We consider the problem of detecting the source of a rumor (information diffusion) in a network based on observations about which set of nodes possess the rumor. In a recent work [10], this question was introduced and studied. The authors proposed rumor centrality as an estimator for detecting the source. They establish it to be the maximum likelihood estimator with respect to the popular Susceptible Infected (SI) model with exponential spreading time for regular trees. They showed that as the size of infected graph increases, for a line (2-regular tree) graph, the probability of source detection goes to 0 while for d-regular trees with d ≥ 3 the probability of detection, say αd, remains bounded away from 0 and is less than 1/2. Their results, however stop short of providing insights for the heterogeneous setting such as irregular trees or the SI model with non-exponential spreading times.

We provide a systematic study of the problem of finding the source of a rumor in a network. We model rumor spreading in a network with the popular susceptible-infected (SI) model and then construct an estimator for the rumor source. This estimator is based upon a novel topological quantity which we term rumor centrality. We establish that this is a maximum likelihood (ML) estimator for a class of graphs. We find the following surprising threshold phenomenon: on trees which grow faster than a line, the estimator always has nontrivial detection probability, whereas on trees that grow like a line, the detection probability will go to 0 as the network grows. Simulations performed on synthetic networks such as the popular small-world and scale-free networks, and on real networks such as an internet AS network and the U.S. electric power grid network, show that the estimator either finds the source exactly or within a few hops of the true source across different network topologies. We compare rumor centrality to another common network centrality notion known as distance centrality. We prove that on trees, the rumor center and distance center are equivalent, but on general networks, they may differ. Indeed, simulations show that rumor centrality outperforms distance centrality in finding rumor sources in networks which are not tree-like.

We provide a systematic study of the problem of finding the source of a computer virus in a network. We model virus spreading in a network with a variant of the popular SIR model and then construct an estimator for the virus source. This estimator is based upon a novel combinatorial quantity which we term rumor centrality. We establish that this is an ML estimator for a class of graphs. We find the following surprising threshold phenomenon: on trees which grow faster than a line, the estimator always has non-trivial detection probability, whereas on trees that grow like a line, the detection probability will go to 0 as the network grows. Simulations performed on synthetic networks such as the popular small-world and scale-free networks, and on real networks such as an internet AS network and the U.S. electric power grid network, show that the estimator either finds the source exactly or within a few hops in different network topologies. We compare rumor centrality to another common network centrality notion known as distance centrality. We prove that on trees, the rumor center and distance center are equivalent, but on general networks, they may differ. Indeed, simulations show that rumor centrality outperforms distance centrality in finding virus sources in networks which are not tree-like.