发表机构
Faculty of Computer Science, RWTH Aachen University; Sibley School of Mechanical and Aerospace Engineering, Cornell University; Division of Decision and Control Systems, School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology; Digital Futures（亚琛工业大学计算机学院; 康奈尔大学西尔比机械与航空航天工程学院; 皇家理工学院电气与计算机学院决策与控制系; 数字未来）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文利用图元研究大规模网络上非线性意见动态，证明图元动态近似有限维系统且均衡点可被连续极限近似，实现基于图元结构的意见分布量化表征。
AI 中文摘要
意见动态已在多个学科中研究数十年，大量理论文献关注共识、极化和聚类等行为。尽管经典模型在模拟中能展现出更复杂的意见模式，但对此类分布的量化尚未被完全理解。为解决这一问题，本文利用图元（graphons）研究大规模网络上非线性意见动态的行为，图元能够捕捉底层网络结构。在该模型中，智能体根据包含交互中饱和效应的非线性规则更新其意见。网络由从图元生成的随机图表示，并在图元上引入了相应的非线性动态模型。我们证明，当网络规模较大时，图元动态近似有限维系统。利用随机图的谱近似结果，我们进一步证明非线性模型的均衡点也可由连续极限的均衡点近似。这一结果使得基于底层图元结构对意见分布进行量化表征成为可能。理论结果通过数值模拟加以说明。
英文摘要
Opinion dynamics have been studied for decades across disciplines, with much of the theoretical literature focusing on behaviors such as consensus, polarization, and clustering. Although classic models can exhibit more complex opinion patterns in simulations, quantifying such distributions is not fully understood. To address this question, in this paper, we study the behavior of nonlinear opinion dynamics over large-scale networks using graphons, which capture the underlying network structure. In the model, agents update their opinions according to a nonlinear rule that includes saturation effects in interactions. The network is represented by random graphs generated from a graphon, and a corresponding nonlinear dynamical model is introduced over the graphon. We show that the graphon dynamics approximate the finite-dimensional system, when the network size is large. Leveraging spectral approximation results for random graphs, we further show that the equilibria of the nonlinear model can also be approximated by those of the continuum limit. This result enables a quantitative characterization of the opinion distribution based on the underlying graphon structure. The theoretical results are illustrated by numerical simulation.
