Opinion Dynamics Incorporating Higher-Order Interactions
Zuobai Zhang, Wanyue Xu, Zhongzhi Zhang, Guanrong Chen
TL;DR
This work addresses the limitation of classic opinion dynamics models by incorporating higher-order, long-range interactions through a higher-order random-walk framework. It extends the Friedkin-Johnsen model with a random-walk matrix polynomial $P^{*}$ and update rule $ oldsymbol{x}^{(t+1)}=oldsymbol{ abla}oldsymbol{s}+( oldsymbol{I}-oldsymbol{ abla}) oldsymbol{P}^{*}oldsymbol{x}^{(t)}$, with $P^{*}=oldsymbol{I}-oldsymbol{D}^{-1}oldsymbol{L}_{oldsymbol{eta}}(oldsymbol{G})=oldsymbol{I}+eta_1oldsymbol{P}+eta_2oldsymbol{P}^2+\cdots+eta_Toldsymbol{P}^T$. The authors prove a unique equilibrium $ oldsymbol{z}^*$ solving $(oldsymbol{I}-(oldsymbol{I}-oldsymbol{ abla})oldsymbol{P}^{*})oldsymbol{z}^*=oldsymbol{ abla}oldsymbol{s}$ and develop a sparsification-based method to approximate $ oldsymbol{z}^*$ in nearly linear time, maintaining spectral similarity to the dense model. Extensive experiments on real networks show that higher-order interactions can markedly alter equilibria relative to the standard FJ model, and the proposed Approx method achieves large-scale efficiency with small mean absolute error. This work thus enables scalable analysis of polarization, disagreement, and related quantities in networks with rich, long-range interactions.
Abstract
The issue of opinion sharing and formation has received considerable attention in the academic literature, and a few models have been proposed to study this problem. However, existing models are limited to the interactions among nearest neighbors, ignoring those second, third, and higher-order neighbors, despite the fact that higher-order interactions occur frequently in real social networks. In this paper, we develop a new model for opinion dynamics by incorporating long-range interactions based on higher-order random walks. We prove that the model converges to a fixed opinion vector, which may differ greatly from those models without higher-order interactions. Since direct computation of the equilibrium opinion is computationally expensive, which involves the operations of huge-scale matrix multiplication and inversion, we design a theoretically convergence-guaranteed estimation algorithm that approximates the equilibrium opinion vector nearly linearly in both space and time with respect to the number of edges in the graph. We conduct extensive experiments on various social networks, demonstrating that the new algorithm is both highly efficient and effective.