Non-Bayesian Social Learning on Random Digraphs with Aperiodically Varying Network Connectivity

Abstract

We study non-Bayesian social learning on random directed graphs and show that under mild assumptions on the connectivity of the network, all the agents almost surely learn the true state of the world asymptotically in time if the sequence of the associated weighted adjacency matrices belongs to Class P * (a broad class of stochastic chains that subsumes uniformly strongly connected chains). We show that though uniform strong connectivity is not necessary for asymptotic learning, it helps ensure that all the agents' beliefs converge to a consensus almost surely even when the true state is not identifiable. We then show how our main result applies to a few variants of the original model such as inertial non-Bayesian learning and learning in the presence of link failures. Besides, we show that our main result is an extension of a few known results that pertain to learning on time-varying graphs. We also show by proof and an example that if the network of influences is balanced in a certain sense, then asymptotic learning occurs almost surely even in the absence of uniform strong connectivity.

Towards a Perron-Frobenius Theorem for Strongly Aperiodic Stochastic Chains

“…In addition, there is a network of interacting agents whose common objective is to learn the identity of the true state through mutual interaction as well as by performing private measurements on the state of the world. We note that [24] generalizes certain known results on distributed learning to networks described by random, independently distributed time-varying directed graphs. Importantly, the sequence of weighted adjacency matrices of all the networks considered therein are assumed to belong to Class P * .…”

“…Importantly, the sequence of weighted adjacency matrices of all the networks considered therein are assumed to belong to Class P * . Hence, along with Definitions 13 and 14, Theorem 1 significantly facilitates our interpretation of the main results of [24].…”

“…Other Related Works: This endeavor evolves out of our work on social learning over timevarying networks [23,24]. Therein, we showed that non-Bayesian social learning can occur asymptotically almost surely over random time-varying networks even if standard connectivity conditions are violated.…”