Polytope samplers for network tomography

Abstract

Network tomography is concerned with reconstruction and estimation of properties of traffic flow that are linked to the observed data through an underdetermined linear system. The likelihood function for such problems can be expressed only as the sum over integer‐valued points in a convex polytope. Typically this set is too large to enumerate, so that statistical inference must proceed by sampling from the polytope. Recent progress has seen the development of polytope sampling algorithms that operate well when the network link‐path incidence matrix is totally unimodular. In this paper we examine whether this property is likely to hold in practical applications. We find that total unimodularity is assured for certain simple networks, but that it can fail in more complex cases. We show that when it does fail, the existing polytope samplers may not generate the requisite irreducible Markov chain. As a remedy, a modified algorithm is proposed in which the basis for the polytope adapts to ensure adequate mixing over the entire polytope. The operation of this algorithm is illustrated by a numerical example.