Graphops上的网络动力学
Network Dynamics on Graphops
摘要：
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。更确切地说，众所周知，在充分混合/均匀/全对全耦合系统中，可以导出平均场极限方程，例如Vlasov-Fokker-Planck方程（VFPE）。介观VFPE描述了在某种状态下找到单个顶点/粒子的概率，形成了微观统计物理和宏观流体类型近似之间的桥梁。在这个框架中的一个主要障碍是将复杂的网络结构纳入极限方程。在许多情况下，只存在启发式近似，或者极限依赖于特定类别的积分算子。在本文中，我们注意到，有一个更优雅的，深刻的更一般的方式，由于最近的进展，在理论的图形限制。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。
英文摘要：
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.
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