Network tomography for integer-valued traffic

TL;DR

This paper develops a new sampling methodology for network tomography of integer-valued traffic, addressing challenges in inference and sampling efficiency, especially when the link-path incidence matrix is totally unimodular.

Contribution

It introduces a modified sampling approach that adapts to the polytope structure, ensuring better mixing in Bayesian inference for network traffic models.

Findings

Sampling algorithms can fail due to inflexibility in directions.

Total unimodularity of the incidence matrix guarantees effective sampling.

The proposed method improves inference on real traffic data sets.

Abstract

A classic network tomography problem is estimation of properties of the distribution of route traffic volumes based on counts taken on the network links. We consider inference for a general class of models for integer-valued traffic. Model identifiability is examined. We investigate both maximum likelihood and Bayesian methods of estimation. In practice, these must be implemented using stochastic EM and MCMC approaches. This requires a methodology for sampling latent route flows conditional on the observed link counts. We show that existing algorithms for doing so can fail entirely, because inflexibility in the choice of sampling directions can leave the sampler trapped at a vertex of the convex polytope that describes the feasible set of route flows. We prove that so long as the network's link-path incidence matrix is totally unimodular, it is always possible to select a coordinate system representation of the polytope for which sampling parallel to the axes is adequate. This motivates a modified sampler in which the representation of the polytope adapts to provide good mixing behavior. This methodology is applied to three road traffic data sets. We conclude with a discussion of the ramifications of the unimodularity requirements for the routing matrix.