Characterization of the Nonequilibrium Steady State of a Heterogeneous Nonlinear q-Voter Model with Zealotry (Supplementary Material)

Supplementary material for Characterization of the Nonequilibrium Steady State of a Heteogeneous Nonlinear q-Voter Model with Zealotry.Authors: Andrew Mellor, Mauro Mobilia, R.K.P. ZiaItems include:An illustration of the model mechanism.Stationary state evolution as a function of z.Explicit expressions of LGA matrices, F and D.Vorticity and Stream Function: Definitions and Figure 3.Comparisons between precision numerical results and simulation data.Comparisons between LGA predictions and simulation results.Abstract:We introduce an heterogeneous nonlinear q-voter model with two types of susceptible voters and zealots, and study its non-equilibrium properties when the population is finite and well mixed. In this two-opinion model, each individual supports one of two parties and is either a susceptible voter of type q1 or q2, or is an inflexible zealot. At each time step, a qi-susceptible voter (i=1,2) consults a group of qi neighbors and adopts their opinion if all group members agree, while zealots are inflexible and never change their opinion. We show that this model violates the detailed balance whenever q1≠q2and has surprisingly rich properties. Here, we focus on the characterization of the model’s non-equilibrium stationary state (NESS) in terms of its probability distribution and currents in the distinct regimes of low and high density of zealotry. We unveil the NESS properties in each of these phases by computing the opinion distribution and the circulation of probability currents, as well as the two-point correlation functions at unequal times (formally related to a “probability angular momentum”). Our analytical calculations obtained in the realm of a linear Gaussian approximation are compared with numerical results.Viewing Notes:To view the supplementary materials, download all files and run index.html in a web browser (preferably Chrome).《带有狂热者的非均匀非线性q投票者模型的非平衡稳态表征补充材料》 作者：Andrew Mellor、Mauro Mobilia、R.K.P. Zia 包含内容： 1. 模型机制示意图 2. 以z为自变量的稳态演化过程 3. LGA矩阵F与D的显式表达式 4. 涡度与流函数：定义及图3 5. 高精度数值结果与仿真数据的对比 6. LGA预测结果与仿真结果的对比 摘要：本文提出一种包含两类易感选民与狂热者的非均匀非线性q投票者模型（q-Voter Model），并研究了有限规模且完全混合群体下该模型的非平衡特性。在这一二意见模型中，每个个体要么支持两党之一，且属于q₁型或q₂型易感选民，要么为僵化的狂热者。每一时间步中，qᵢ型易感选民（i=1,2）会咨询一组同类型邻居，若所有群体成员意见一致，则采纳该意见；而狂热者始终僵化，不会改变自身立场。研究表明，当q₁≠q₂时，该模型违背细致平衡条件，且具备极为丰富的特性。本文聚焦于该模型的非平衡稳态（Non-equilibrium stationary state, NESS）表征，以低、高狂热者密度的不同状态区间内的概率分布与概率流为切入点，通过计算意见分布、概率环流以及非等时两点关联函数（形式上与"概率角动量"相关），揭示了各相中的非平衡稳态特性。本文将线性高斯近似框架下得到的解析计算结果与数值结果进行了对比。 查看说明：如需查看补充材料，请下载全部文件并在网页浏览器（推荐使用Chrome）中运行index.html。