Population Protocols for Majority in Arbitrary Networks

We study the problem of determining the majority type in an arbitrary connected network, each vertex of which has initially two possible types. The vertices may have a few additional possible states and can interact in pairs only if they share an edge. Any (population) protocol is required to stabilize in the initial majority. We first present and analyze a protocol with 4 states per vertex that always computes the initial majority value, under any fair scheduler.This protocol is optimal, in the sense that there does not exist any population protocol that always computes majority with fewer than 4 states per vertex. However, this does not rule out the existence of a protocol with 3 states per vertex that is correct with high probability (whp). To this end, we examine an elegant and very natural majority protocol with 3 states per vertex, introduced in Angluin et al. (Distrib Comput 21:87–102, 2008), where its performance has been analyzed for the clique graph. In particular, we study the performance of this protocol in arbitrary networks, under the probabilistic scheduler. We prove that, when the two initial states are put uniformly at random on the vertices, the protocol of Angluin et al. (Distrib Comput 21:87–102, 2008) converges to the initial majority with probability higher than the probability of converging to the initial minority. In contrast, we show that the resistance of the protocol to failure when the underlying graph is a clique causes the failure of the protocol in general graphs.