Optimal coteries and voting schemes

We consider coteries (i.e. intersecting, Sperner systems) on arbitrary networks with given node reliability, which maximize the network availability (i.e. the probability that the set of operating nodes has a connected subgraph which contains a set from the coterie). We show that the singleton coterie is always optimal on any network when the node reliability is p≤ 1 2. For p≥ 1 2, we give optimal coteries for the complete multipartite networks. The resulting optimal coteries when restricted to a special subclass of complete bipartite networks are shown to exceed the availability of any voting scheme, for p 1 2.