Permutation Tests for Infection Graphs

We formulate and analyze a hypothesis testing problem for inferring the edge structure of an infection graph. Our model is as follows: A disease spreads over a network via contagion and random infection, where uninfected nodes contract the disease at a time corresponding to an independent exponential random variable and infected nodes transmit the disease to uninfected neighbors according to independent exponential random variables with an unknown rate parameter. A subset of nodes is also censored, meaning the infection statuses of the nodes are unobserved. Given the statuses of all nodes in the network, the goal is to determine the underlying graph. Our procedure consists of a permutation test, and we derive a condition in terms of automorphism groups of the graphs corresponding to the null and alternative hypotheses that ensures the validity of our test. Notably, the permutation test does not involve estimating unknown parameters governing the infection process; instead, it leverages differences in the topologies of the null and alternative graphs. We derive risk bounds for our testing procedure in settings of interest; provide extensions to situations involving relaxed versions of the algebraic condition; and discuss multiple observations of infection spreads. We conclude with experiments validating our results.