Phase transition of social learning collectives and the echo chamber

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 $1+\gamma$ in the limit $N,M\to \infty ,$ and $\gamma =1+\frac{(1-p)q_I}{p{q}_O}$. 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 $\propto N^{2-\gamma }$ with $N$). There is a threshold value $N_s$ for the system size that scales as $ln{N}_s\propto 1/(\gamma -1)$. 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. $\text{E}\left(N_1\right)$ decreases to zero as $p\to 1$, which is referred to as the “echo chamber.”