What are higher-order networks?

TL;DR

This survey explores higher-order networks that extend traditional graph models to capture complex, multi-way relationships in data, highlighting their importance, definitions, and applications across disciplines.

Contribution

It clarifies the concept of higher-order networks, discusses their significance, and reviews their applications, providing a comprehensive overview of recent developments in the field.

Findings

Higher-order networks capture multi-way relationships beyond pairwise edges.

They offer more expressive modeling tools for complex systems.

Recent studies demonstrate their effectiveness in various applications.

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.