Abstract
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 diffusion strategy, which blends the agent's data with the beliefs of its neighbors. We focus on weakly-connected graphs, where the network is partitioned into sending and receiving sub-networks, and we allow for heterogeneous models across the agents. First, we examine what agents learn (social learning) 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 how 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 (topology learning). An unexpected interplay between social and topology learning emerges: given H hypotheses and S sending sub-networks, topology learning can be feasible when H ≥ S. 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 diversity in the statistical models of the sending sub-networks.
Highlights
In a social learning problem, several agents linked through a network topology form their individual opinions about a phenomenon of interest by observing the beliefs of their neighboring agents [3]–[10]
One useful conclusion stemming from our analysis is to reveal some unexpected interplay between these two coexisting learning problems — see Section VIII further ahead
A similar behavior is observed, for example, over social networks where an influential agent scarcely consults information from his/her followers; or in the presence of stubborn agents, which insist on their opinion regardless of the evidence provided by their own observations or by neighboring agents
Summary
In a social learning problem, several agents linked through a network topology form their individual opinions about a phenomenon of interest (learning process) by observing the beliefs of their neighboring agents (social interaction) [3]–[10]. MATTA ET AL.: INTERPLAY BETWEEN TOPOLOGY AND SOCIAL LEARNING OVER WEAK GRAPHS to learn the strength of connections (weighted topology) from the sending components to the receiving agents This second question is useful in identifying the main sources for opinion formation over a network. The main difference in this work from [11], [12] is that we consider the following general setting: i) diffusion-type algorithms with linear combination of log-beliefs (as opposed to the beliefs themselves), ii) heterogeneous data and inference model, and iii) inverse topology learning. We are interested to learn the topology linking the receiving agents to the sending components This question is interesting because it would allow us to identify the main sources of information in a network and how they influence opinion formation.
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