Dynamics of Contact Processes on Simplicial Complexes

近几十年来，图/网络上的相互作用粒子系统已经渗透到许多科学领域。建模思想是为每个顶点/节点关联一个状态，然后通过指定沿边/链接的顶点之间的相互作用来定义图上的动态系统。众所周知的例子是接触过程、选民模型或伊辛模型，它们都是在晶格型图（例如，在d维整数晶格上）的经典情况下提出的。在这个项目中，我们将研究由简单复合体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：以给定的速率恢复受感染的顶点，以及以与受感染的邻居数量成比例的速率感染易感顶点。然而，特别是在社会传染模型的背景下，仅仅允许沿边缘的两个顶点的二元交互通常是太简单了。在这个项目中，我们将学习规则，顶点如何在高维简单体中相互作用。在这些简单接触过程模型中，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简单结构对动力学的相关影响。特别是，我们将重点关注接触过程的三种主要结构：(1)简单晶格类配合物，(2)简单随机无标度配合物，(3)简单自适应配合物。对于(1)和(2)，我们期望通过耦合、对偶性、再生时间等概率技术获得几个严格的分析结果。对于自适应简单复合体，其中复合体的动力学和动力学是耦合的，我们将把数值模拟与形式矩闭包方案结合起来，推导近似的微分方程来研究(3)。综上所述，该项目将在高维几何结构、相互作用粒子系统、随机动力学和各种应用之间建立新的联系。

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.