Inference of a Rumor's Source in the Independent Cascade Model

We consider the so-called Independent Cascade Model for rumor spreading or epidemic processes popularized by Kempe et al. [2003]. In this model, a small subset of nodes from a network are the source of a rumor. In discrete time steps, each informed node "infects" each of its uninformed neighbors with probability p. While many facets of this process are studied in the literature, less is known about the inference problem: given a number of infected nodes in a network, can we learn the source of the rumor? In the context of epidemiology this problem is often referred to as patient zero problem. It belongs to a broader class of problems where the goal is to infer parameters of the underlying spreading model, see, e.g., Lokhov [NeurIPS'16] or Mastakouri et al. [NeurIPS'20]. In this work we present a maximum likelihood estimator for the rumor's source, given a snapshot of the process in terms of a set of active nodes X after t steps. Our results show that, for cycle-free graphs, the likelihood estimator undergoes a non-trivial phase transition as a function t. We provide a rigorous analysis for two prominent classes of acyclic network, namely d-regular trees and Galton-Watson trees, and verify empirically that our heuristics work well in various general networks.