  We consider the problem of identifying the topology of a weighted, undirected
network $\mathcal G$ from observing snapshots of multiple independent consensus
dynamics. Specifically, we observe the opinion profiles of a group of agents
for a set of $M$ independent topics and our goal is to recover the precise
relationships between the agents, as specified by the unknown network $\mathcal
G$. In order to overcome the under-determinacy of the problem at hand, we
leverage concepts from spectral graph theory and convex optimization to unveil
the underlying network structure. More precisely, we formulate the network
inference problem as a convex optimization that seeks to endow the network with
certain desired properties -- such as sparsity -- while being consistent with
the spectral information extracted from the observed opinions. This is
complemented with theoretical results proving consistency as the number $M$ of
topics grows large. We further illustrate our method by numerical experiments,
which showcase the effectiveness of the technique in recovering synthetic and
real-world networks.
