Capacity Bounds for Networks with Correlated Sources and Characterisation of Distributions by Entropies

TL;DR

This paper investigates how well entropy functions can characterize correlations among sources in network capacity problems, proposing auxiliary variables to improve bounds and extending the approach to other entropy measures.

Contribution

It introduces a method using auxiliary random variables to better characterize source correlation via entropy functions, aiding in bounding network capacity with correlated sources.

Findings

Auxiliary random variables improve correlation characterization.

Implicit and explicit capacity bounds derived for networks with correlated sources.

Applicable to Shannon, Renyi, and Tsallis entropy measures.

Abstract

Characterising the capacity region for a network can be extremely difficult. Even with independent sources, determining the capacity region can be as hard as the open problem of characterising all information inequalities. The majority of computable outer bounds in the literature are relaxations of the Linear Programming bound which involves entropy functions of random variables related to the sources and link messages. When sources are not independent, the problem is even more complicated. Extension of Linear Programming bounds to networks with correlated sources is largely open. Source dependence is usually specified via a joint probability distribution, and one of the main challenges in extending linear program bounds is the difficulty (or impossibility) of characterising arbitrary dependencies via entropy functions. This paper tackles the problem by answering the question of how well entropy functions can characterise correlation among sources. We show that by using carefully chosen auxiliary random variables, the characterisation can be fairly "accurate" Using such auxiliary random variables we also give implicit and explicit outer bounds on the capacity of networks with correlated sources. The characterisation of correlation or joint distribution via Shannon entropy functions is also applicable to other information measures such as Renyi entropy and Tsallis entropy.