Rooting out the rumor culprit from suspects

Suppose that a rumor originating from a single source among a set of suspects spreads in a network, how to root out this rumor source? With the a priori knowledge of suspect nodes and a snapshot observation of infected nodes, we construct a maximum a posteriori (MAP) estimator to identify the rumor source using the susceptible-infected (SI) model. We propose to use a notion of local rumor center to characterize P_c(n), the correct detection probability of the source estimator upon observing n infected nodes, in both the finite and asymptotic regimes, for regular trees of node degree δ. First, when all nodes are suspects, lim_n→∞P_c(n) grows from 0.25 to 0.307 as δ increases from three to infinity, a result first established in Shah and Zaman (2011, 2012) via a different approach; furthermore, P_c(n) monotonically decreases with n and increases with δ even in the finite-n regime. Second, when the suspect nodes form a connected subgraph of the network, lim_n→∞P_c(n) significantly exceeds the a priori probability if δ ≥ 3, and reliable detection is achieved as δ becomes sufficiently large; furthermore, P_c(n) monotonically decreases with n and increases with δ. Third, when there are only two suspect nodes, lim_n→∞P_c(n) is at least 0.75 if δ ≥ 3; and P_c(n) increases with the distance between the two suspects. Fourth, when there are multiple suspect nodes, among all possible connection patterns, that all the suspects form a single connected subgraph yields the smallest P_c(n). Our analysis leverages ideas from the Pólya's urn model in probability theory and sheds insight into the behavior of the rumor spreading process not only in the asymptotic regime but also for the general finite-n regime.