Multidimensional Opinion Dynamics with Heterogeneous Bounded Confidences and Random Interactions

The paper proves almost-sure finite-time convergence and a negative-exponential mean-square rate for the heterogeneous, multidimensional RIBC model under the uniform full-support and independence assumption on Et, with the same absorbing-set characterization and the same explicit Tn bound and geometric tail as in the candidate solution. The proof strategy in the paper—reduce to a controllable (CIBC) system, design a merging schedule to achieve a uniform Tn, and then lift back via block-independence—matches the candidate’s outline essentially step-for-step; differences (e.g., naming Borel–Cantelli vs. using a direct geometric bound, or the specific control algorithmic details) are cosmetic. See Theorem 2 and its proof sketch, the CIBC bridge, and the Tn bound and final probabilistic estimate in the paper .