Learning in Hierarchical Social Networks

TL;DR

This paper analyzes how hierarchical social networks can efficiently aggregate information for binary hypothesis testing, providing bounds on error probabilities and exploring message-passing schemes with non-binary messages.

Contribution

It introduces bounds on error probabilities in hierarchical networks and characterizes the impact of message alphabet size on error decay rates.

Findings

Error probabilities decrease exponentially with the number of leaf agents.

Hierarchical structure influences the convergence rate of error probabilities.

Non-binary message schemes improve error exponent compared to binary messages.

Abstract

We study a social network consisting of agents organized as a hierarchical M-ary rooted tree, common in enterprise and military organizational structures. The goal is to aggregate information to solve a binary hypothesis testing problem. Each agent at a leaf of the tree, and only such an agent, makes a direct measurement of the underlying true hypothesis. The leaf agent then makes a decision and sends it to its supervising agent, at the next level of the tree. Each supervising agent aggregates the decisions from the M members of its group, produces a summary message, and sends it to its supervisor at the next level, and so on. Ultimately, the agent at the root of the tree makes an overall decision. We derive upper and lower bounds for the Type I and II error probabilities associated with this decision with respect to the number of leaf agents, which in turn characterize the converge rates of the Type I, Type II, and total error probabilities. We also provide a message-passing scheme involving non-binary message alphabets and characterize the exponent of the error probability with respect to the message alphabet size.