Takeover times for a simple model of network infection

Abstract

We study a stochastic model of infection spreading on a network. At each time step a node is chosen at random, along with one of its neighbors. If the node is infected and the neighbor is susceptible, the neighbor becomes infected. How many time stepsTdoes it take to completely infect a network ofNnodes, starting from a single infected node? An analogy to the classic “coupon collector” problem of probability theory reveals that the takeover timeTis dominated by extremal behavior, either when there are only a few infected nodes near the start of the process or a few susceptible nodes near the end. We show that forN≫ 1, the takeover timeTis distributed as a Gumbel for the star graph; as the sum of two Gumbels for a complete graph and an Erdős-Rényi random graph; as a normal for a one-dimensional ring and a two-dimensional lattice; and as a family of intermediate skewed distributions ford-dimensional lattices withd≥ 3 (these distributions approach the sum of two Gumbels asdapproaches infinity). Connections to evolutionary dynamics, cancer, incubation periods of infectious diseases, first-passage percolation, and other spreading phenomena in biology and physics are discussed.