  We study the problem of estimating the source of a network cascade. The
cascade starts from a single vertex at time 0 and spreads over time, but only a
noisy version of the propagation is observable. The goal is then to design a
stopping time and estimator that will estimate the source well while ensuring
the cost of the cascade to the system is not too large. We rigorously formulate
a Bayesian approach to the problem. If vertices can be labelled by vectors in
Euclidean space (which is natural in geo-spatial networks), the optimal
estimator is the conditional mean estimator, and we derive an explicit form for
the optimal stopping time under minimal assumptions on the cascade dynamics. We
study the performance of the optimal stopping time on lattices, and show that a
computationally efficient but suboptimal stopping time which compares the
posterior variance to a threshold has near-optimal performance.
