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.Comment: 43 pages, 13 figure
