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

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 Rényi entropy and Tsallis entropy.

Common information, matroid representation, and secret sharing for matroid ports

“…Linear information and rank inequalities are fundamental in the linear programming technique that has been used to find bounds on the information ratio of secret sharing schemes [6,7,36,44,48] and on the achievable rates in network coding [17,56,60]. An improvement to that technique has been recently proposed [22].…”

Common Information, Matroid Representation, and Secret Sharing for Matroid Ports

“…Linear information and rank inequalities are fundamental in the linear programming technique that has been used to find bounds on the information ratio of secret sharing schemes [7,8,34,42,46] and on the achievable rates in network coding [17,53,56]. An improvement to that technique has been recently proposed [20].…”

On the Partition Bound for Undirected Unicast Network Information Capacity

“…Note that there are 2 7−1 −1 = 63 possible distinct cut-sets of the form cs(α, α c ) in the network (similar bi-partitions of a set are also considered in [13] but in a different context). Without loss of generality, assume that α contains the node v 1 .…”