SCIENCE CHINA Information Sciences, Volume 63, Issue 1: 112202(2020) https://doi.org/10.1007/s11432-019-9890-5

## Asymptotic properties of distributed social sampling algorithm

• AcceptedApr 29, 2019
• PublishedDec 23, 2019
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### 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. First, 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 sample algorithm is addressed. Finally, we provide a numerical simulation to validate the theoretical results of this paper.

### Acknowledgment

This work was supported by National Key Research and Development Program of China (Grant No. 2016YFB0901900) and National Natural Science Foundation of China (Grant No. 61573345).

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• Figure 1

(Color online) Trace of estimate of $Q_{i,k}$ for $i\in~\left[~N\right]$ over every single opinion state $m\in\left[~M\right]$ with $M=4$. protectłinebreak (a) $m=1$, the estimate of the first element of true empirical distribution $q^{\ast}$; (b) $m=2$, the estimate of the second element of $q^{\ast}$; (c) $m=3$, the estimate of the third element of $q^{\ast}$; (d) $m=4$, the estimate of the last element of $q^{\ast}$.

• Figure 2

(Color online) Histogram and limit distribution for $\left(~Q_{i,k}-q^{\ast}\right)/\sqrt{\delta_{k}}$ at $k=500$. (a) Node 5; (b) node 6; protectłinebreak (c) node 10; (d) node 24.

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