Normalized Friedkin–Johnsen Opinion Dynamics

As online social platforms increasingly serve as central arenas for opinion aggregation, numerous opinion dynamics models have been extensively developed to characterize how opinions propagate and to predict emerging social phenomena. Among various models, the Friedkin–Johnsen (FJ) model stands out as one of the most extensively studied frameworks, with broad applications across diverse domains and recent inspiration on the design of graph neural network architectures. However, the conventional FJ row-stochastic pairwise interaction has been shown to induce imbalanced influence patterns, often weakening the impact of high-degree nodes. To address this issue, we propose a normalized FJ model based on the normalized Laplacian, which symmetrically incorporates degree information from both endpoints. We establish its convergence and analyze its structural properties and intrinsic balance, along with a game-theoretic interpretation. We further develop efficient algorithms for computing equilibrium opinions and related metrics. Experiments on large-scale networks demonstrate the effectiveness and scalability of our approach.