A covert communications network is a connected, overlay peer-to-peer network used to support communications within a group in which the survival of the group depends on: (1) anonymity of communications; and (2) concealment of network membership to both other members of the group and external eavesdroppers. These requirements are much more stringent than for typical privacy and anonymity systems. We consider the topologies of resilient covert communications networks that: (1) minimize the impact on the network in the event of a subverted node; and (2) maximize the connectivity of the survivor network with the removal of the subverted node and its closed neighborhood. We analyze the properties of resilient covert networks, propose a measurement for determining the suitability of a topology for use in a covert communication network, determine the properties of an optimal covert network topology, and analyze several optimal topologies.