  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.
