Stability Analysis of Echo Chambers: A Stochastic Compartmental Model of Information Diffusion in Scale-Free Networks

The digital information ecosystem is increasingly characterized by the rapid proliferation of misinformation and the spontaneous emergence of polarized communities known as echo chambers. While classical epidemiological models have been widely adapted to describe information diffusion, they frequently fail to account for the unique sociotechnical mechanisms governing digital interaction: specifically, the interplay between individual cognitive resistance and algorithmic curation. This paper introduces a modified Susceptible-Critical-Infected-Recovered (SCIR) compartmental model designed to analyze the stability of echo chambers within scale-free networks. We explicitly incorporate two novel parameters: Cognitive Immunity (σ), representing the psychological capacity to critically evaluate and reject misinformation, and Algorithmic Bias (α), quantifying the platform-driven amplification of homophilic interactions. By applying heterogeneous mean-field theory, we derive the exact basic reproduction number, R0, for this system. Our stability analysis reveals that while scale-free topologies typically exhibit a vanishing epidemic threshold, high levels of algorithmic bias can stabilize endemic states of misinformation (echo chambers) even in the presence of moderate cognitive immunity. Conversely, we demonstrate that a critical threshold of cognitive immunity exists, beyond which the information cascade collapses, regardless of algorithmic amplification. These findings provide a rigorous mathematical framework for evaluating the efficacy of ”prebunking” interventions versus algorithmic regulation in mitigating the spread of digital disinformation.