  We study the problem of identifying the source of a stochastic diffusion
process spreading on a graph based on the arrival times of the diffusion at a
few queried nodes. In a graph $G=(V,E)$, an unknown source node $v^* \in V$ is
drawn uniformly at random, and unknown edge weights $w(e)$ for $e\in E$,
representing the propagation delays along the edges, are drawn independently
from a Gaussian distribution of mean $1$ and variance $\sigma^2$. An algorithm
then attempts to identify $v^*$ by querying nodes $q \in V$ and being told the
length of the shortest path between $q$ and $v^*$ in graph $G$ weighted by $w$.
We consider two settings: non-adaptive, in which all query nodes must be
decided in advance, and adaptive, in which each query can depend on the results
of the previous ones. Both settings are motivated by an application of the
problem to epidemic processes (where the source is called patient zero), which
we discuss in detail.
  We characterize the query complexity when $G$ is an $n$-node path. In the
non-adaptive setting, $\Theta(n\sigma^2)$ queries are needed for $\sigma^2 \leq
1$, and $\Theta(n)$ for $\sigma^2 \geq 1$. In the adaptive setting, somewhat
surprisingly, only $\Theta(\log\log_{1/\sigma}n)$ are needed when $\sigma^2
\leq 1/2$, and $\Theta(\log \log n)+O_\sigma(1)$ when $\sigma^2 \geq 1/2$. This
is the first mathematical study of source identification with time queries in a
non-deterministic diffusion process.
