On identifying the causative network of an epidemic

The history of infections and epidemics holds famous examples where understanding, containing and ultimately treating an outbreak began with understanding its mode of spread. The key question then, is: which network of interactions is the main cause of the spread? And can we determine the causative network without any knowledge of the epidemic, other than the identify of a minuscule subsample of infected nodes? This comes down to understanding the diagnostic power of network information. Specifically, in this paper we consider an epidemic that spreads on one of two networks. At some point in time, we see a small random subsample (perhaps a vanishingly small fraction) of those infected. We derive sufficient conditions two networks must have for this problem to be identifiable. We provide an efficient algorithm that solves the hypothesis testing problem on such graphs, and we characterize a regime in which our algorithm succeeds. Finally, we show that the condition we need for this identifiability property is fairly mild, and in particular, is satisfied by two common graph topologies: the grid, and the Erdös-Renyi graphs.