Dynamic models of random simplicial complexes

随机单纯复形的动态模型

• 批准号: EP/P026729/1
• 负责人: Nikolaos Fountoulakis
• 金额: $ 34.31万
• 依托单位: University of Birmingham
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2017
• 资助国家: 英国
• 起止时间: 2017 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FP026729%2F1
• 关键词: Dynamic; models; random; simplicial; complexes

Networks have recently become a central notion in a variety of scientific disciplines such as computer science, data science, biology and economics. Their importance increased in the last decade as the development of the Internet and the mobile networks have had a significant impact on society and various sectors of the economy. This makes it important to develop an appropriate mathematical framework to understand their formation and evolution. In particular, it turns out that an intrinsic characteristic of network evolution is randomness. In fact, large networks involve randomess both in their formation and their evolution. One of the main challenges is to gain insight into how randomness actually gives rise to structure and self-organisation. The most common representation of networks is through graphs, where the interaction between individuals is represented by an edge that joins the corresponding nodes. However, in reality interactions are far more complex and, in fact, involve a multitude of individuals. In this project, we will use the mathematical framework of random discrete topological spaces to develop a systematic theory of the evolution of networks whose growth is characterised by multidimensional interactions. Such spaces have also been used in an entirely different context, as part of the development of the theory of quantum gravity. There, they were used as an expression of the granular structure of space-time, in an attempt to unify general relativity with quantum mechanics. In our project, we will consider even more general models which allow for dynamically growing structures. The objectives focus on the study of large-scale properties of these objects, based on tools and notions from combinatorics, probability theory and topology. Furthermore, we will explore how these features are linked to dynamics on these spaces, such as random walks and their mixing time. We will also determine the key combinatorial characteristics of these networks and investigate the emergence of global-scale phenomena in them. In particular, we will be aiming to determine those conditions on the growth mechanism which ensure the emergence of power-law degree distributions in the network as well as its small-world properties. Furthermore, we will aim to discover those conditions that characterise the emergence of condensation phenomena in these networks. Such phenomena are also relevant in statistical physics. In this context, they are linked to the emergence of irregularities in the distribution of the degrees and the existence of extremely highly connected nodes.

网络最近已经成为计算机科学、数据科学、生物学和经济学等各种科学学科的核心概念。在过去十年中，随着互联网和移动的网络的发展对社会和经济的各个部门产生了重大影响，它们的重要性有所增加。这使得开发一个适当的数学框架来理解它们的形成和演变变得非常重要。特别是，网络演化的一个内在特征是随机性。事实上，大型网络的形成和演化都具有随机性。主要挑战之一是深入了解随机性实际上如何产生结构和自组织。网络最常见的表示是通过图，其中个体之间的交互由连接相应节点的边表示。然而，实际上，互动要复杂得多，而且实际上涉及众多个人。在这个项目中，我们将使用随机离散拓扑空间的数学框架来发展一个系统的网络演化理论，其增长的特点是多维的相互作用。这样的空间也被用于一个完全不同的背景下，作为量子引力理论发展的一部分。在那里，它们被用来表达时空的颗粒结构，试图统一广义相对论和量子力学。在我们的项目中，我们将考虑更通用的模型，这些模型允许动态增长的结构。目标集中在这些对象的大规模属性的研究，基于组合学，概率论和拓扑学的工具和概念。此外，我们将探索这些特征如何与这些空间上的动力学联系起来，例如随机游动及其混合时间。我们还将确定这些网络的关键组合特征，并研究其中全球规模现象的出现。特别是，我们的目标是确定那些条件的增长机制，确保出现幂律度分布的网络，以及它的小世界属性。此外，我们的目标是发现这些网络中出现凝聚现象的条件。这些现象在统计物理学中也是相关的。在这种情况下，它们与度分布的不规则性和极高连接节点的存在有关。

项目成果

Vertex-isoperimetric stability in the hypercube

超立方体中的顶点等周稳定性

• DOI: 10.1016/j.jcta.2019.105186
• 发表时间: 2020
• 期刊: Journal of Combinatorial Theory, Series A
• 作者: Przykucki M

High-dimensional bootstrap processes in evolving simplicial complexes

演化单纯复形中的高维引导过程

• DOI: 10.48550/arxiv.1910.10139
• 发表时间: 2019
• 期刊: arXiv e-prints
• 作者: Fountoulakis Nikolaos

Dynamical Models for Random Simplicial Complexes

随机单纯复形的动力学模型

• DOI: 10.48550/arxiv.1910.12715
• 发表时间: 2019
• 期刊: arXiv e-prints
• 作者: Fountoulakis Nikolaos

Algebraic and combinatorial expansion in random simplicial complexes

随机单纯复形中的代数和组合展开式

• 发表时间: 2022
• 期刊: Random Structures and Algorithms
• 影响因子: 1
• 作者: Fountoulakis N.

Best response dynamics on random graphs

随机图上的最佳响应动态

• DOI: 10.1016/j.geb.2021.11.003
• 发表时间: 2022
• 期刊: Games and Economic Behavior
• 影响因子: 1.1
• 作者: Chellig J