Consensus of Interacting Particle Systems on Erdos-Renyi Graphs

Interacting Particle Systems-exemplified by the voter model, iterative majority, and iterative k-majority processes-have found use in many disciplines including distributed systems, statistical physics, social networks, and Markov chain theory. In these processes, nodes update their "opinion" according to the frequency of opinions amongst their neighbors. We propose a family of models parameterized by an update function that we call Node Dynamics: every node initially has a binary opinion. At each round a node is uniformly chosen and randomly updates its opinion with the probability distribution specified by the value of the update function applied to the frequencies of its neighbors' opinions. In this work, we prove that the Node Dynamics converge to consensus in time Theta(n log n) in complete graphs and dense Erdos-Renyi random graphs when the update function is from a large family of "majority-like" functions. Our technical contribution is a general framework that upper bounds the consensus time. In contrast to previous work that relies on handcrafted potential functions, our framework systematically constructs a potential function based on the state space structure.