论文标题
图形机上的网络动力学
Network Dynamics on Graphops
论文作者
论文摘要
在此简短说明中，我们报告了一个形式的数学观察结果：我们将在分析相互作用的粒子系统中违反一个多世纪历史的障碍。更确切地说，众所周知，在混合/均匀/全耦合系统中，可能会导出均值场限制方程，例如vlasov-fokker-planck方程（VFPES）。 A介质VFPE描述了在特定状态下找到单个顶点/粒子的概率，从而在微观统计物理学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纳入限制方程式。在许多情况下，仅存在启发式近似值，或者限制依赖于特定类别的整体操作员。在本文中，我们注意到，由于图形限制理论的最新进展，有一种更加优雅，更加通用的方式。特别是，我们展示了如何通过Graphops（Graph Operator）轻松地进入VFPE的复杂网络动力学。
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.
