On the number of matrices and a random matrix with prescribed row and column sums and 0-1 entries

TL;DR

This paper provides an asymptotic estimate for the number of 0-1 matrices with fixed row and column sums and shows that a uniformly sampled matrix is typically close to a maximum-entropy matrix, extending to matrices with fixed zeros.

Contribution

It introduces a method to estimate the count of such matrices and demonstrates the typical structure of a random matrix in this set, based on entropy maximization.

Findings

Asymptotic estimate for the number of matrices with given margins

High probability that a random matrix is close to the maximum-entropy matrix

Extension to matrices with prescribed zeros

Abstract

We consider the set Sigma(R,C) of all mxn matrices having 0-1 entries and prescribed row sums R=(r_1, ..., r_m) and column sums C=(c_1, ..., c_n). We prove an asymptotic estimate for the cardinality |Sigma(R, C)| via the solution to a convex optimization problem. We show that if Sigma(R, C) is sufficiently large, then a random matrix D in Sigma(R, C) sampled from the uniform probability measure in Sigma(R,C) with high probability is close to a particular matrix Z=Z(R,C) that maximizes the sum of entropies of entries among all matrices with row sums R, column sums C and entries between 0 and 1. Similar results are obtained for 0-1 matrices with prescribed row and column sums and assigned zeros in some positions.