  We study the problem of parameter estimation based on infection data from an
epidemic outbreak on a graph. We assume that successive infections occur via
contagion; i.e., transmissions can only spread across existing directed edges
in the graph. Our stochastic spreading model allows individual nodes to be
infected more than once, and the probability of the transmission spreading
across a particular edge is proportional to both the cumulative number of times
the source nodes has been infected in previous stages of the epidemic and the
weight parameter of the edge. We propose a maximum likelihood estimator for
inferring the unknown edge weights when full information is available
concerning the order and identity of successive edge transmissions. When the
weights take a particular form as exponential functions of a linear combination
of known edge covariates, we show that maximum likelihood estimation amounts to
optimizing a convex function, and produces a solution that is both consistent
and asymptotically normal. Our proofs are based on martingale convergence
theorems and the theory of weighted P\'{o}lya urns. We also show how our theory
may be generalized to settings where the weights are not exponential. Finally,
we analyze the case where the available infection data comes in the form of an
unordered set of edge transmissions. We propose two algorithms for weight
parameter estimation in this setting and derive corresponding theoretical
guarantees. Our methods are validated using both synthetic data and real-world
data from the Ebola spread in West Africa.
