Phase transition of social learning collectives and the echo chamber

Shintaro Mori, Kazuaki Nakayama, and Masato Hisakado

Phys. Rev. E 94, 052301 – Published 2 November 2016

Abstract

We study a simple model for social learning agents in a restless multiarmed bandit. There are $N$ agents, and the bandit has $M$ good arms that change to bad with the probability $q_{c} / N$ . If the agents do not know a good arm, they look for it by a random search (with the success probability $q_{I}$ ) or copy the information of other agents' good arms (with the success probability $q_{O}$ ) with probabilities $1 - p$ or $p$ , respectively. The distribution of the agents in $M$ good arms obeys the Yule distribution with the power-law exponent $1 + γ$ in the limit $N, M \to \infty,$ and $γ = 1 + \frac{(1 - p) q_{I}}{p q_{O}}$ . The system shows a phase transition at $p_{c} = \frac{q_{I}}{q_{I} + q_{o}}$ . For $p < p_{c} (> p_{c})$ , the variance of $N_{1}$ per agent is finite (diverges as $\propto N^{2 - γ}$ with $N$ ). There is a threshold value $N_{s}$ for the system size that scales as $ln N_{s} \propto 1 / (γ - 1)$ . The expected value of the number of the agents with a good arm $N_{1}$ increases with $p$ for $N > N_{s}$ . For $p > p_{c}$ and $N < N_{s}$ , all agents tend to share only one good arm. If the shared arm changes to be bad, it takes a long time for the agents to find another good one. $E (N_{1})$ decreases to zero as $p \to 1$ , which is referred to as the “echo chamber.”