Interplay Between Topology and Social Learning Over Weak Graphs

This work examines a distributed learning problem where the agents of a network form their beliefs about certain hypotheses of interest. Each agent collects streaming (private) data and updates continually its belief by means of a <italic>diffusion</italic> strategy, which blends the agent's data with the beliefs of its neighbors. We focus on <italic>weakly-connected</italic> graphs, where the network is partitioned into sending and receiving sub-networks, and we allow for heterogeneous models across the agents. First, we examine <italic>what</italic> agents learn (<italic>social learning</italic>) and provide an analytical characterization for the long-term beliefs at the agents. Among other effects, the analysis predicts when a leader-follower behavior is possible, where some sending agents control the beliefs of the receiving agents by forcing them to choose a particular and possibly fake hypothesis. Second, we consider the dual or reverse learning problem that reveals <italic>how</italic> agents learn: given the beliefs collected at a receiving agent, we would like to discover the influence that any sending sub-network might have exerted on this receiving agent (<italic>topology learning</italic>). An unexpected interplay between social and topology learning emerges: given <inline-formula><tex-math notation="LaTeX">$H$</tex-math></inline-formula> hypotheses and <inline-formula><tex-math notation="LaTeX">$S$</tex-math></inline-formula> sending sub-networks, topology learning can be feasible when <inline-formula><tex-math notation="LaTeX">$H\geq S$</tex-math></inline-formula>. The latter being only a necessary condition, we then examine the feasibility of topology learning for two useful classes of problems. The analysis reveals that a critical element to enable topology learning is a sufficient degree of <italic>diversity</italic> in the statistical models of the sending sub-networks.