In this brief note, we report a formal mathematical observation: we are about to breach a major century-old barrier in the analysis of interacting particle systems. More precisely, it is well-known that in well-mixed/homogeneous/all-to-all-coupled systems, one may derive mean-field limit equations such as Vlasov-Fokker-Planck equations (VFPEs). A mesoscopic VFPE describes the probability of finding a single vertex/particle in a certain state, forming a bridge between microscopic statistical physics and macroscopic fluid-type approximations. One major obstacle in this framework is to incorporate complex network structures into limiting equations. In many cases, only heuristic approximations exist, or the limits rely on particular classes of integral operators. In this paper, we notice that there is a much more elegant, and profoundly more general, way available due to recent progress in the theory of graph limits. In particular, we show how one may easily enter complex network dynamics via graphops (graph operators) into VFPEs.

中文翻译：

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Graphops 上的网络动力学

在这篇简短的笔记中，我们报告了一个正式的数学观察：我们即将突破相互作用粒子系统分析中的一个主要的百年障碍。更准确地说，众所周知，在充分混合/均匀/全部耦合的系统中，可以推导出平均场极限方程，例如 Vlasov-Fokker-Planck 方程 (VFPE)。细观 VFPE 描述了在特定状态下找到单个顶点/粒子的概率，在微观统计物理和宏观流体类型近似之间形成桥梁。该框架中的一个主要障碍是将复杂的网络结构合并到限制方程中。在许多情况下，只存在启发式近似，或者限制依赖于特定类别的积分运算符。在这篇论文中，我们注意到有一个更优雅、更广泛的、由于图极限理论的最新进展，可用的方法。特别是，我们展示了如何通过 graphops（图运算符）轻松地将复杂的网络动态输入到 VFPE 中。