  Information spreads across social and technological networks, but often the
network structures are hidden from us and we only observe the traces left by
the diffusion processes, called cascades. Can we recover the hidden network
structures from these observed cascades? What kind of cascades and how many
cascades do we need? Are there some network structures which are more difficult
than others to recover? Can we design efficient inference algorithms with
provable guarantees?
  Despite the increasing availability of cascade data and methods for inferring
networks from these data, a thorough theoretical understanding of the above
questions remains largely unexplored in the literature. In this paper, we
investigate the network structure inference problem for a general family of
continuous-time diffusion models using an $l_1$-regularized likelihood
maximization framework. We show that, as long as the cascade sampling process
satisfies a natural incoherence condition, our framework can recover the
correct network structure with high probability if we observe $O(d^3 \log N)$
cascades, where $d$ is the maximum number of parents of a node and $N$ is the
total number of nodes. Moreover, we develop a simple and efficient
soft-thresholding inference algorithm, which we use to illustrate the
consequences of our theoretical results, and show that our framework
outperforms other alternatives in practice.
