# Broken Detailed Balance and Non-Equilibrium Dynamics in Noisy Social Learning Models

## TL;DR

This paper introduces a continuous-time social learning model with noisy information sources, revealing how noise prevents consensus and leads to a non-equilibrium steady state with persistent opinion correlations.

## Contribution

It presents a novel Degroot-type model incorporating feedback and noise, demonstrating the emergence of non-equilibrium steady states in social opinion dynamics.

## Key findings

- Noisy information destroys consensus formation.
- The model exhibits non-equilibrium steady states with persistent opinion correlations.
- Synchronization of opinions occurs under common noise conditions.

## Abstract

We propose new Degroot-type social learning models with feedback in a continuous time, to investigate the effect of a noisy information source on consensus formation in a social network. Unlike the standard Degroot framework, noisy information models destroy consensus formation. On the other hand, the noisy opinion dynamics converge to the equilibrium distribution that encapsulates correlations among agents' opinions. Interestingly, such an equilibrium distribution is also a non-equilibrium steady state (NESS) with a non-zero probabilistic current loop. Thus, noisy information source leads to a NESS at long times that encodes persistent correlated opinion dynamics of learning agents. Our model provides a simple realization of NESS in the context of social learning. Other phenomena such as synchronization of opinions when agents are subject to a common noise are also studied.