  We consider the problem of detecting the source of a rumor which has spread
in a network using only observations about which set of nodes are infected with
  the rumor and with no information as to \emph{when} these nodes became
infected. In a recent work \citep{ref:rc} this rumor source detection problem
was introduced and studied. The authors proposed the graph score function {\em
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 times for regular trees. They
showed that as the size of the infected graph increases, for a path graph
(2-regular tree), the probability of source detection goes to $0$ while for
$d$-regular trees with $d \geq 3$ the probability of detection, say $\alpha_d$,
remains bounded away from $0$ and is less than $1/2$. However, their results
stop short of providing insights for the performance of the rumor centrality
estimator in more general settings such as irregular trees or the SI model with
non-exponential spreading times.
  This paper overcomes this limitation and establishes the effectiveness of
rumor centrality for source detection for generic random trees and the SI model
with a generic spreading time distribution. The key result is an interesting
connection between a continuous time branching process and the effectiveness of
rumor centrality. Through this, it is possible to quantify the detection
probability precisely. As a consequence, we recover all previous results as a
special case and obtain a variety of novel results including the {\em
universality} of rumor centrality in the context of tree-like graphs and the SI
model with a generic spreading time distribution.
