  We consider the task of estimating a network cascade as fast as possible. The
cascade is assumed to spread according to a general Susceptible-Infected
process with heterogeneous transmission rates from an unknown source in the
network. While the propagation is not directly observable, noisy information
about its spread can be gathered through multiple rounds of error-prone
diagnostic testing. We propose a novel adaptive procedure which quickly outputs
an estimate for the cascade source and the full spread under this observation
model. Remarkably, under mild conditions on the network topology, our procedure
is able to estimate the full spread of the cascade in an $n$-vertex network,
before $\mathrm{poly log}(n)$ vertices are affected by the cascade. We
complement our theoretical analysis with simulation results illustrating the
effectiveness of our methods.
