  In many real-world scenarios, it is nearly impossible to collect explicit
social network data. In such cases, whole networks must be inferred from
underlying observations. Here, we formulate the problem of inferring latent
social networks based on network diffusion or disease propagation data. We
consider contagions propagating over the edges of an unobserved social network,
where we only observe the times when nodes became infected, but not who
infected them. Given such node infection times, we then identify the optimal
network that best explains the observed data. We present a maximum likelihood
approach based on convex programming with a l1-like penalty term that
encourages sparsity. Experiments on real and synthetic data reveal that our
method near-perfectly recovers the underlying network structure as well as the
parameters of the contagion propagation model. Moreover, our approach scales
well as it can infer optimal networks of thousands of nodes in a matter of
minutes.
