Phase transition of social learning collectives and the echo chamber

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→∞, and γ=1+(1-p)q_{I}/pq_{O}. The system shows a phase transition at p_{c}=q_{I}/q_{I}+q_{o}. For pp_{c}), the variance of N_{1} per agent is finite (diverges as ∝N^{2-γ} with N). There is a threshold value N_{s} for the system size that scales as lnN_{s}∝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