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 steps does it take to completely infect a network of nodes, starting from a single infected node? An analogy to the classic “coupon collector” problem of probability theory reveals that the takeover time is 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 for , the takeover time is distributed as a Gumbel distribution for the star graph, as the convolution of two Gumbel distributions 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 for -dimensional lattices with (these distributions approach the convolution of two Gumbel distributions as approaches infinity). Connections to evolutionary dynamics, cancer, incubation periods of infectious diseases, first-passage percolation, and other spreading phenomena in biology and physics are discussed.
Highlights
Contagion is a topic of broad interdisciplinary interest
Related phenomena arise in probability theory and statistical physics in the setting of first-passage percolation [18,19], and in evolutionary dynamics in connection with the spread of mutations through a resident population [20,21,22,23,24]
We show that for N 1, the takeover time T is distributed as a Gumbel distribution for the star graph, and as the convolution of two Gumbel distributions for a complete graph and an Erdos-Rényi random graph
Summary
Contagion is a topic of broad interdisciplinary interest. Originally studied in the context of infectious diseases [1,2,3,4], contagion has been used as a metaphor for diverse processes that spread by contact between neighbors. When a big network is almost fully infected, it becomes increasingly difficult to find the last few susceptible individuals to infect These intuitions led us to suspect that the problem of calculating the distribution of takeover times might be amenable to the techniques used to study the classic “coupon collector” problem in probability theory [26,27]. Erdos and Rényi proved that for large N, the distribution of waiting times for the coupon collection problem approaches a Gumbel distribution [28] This type of distribution is right skewed and is one of the three universal extreme value distributions [29,30]. We discuss the possible relevance of our results to fixation times in evolutionary dynamics, population genetics, and cancer biology, and to the longstanding (yet theoretically unexplained) clinical observation that incubation periods for infectious diseases frequently have right-skewed distributions
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