Network equivalence for a joint compound-arbitrarily-varying network model

We consider the problem of finding the capacity of noisy networks under the presence of Byzantine adversaries, modeled by a joint compound channel and arbitrarily varying channel (AVC) model. This extends our earlier work which considers these models only in isolation. The motivation for this setup is that typically the adversary first selects an arbitrary subset of edges from the network and then specifies adversarial transmissions to each of the selected edges. We show that in some cases equivalence between this network and another network holds in the sense that for a fixed selection of adversarial edges the noisy links can be replaced by noiseless bit-pipes with a capacity equal to the random coding capacity of the corresponding AVC. In particular, the capacity region for the noisy network can be outer bounded by the intersection of the individual capacity regions for the noiseless case, for each adversarial edge selection. Moreover, if the network is fully connected, we also show that this upper bound is equivalent to the capacity of the noisy network. We also provide necessary and sufficient condition for full connectivity, making use of a new condition for an AVC termed overwritability.