We address the problem of finding the capacity of networks with independent point-to-point channels where a subset of these channels is replaced either by a compound channel (CC) or an arbitrarily varying channel (AVC). These channels represent a good model for the presence of a Byzantine adversary which controls a subset of links or nodes in the network. We show that equivalence between this network and another network hold in the sense that all links can be replaced by noiseless bit-pipes with the same capacity as the noisy CC or nonsymmetrizable AVC, leading to identical capacity regions for both networks. We then strengthen these results by showing that an additional feedback path between the output and input of a CC or an additional forward path for the AVC extends the equivalent capacity region for both the noisy and the derived noiseless network. This explicitly includes the symmetrizable AVC case.