Rumors in a Network: Whos the Culprit?

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 a variant of the popular SIR model and then construct an estimator for the rumor source. This estimator is based upon a novel topological quantity which we term \textbf{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 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.

💡 Research Summary

The paper tackles the fundamental problem of locating the origin of a rumor (or any contagion) that spreads through a network. The authors first adopt a variant of the classic Susceptible‑Infected‑Recovered (SIR) model that is more appropriate for rumors: nodes become “infected” (i.e., they have heard the rumor) and remain infectious indefinitely, while a “recovery” step is essentially absent. Transmission occurs in discrete time steps; at each step every infected node independently selects a random neighbor that has not yet heard the rumor and passes it on. Crucially, the order in which nodes become infected is assumed to be uniformly random among all feasible infection sequences. This stochastic description captures the fact that, in many online social systems, once a piece of information is posted it does not disappear, and the only uncertainty lies in the timing and order of subsequent shares.

From this model the authors derive a novel topological statistic they call rumor centrality. For a candidate node (v) in a tree‑shaped infection subgraph, let the subtrees hanging from (v) have sizes (|S_1|,\dots,|S_k|) (where (k) is the degree of (v) in the infection tree). The number of admissible infection orderings that are consistent with (v) being the true source is

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