  We study the information-theoretic lower bound of the sample complexity of
the correct recovery of diffusion network structures. We introduce a
discrete-time diffusion model based on the Independent Cascade model for which
we obtain a lower bound of order $\Omega(k \log p)$, for directed graphs of $p$
nodes, and at most $k$ parents per node. Next, we introduce a continuous-time
diffusion model, for which a similar lower bound of order $\Omega(k \log p)$ is
obtained. Our results show that the algorithm of Pouget-Abadie et al. is
statistically optimal for the discrete-time regime. Our work also opens the
question of whether it is possible to devise an optimal algorithm for the
continuous-time regime.
