OPINION DYNAMICS MODELS WITH MEMORY IN COOPETITIVE SOCIAL NETWORKS: ANALYSIS, APPLICATION AND SIMULATION

The paper’s main polarization result under the memory-based communication rule introduces the delay-eliminating auxiliary state z_i(t) and proves ż_i = A z_i + Ā u_i, with Ā = σΛ + (1 − σ)e^{−Ah}Λ, exactly as the candidate does (their Ā is the paper’s A in (3.2); see (3.1)–(3.2) and (7.10) . Using structural balance, the signed interactions are gauged into an unsigned Laplacian form, and a Jordan/Kronecker decomposition reduces stability to modal conditions in which the disagreement modes are governed by blocks A − λ_i Ā F; this is stated in Theorem 3.1 and derived via the determinant factorization ∏_{i=2}^N |sI − (A − λ_i Ā F)| in the appendix . The paper’s definition of “equal polarization” allows a (possibly nonstationary) limiting trajectory x*(t), matching the candidate’s convergence to e^{At}c up to the same gauge transform . The candidate’s steps—delay elimination, gauge transform, stacked LTI form, modal reduction, decay of disagreement modes, and reconstruction to x—mirror the paper’s proof structure and conclusions; the only discrepancy is a notational swap (A vs Ā) that the candidate notes. Overall, both arguments are correct and essentially the same in substance.