Dynamic models of random simplicial complexes

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中心概念。在过去十年中，随着互联网和移动网络的发展对社会和经济各部门产生了重大影响，它们的重要性增加了。这使得开发一个适当的数学框架来理解它们的形成和演化变得重要。特别是，事实证明，网络进化的一个内在特征是随机性。事实上，大型网络在形成和演化过程中都包含着随机性。主要挑战之一是深入了解随机性实际上如何导致结构和自组织。最常见的网络表示是通过图，其中个体之间的交互由连接相应节点的边来表示。然而，在现实中，互动要复杂得多，事实上，涉及到许多个人。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以多维相互作用为特征的网络演化的系统理论。作为量子引力理论发展的一部分，这样的空间也被用于完全不同的背景下。在那里，它们被用来表达时空的颗粒结构，试图将广义相对论与量子力学统一起来。在我们的项目中，我们将考虑更一般的模型，允许动态增长的结构。目标是基于组合学、概率论和拓扑学的工具和概念来研究这些对象的大规模性质。此外，我们将探索这些特征是如何与这些空间上的动力学联系在一起的，例如随机行走及其混合时间。我们还将确定这些网络的关键组合特征，并调查它们中出现的全球尺度现象。具体地说，我们将致力于确定那些关于增长机制的条件，这些条件确保网络中出现幂律度分布以及其小世界性质。此外，我们将致力于发现这些网络中出现凝结现象的特征条件。这种现象在统计物理学中也是相关的。在这方面，它们与学位分布中出现的不规则性和高度连接的节点的存在有关。

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.