Beyond the Cut-Set Bound: Uncertainty Computations in Network Coding With Correlated Sources

Cut-set bounds are not, in general, tight for all classes of network communication problems. In this paper, we introduce a new technique for proving converses for the problem of transmission of correlated sources in networks, which results in bounds that are tighter than the corresponding cut-set bounds. We also define the concept of “uncertainty region” which might be of independent interest. We provide a full characterization of this region for the case of two correlated random variables. The bounding technique works as follows: on one hand, we show that if the communication problem is solvable, the uncertainty of certain random variables in the network with respect to imaginary parties that have partial knowledge of the sources must satisfy some constraints that depend on the network architecture. On the other hand, the same uncertainties have to satisfy constraints that only depend on the joint distribution of the sources. Matching these two leads to restrictions on the statistical joint distribution of the sources in communication problems that are solvable over a given network architecture. Our technique also provides nontrivial outer bounds for communication problems with secrecy constraints.