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.