  Many optimization, inference and learning tasks can be accomplished
efficiently by means of decentralized processing algorithms where the network
topology (i.e., the graph) plays a critical role in enabling the interactions
among neighboring nodes. There is a large body of literature examining the
effect of the graph structure on the performance of decentralized processing
strategies. In this article, we examine the inverse problem and consider the
reverse question: How much information does observing the behavior at the nodes
of a graph convey about the underlying topology? For large-scale networks, the
difficulty in addressing such inverse problems is compounded by the fact that
usually only a limited fraction of the nodes can be probed, giving rise to a
second important question: Despite the presence of unobserved nodes, can
partial observations still be sufficient to discover the graph linking the
probed nodes? The article surveys recent advances on this challenging learning
problem and related questions.
