Fundamental Limits of Communications in Interference Networks-Part II: Information Flow in Degraded Networks

In this second part of our multi-part papers, the information flow in degraded interference networks is studied. A full characterization of the sum-rate capacity for the degraded networks with any possible configuration is established. Here, network with any possible configuration is referred to as a general single-hop communication scenario with any number of transmitters, any number of receivers and any possible distribution of messages among transmitters and receivers. It is shown that a successive decoding scheme is sumrate optimal for these networks. Also, it is proved that the transmission of only a certain subset of messages is sufficient to achieve the sumrate capacity in such networks. Algorithms are presented to determine this subset of messages explicitly. According to these algorithms, the optimal strategy to achieve the sum-rate capacity in degraded networks is that the transmitters try to send information for the stronger receivers and, if possible, avoid sending the messages with respect to the weaker receivers. This is indeed a fundamental characteristic of the information flow in degraded networks. The algorithms are easily understood using our graphical illustrations for the achievability schemes based on directed graphs. The sum-rate expression for the degraded networks is then used to derive a unified outer bound on the sum-rate capacity of arbitrary non-degraded networks. Several variations of the degraded networks are identified for which the derived outer bound is sum-rate optimal. Specifically, noisy interference regimes are derived for certain classes of multi-user/multi-message interference networks. Also, for the first time, network scenarios are identified where the incorporation of both successive decoding and treating interference as noise achieves their sum-rate capacity. Finally, by taking insight from our results for degraded networks, we establish a unified outer bound on the entire capacity region of the general interference networks. These outer bounds for a broad range of network scenarios are tighter than the existing cut-set bound.