[论文解读] What are higher-order networks?

Christian Bick, Elizabeth Gross+2 more|arXiv (Cornell University)|Apr 20, 2021

Complex Network Analysis Techniques被引用 32

一句话总结

This survey clarifies what higher-order networks are, why they matter, and how they can be represented and used across topology, statistics of relational data, and network dynamics.

ABSTRACT

Network-based modeling of complex systems and data using the language of graphs has become an essential topic across a range of different disciplines. Arguably, this graph-based perspective derives its success from the relative simplicity of graphs: A graph consists of nothing more than a set of vertices and a set of edges, describing relationships between pairs of such vertices. This simple combinatorial structure makes graphs interpretable and flexible modeling tools. The simplicity of graphs as system models, however, has been scrutinized in the literature recently. Specifically, it has been argued from a variety of different angles that there is a need for higher-order networks, which go beyond the paradigm of modeling pairwise relationships, as encapsulated by graphs. In this survey article we take stock of these recent developments. Our goals are to clarify (i) what higher-order networks are, (ii) why these are interesting objects of study, and (iii) how they can be used in applications.

研究动机与目标

• 在多个社区中澄清 higher-order networks 的概念与术语。
• 解释数学表示方法，如 hypergraphs 与 abstract simplicial complexes。
• 回顾用于数据的拓扑与几何工具、关系数据的统计建模，以及 higher-order 结构上的动力学。

提出的方法

• 给出 graphs、hypergraphs、以及 simplicial complexes 的统一定义。
• 描述拓扑数据分析（topological data analysis）概念，如同源性（homology）和持久同源性（persistent homology），以及它们与 higher-order 结构的联系。
• 讨论使用 hypergraphs 和 simplicial complexes 的关系数据的统计建模方法。
• 考察 higher-order network 动力系统，以及在何种条件下需要 higher-order 表示，或可归结为 dyadic 模型。

实验结果

研究问题

• RQ1 什么构成了 higher-order network，以及它如何扩展传统基于图的表示？
• RQ2 如何使用 higher-order 结构来理解数据的拓扑与几何？
• RQ3 在 hypergraphs 或 simplicial complexes 中，哪些统计模型适用于关系数据？
• RQ4 higher-order 交互如何影响网络动力系统，以及何时可以将其简化为 dyadic 动力学？

主要发现

• Higher-order networks 通过 hypergraphs 和 simplicial complexes 实现非 dyadic 的多方相互作用，从而对图进行推广。
• 拓扑工具如 homology 和 persistent homology 可以从 higher-order 结构中提取有意义的数据形状信息。
• 关系数据可以直接用 hypergraphs 或 simplicial complexes 来建模，而不是简化为成对图，从而实现更丰富的概率模型。
• higher-order network 动力学可能表现出不同于 dyadic 系统的行为，但坐标变换有时可以产生有效的 dyadic 表述。
• 该文章提供一个将拓扑、统计和动力学联系起来的连贯框架，用于 higher-order networks。