Opinion Dynamics in Social Networks with Edge-Heterogeneous Confidence Bounds: Clustering, Polarization, and Implications for Online Platforms

Opinion polarization, echo chambers, and the rapid formation of opinion clusters have become defining features of debates on contemporary online social platforms. To explain these phenomena from a control-theoretic perspective, this paper investigates opinion dynamics in social networks with edge-heterogeneous confidence bounds, focusing on clustering and polarization behaviors driven by pair-dependent trust and asymmetric influence. Two discrete-time models are proposed, including an unsigned bounded-confidence model and a more general signed model that incorporates both supportive and oppositional interactions. The interaction structures are described by time-varying unsigned and signed digraphs, respectively, in which heterogeneous interpersonal influence is characterized by edge-dependent confidence bounds that naturally encode platform-mediated trust. For the proposed models, rigorous sufficient conditions are established for invariant cluster consensus and structurally balanced polarization. Numerical simulations, including a case study on the Slashdot Zoo signed social network with 50 controversial users, illustrate the theoretical results and demonstrate their relevance for understanding opinion evolution on internet-scale platforms.