Theory and applications of higher-order network models

网络（或图形）提供了一个有用的建模工具来描述非常不同的复杂系统，从大脑和生态系统到社会组织、金融市场、交通系统和电网，等等。由于这些复杂的系统由相互作用或相互关联的实体组成，网络模型将它们表示为图形，其中组件被描述为顶点，边缘捕获它们之间的成对（二元）交互。这些模型的主要优势之一是它们能够将系统结构与其动态联系起来；例如，一群人的交流网络可以提供关于一个新想法如何在这样一个社区的成员中传播的见解，因为网络是允许信息流动的平均数。另一方面，当前网络模型的一个基本缺点是它们无法适当地捕获高阶交互，即那些发生在两个以上组件之间的交互。例如，考虑三个人，分别叫A、B和C，他们一起合作。标准网络将使用三对（A， B）、（B， C）和（C， A）来模拟这种三元交互。然而，使用单一的交互（a，B，C）可以更好地表示这些人之间的关系，这是标准网络模型无法存储的。随着大数据的出现，越来越多的证据表明，这些数据经常呈现出基于群体的互动的几个实例，而标准的网络模型无法捕捉到这些实例，这一点已经变得越来越明显。这个项目的目的是建模、描述和应用新的网络模型，有效地捕获复杂系统中发生的高阶相互作用；这将反过来使我们更好地理解这种系统的行为和动力学。高阶网络模型数量有限的一个关键原因是数据通常以成对格式收集：在数据收集阶段忽略了高阶信息。然而，这种限制有时可以通过使用伴随实验信息的元数据来克服；例如，两个以上作者之间的合作关系可以通过查看他们共同撰写的论文的作者列表来恢复。将使用的主要建模工具是代数拓扑的简单复合体。单纯形在高维空间中泛化线段，因此可以有效地表示多体相互作用。一个简单复合体就是“连接”的简单体的集合。所开发的理论将为分析工具提供理论框架，该工具将允许量化网络中组件的重要性，以及在高阶相互作用方面识别大规模结构（例如，密集连接组件的集群）。该理论将配备一套计算效率高的算法，以确保所介绍的技术的广泛适用性，特别是对于大型数据集。开发的技术将针对标准网络模型和其他可用的网络高阶表示（包括张量表示）进行测试。该课程将有助于理解如何在复杂系统的网络建模中整合和利用高阶信息。它将开发用于分析高阶交互的数学、统计和计算工具，并为关系数据开辟新的数据分析。所开发的理论和算法的广泛适用性将允许在公开可用的现实世界数据上进行测试。

Networks (or graphs) provide a useful modelling tool to describe very different complex systems, from brains and ecosystems to social organisations, financial markets, transportation systems, and power grids, to name a few. Since these complex systems consist of entities that interact or relate to each other, network models represent them as graphs where the components are described as vertices and the edges capture pairwise (dyadic) interactions between them. One of the main strengths of these models is in their ability to relate the structure of systems to their dynamic; for example, a communication network of a community of people may provide insights into how a new idea spreads across the members of such a community, since the network is the mean that allows the information to flow. On the other hand, a fundamental shortcoming of current network models is their inability to appropriately capture higher-order interactions, i.e., those interactions that occur between more than two components. For example, consider three people, named A, B, and C, that collaborate together. A standard network would model this triadic interaction using three pairs, namely (A, B), (B, C), and (C, A). However, the relationship between these people would be better represented using a single interaction, (A,B,C), that standard network model cannot store. This exemplifies something that has become more and more apparent with the advent of big data and the growing evidence that these often present several instances of group-based interactions, which standard network models fail to capture. The aim of this project is to model, describe, and apply new classes of network models that effectively capture higher-order interactions occurring in complex systems; this will in turn allow for a better understading of the behavior and dynamics of such systems. A key reason for the limited number of higher-order network models is that data is often collected in a pairwise format: higher-order information is disregarded at the data-collection stage. However, this limitation can sometimes be overcome by using the metadata accompaining the experimental information; For example, collaborative relationships among more than two authors can be recovered by looking at the authors list of the papers they have co-authored. The main modelling tool that will be used is simplicial complexes from algebraic topology. Simplices generalize line segments in higher-dimensional spaces and thus can be effectively used to represent many-body interactions. A simplicial complex is then just a collection of 'connected' simplices. The theory developed will provide a theoretical framework for analytical tools that will allow to quantify the importance of components within the network, as well as to identify large-scale structures (e.g., clusters of densely connected components) in terms of higher-order interactions. The theory will be equipped with a set of computationally efficient algorithms that will ensure wide applicability of the introduced techniques, especially for large datasets. The developed techniques will be tested against standard network models and other available higher-order representations of networks, including tensor representations.This studentship will contribute to understanding how to incorporate and take advantage of higher-order information in networks modelling complex systems. It will develop mathematical, statistical, and computational tools for analysis of higher-order interactions and open up novel data analytics for relational data. The wide applicability of the theory and algorithmics developed will allow testing on publicly available real-world data.