Epicenter of random epidemic spanning trees on finite graphs*

Epidemic source detection is the problem of identifying the network node that originated an epidemic from a partial observation of the epidemic process. The problem finds applications in different contexts, such as detecting the origin of rumors in online social media, and has been studied under various assumptions. Different from prior studies, this work considers an epidemic process on a finite graph that starts on a random node (epidemic source) and terminates when all nodes are infected, yielding a rooted and directed epidemic tree that encodes node infections (i.e., a directed spanning tree of the graph with every edge directed away from the epidemic source). Assuming knowledge of the underlying graph and the undirected spanning tree (i.e., the infection edges but not their directions), can the epidemic source be accurately identified? This work tackles this problem by introducing the epicenter, an efficient estimator for the epidemic source, and thus, the direction of every edge in the epidemic tree. When the underlying graph is vertex-transitive the epicenter can be computed in linear time and it coincides with the well-known distance center of the epidemic tree. Moreover, on a complete graph the epicenter is also the most likely estimator for the source. Finally, the accuracy of the epicenter is evaluated numerically on five different graph models and the performance strongly depends on the graph structure, varying from 31% (on complete graphs) to 13% (on sparse power-law graphs). However, for all graph models considered the epicenter exhibited an accuracy higher than the distance center, being three times more accurate on sparse power-law graphs.