The power of adaptivity in source identification with time queries on the path

We study the problem of identifying the source of a stochastic diffusion process spreading on a graph based on the arrival times of the diffusion at a few queried nodes. In a graph G=(V,E), an unknown source node v⁎∈V is drawn uniformly at random, and unknown edge weights w(e) for e∈E, representing the propagation delays along the edges, are drawn independently from a Gaussian distribution of mean 1 and variance σ2. An algorithm then attempts to identify v⁎ by querying nodes q∈V and being told the length of the shortest path between q and v⁎ in graph G weighted by w. We consider two settings: non-adaptive, in which all query nodes must be decided in advance, and adaptive, in which each query can depend on the results of the previous ones. Both settings are motivated by an application of the problem to epidemic processes (where the source is called patient zero), which we discuss in detail.