# Asymptotic Properties of Distributed Social Sampling Algorithm

## TL;DR

This paper analyzes the asymptotic behavior of a distributed social sampling algorithm, proving convergence, consistency, and normality of the estimates in social networks.

## Contribution

It introduces a stochastic approximation framework for analyzing the algorithm's properties and establishes convergence, consistency, and asymptotic normality results.

## Key findings

- Agents' estimates converge to a common distribution.
- Strong consistency with almost sure convergence to the initial empirical distribution.
- Asymptotic normality of estimation error.

## Abstract

Social sampling is a novel randomized message passing protocol inspired by social communication for opinion formation in social networks. In a typical social sampling algorithm, each agent holds a sample from the empirical distribution of social opinions at initial time, and it collaborates with other agents in a distributed manner to estimate the initial empirical distribution by randomly sampling a message from current distribution estimate. In this paper, we focus on analyzing the theoretical properties of the distributed social sampling algorithm over random networks. Firstly, we provide a framework based on stochastic approximation to study the asymptotic properties of the algorithm. Then, under mild conditions, we prove that the estimates of all agents converge to a common random distribution, which is composed of the initial empirical distribution and the accumulation of quantized error. Besides, by tuning algorithm parameters, we prove the strong consistency, namely, the distribution estimates of agents almost surely converge to the initial empirical distribution. Furthermore, the asymptotic normality of estimation error generated by distributed social sampling algorithm is addressed. Finally, we provide a numerical simulation to validate the theoretical results of this paper.