# Takeover times for a simple model of network infection

## TL;DR

This paper analyzes the time it takes for a simple stochastic infection model to spread across different network types, revealing how the distribution of takeover times varies with network structure.

## Contribution

It provides a detailed characterization of the distribution of infection takeover times across various network topologies, linking extremal behavior to classical probability distributions.

## Key findings

- Gumbel distribution for star graphs
- Sum of Gumbels for complete and Erdős-Rényi graphs
- Normal distribution for 1D and 2D lattices

## 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 $T$ does it take to completely infect a network of $N$ nodes, starting from a single infected node? An analogy to the classic "coupon collector" problem of probability theory reveals that the takeover time $T$ 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 $N \gg 1$, the takeover time $T$ is distributed as a Gumbel for the star graph; as the sum of two Gumbels for a complete graph and an Erd\H{o}s-R\'{e}nyi random graph; as a normal for a one-dimensional ring and a two-dimensional lattice; and as a family of intermediate skewed distributions for $d$-dimensional lattices with $d \ge 3$ (these distributions approach the sum of two Gumbels as $d$ 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.