Bounded-Confidence Models of Multidimensional Opinions with Topic-Weighted Discordance

The paper proves convergence of the topic-weighted HK and DW models by verifying that each update can be written as a product of row-stochastic matrices satisfying Lorenz’s sufficient conditions (positive diagonal, symmetric support, and a uniform lower bound on positive entries), thereby giving existence of a limit X* (see Theorem 3.1 and its use in Theorems 3.2–3.3: matrices (3.3)–(3.4) and the “readily checked” conditions ; explicit forms in and ; model definitions in and discordance in ). It then proves absorbing-separation at the limit via an ε–δ contradiction for HK (Lemma 3.4 and Theorem 3.5) and a pathwise almost-sure contradiction for DW (Theorem 3.6), both consistent and technically sound (, ). The candidate solution establishes the same results by invoking Hendrickx–Tsitsiklis’ cut-balance theorem and the infinite-flow graph characterization. The assumptions it checks (row-stochasticity, uniformly positive diagonals, type-symmetry/cut-balance) do hold for the paper’s models; its absorbing-state arguments via infinite flow are valid. Thus, both are correct, with different proof routes (Lorenz vs. cut-balance/infinite-flow). Minor clarifications the paper could add: explicitly state independence of random edge-topic sampling for the DW almost-sure claim and spell out the uniform lower bound δ (e.g., δ = μ/M) when invoking Theorem 3.1.