Dynamics of Contact Processes on Simplicial Complexes

近几十年来，图/网络上相互作用的粒子系统已经渗透到许多科学中。其建模思想是将状态关联到每个顶点/节点，然后通过指定沿边/链接的顶点之间的交互来定义图上的动态系统。众所周知的例子是接触过程、投票者模型或伊辛模型，它们都是在格型图的经典情况下提出的，例如，在d维整数格子上。在这个项目中，我们将研究由单纯复形给出的更一般几何结构上的接触过程。整数格上的经典接触过程由两个更新规则定义：以给定的速率恢复感染顶点和以与感染邻居数量成正比的速率感染敏感顶点。然而，尤其是在社会传染建模的背景下，仅仅允许沿一条边的两个顶点进行二元互动往往过于简单。在这个项目中，我们将研究规则，即顶点如何跨越更高维度的简化进行交互。在这些单纯的接触过程模型中，我们将研究马尔可夫过程的数学基础，不变测度的存在性，以及单纯结构对动力学的相关影响。特别是，我们将重点讨论接触过程的三种主要结构：(1)单纯格型复形，(2)单纯随机无标度复形，和(3)单纯自适应复形。对于(1)和(2)，我们期望通过耦合、对偶、再生次数等概率技术得到几个严格的解析结果。对于自适应单纯形复形，其中复形和复形上的动力学是耦合的，我们将结合数值模拟和形式矩闭合格式来推导近似微分方程来研究(3)。总而言之，拟议的项目将导致高维几何结构、相互作用的粒子系统、随机动力学和各种应用之间的基本新联系。

Interacting particle systems on graphs/networks have permeated many sciences in recent decades. The modelling idea is to associate to each vertex/node a state, and then to define a dynamical system on the graph by specifying the interaction between vertices along the edges/links. Well-known examples are the contact process, the voter model, or the Ising model, all posed in the classical case on lattice-type graphs, e.g., on the d-dimensional integer lattice. In this project, we are going to study the contact process on more general geometric structures given by simplicial complexes. The classical contact process on the integer lattice is defined by two update rules: the recovery of an infected vertex at a given rate, and the infection of a susceptible vertex at a rate proportional to the number of infected neighbors. Yet, particularly in the context of social contagion modelling, just allowing binary interactions of two vertices along an edge is often to simple. In this project we are going to study rules, how vertices can interact across higher-dimensional simplices. In these simplicial contact process models, we are going to study the mathematical basis of the Markov process, the existence of invariant measures, and the related influence of the simplicial structure on dynamics. In particular, we are going to focus on three main structures for the contact process: (1) simplicial lattice-like complexes, (2) simplicial random scale-free complexes, and (3) simplicial adaptive complexes. For (1) and (2), we expect to obtain several rigorous analytical results via probabilistic techniques such as coupling, duality, regeneration times, etc. For adaptive simplicial complexes, where dynamics of and on the complex is coupled, we are going to combine numerical simulation with formal moment closure schemes to derive approximating differential equations to study (3). In summary, the proposed project is going to lead to fundamental new links between higher-dimensional geometric structures, interacting particle systems, stochastic dynamics, and various applications.