A generalised mean-field approximation for the Deffuant opinion dynamics model on networks.

The paper derives a class-based mean-field (MF) system for the Deffuant model that explicitly accounts for network structure via class proportions q_k, normalized edge probabilities π_kl, and graph density γ. In the derivation, the probability in a time step dt=2/N that a class-k node interacts with a class-l node is 2 q_l π_kl/(N γ); passing to the continuous-time limit yields the effective interaction rate λ_kl = q_l π_kl/γ. The gain term uses the change of variables z = y + (x−y)/μ, giving a Jacobian 1/μ and the domain |x−y| < μ ε, while the loss term integrates over |x−y| < ε; assembling these yields the class-based MF equation (their Eq. (6)). Under the configuration-model assumption π_kl = k l/(N⟨k⟩), the prefactor reduces to (k l q_l)/⟨k⟩^2 (their Eq. (5)), and the single-class case with μ=1/2 recovers the original well-mixed MF equation, independent of π_11. All these steps appear explicitly in the paper’s derivation and statements . The candidate solution reproduces the same logic: it (i) computes λ_kl=q_l π_kl/γ, (ii) sets up gain–loss with the 1/μ Jacobian and domains, (iii) assembles the class-based MF system, (iv) specializes to configuration-model networks, and (v) reduces to the original MF for a single class. Minor differences are expository (they express selection probabilities as hazards), and the paper notes finite-N caveats (no self-edges; N→∞) that the model glosses over . Overall, the proofs are essentially the same and correct.