  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.
