Dynamics of Contact Processes on Simplicial Complexes

图/网络上的相互作用粒子系统在最近几十年已经渗透到许多科学领域。建模思想是将每个顶点/节点与状态相关联，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，投票模型或伊辛模型，所有这些都是在格型图的经典情况下提出的，例如，在d维整数格上。在这个项目中，我们将研究由单纯复形给出的更一般的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：以给定的速率恢复被感染的顶点，以及以与被感染的邻居数目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，仅仅允许两个顶点沿沿着边的二元交互往往过于简单。在这个项目中，我们将研究规则，顶点如何在高维单纯形中相互作用。在这些单纯接触过程模型中，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。特别地，我们将关注接触过程的三种主要结构：（1）单纯格状复形，（2）单纯随机无标度复形，（3）单纯自适应复形。对于（1）和（2），我们希望通过概率技术，如耦合，对偶，再生时间等，获得一些严格的分析结果。对于自适应单纯复合体，其中复合体的动力学是耦合的，我们将联合收割机结合数值模拟与形式矩封闭方案，以获得近似微分方程研究（3）。总之，该项目将导致更高维度的几何结构，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.