recurrent or transient behavior of the underlying random walk, the urn models have universal scaling and centering giving appropriate normal distribution at the limit. The work also provides similar results for urn models corresponding to other infinite lattices.

On the Expected Total Number of Infections for Virus Spread on a Finite Network

In this paper we consider a simple virus infection spread model on a finite population of n agents connected by some neighborhood structure. Given a graph G on n vertices, we begin with some fixed number of initial infected vertices. At each discrete time step, an infected vertex tries to infect its neighbors with probability � 2 (0,1) independently of others and then it dies out. The process continues till all infected vertices die out. We focus on obtaining proper lower bounds on the expected number of ever infected vertices. We obtain a simple lower bound, using breadth-first search algorithm and show that for a large class of graphs which can be classified as the ones which locally “look like” a tree in sense of the local weak convergence [1], this lower bound gives better approximation than some of the known approximations through matrix-method based upper bounds [3]. AMS 2000 Subject Classifications: Primary: 60K35, 05C80; secondary: 60J85, 90B15

Variance estimation for tree-order restricted models

In this article, we discuss the estimation of the common variance of several normal populations with tree-order restricted means. We discuss the asymptotic properties of the maximum-likelihood estimator (MLE) of the variance as the number of populations tends to infinity. We consider several cases of various orders of the sample sizes and show that the MLE of the variance may or may not be consistent or be asymptotically normal.