We formulate and analyze a novel hypothesis testing problem for inferring the edge structure of an infection graph. In our model, a disease spreads over a network via contagion or random infection, where the times between successive contagion events are independent exponential random variables with unknown rate parameters. A subset of nodes is also censored uniformly at random. Given the observed infection statuses of nodes in the network, the goal is to determine the underlying graph. We present a procedure based on permutation testing, and we derive sufficient conditions for the validity of our test in terms of automorphism groups of the graphs corresponding to the null and alternative hypotheses. Our test is easy to compute and does not involve estimating unknown parameters governing the process. We also derive risk bounds for our permutation test in a variety of settings, and relate our test statistic to approximate likelihood ratio testing and maximin tests. For graphs not satisfying the necessary symmetries, we provide an additional method for testing the significance of the graph structure, albeit at a higher computational cost.