What are higher-order networks? (English) Zbl 1519.05226
Summary: 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.
MSC:
| 05C82 | Small world graphs, complex networks (graph-theoretic aspects) |
| 34B45 | Boundary value problems on graphs and networks for ordinary differential equations |
| 68R10 | Graph theory (including graph drawing) in computer science |
| 55U10 | Simplicial sets and complexes in algebraic topology |
| 05C65 | Hypergraphs |
Keywords:
networks; graphs; hypergraphs; simplicial complexes; topology; dynamics; statistics; relational data; data analysisSoftware:
Eirene; SimpleHypergraphs.jl; PersistenceImages; simpcomp; hypergraph; Flagser; TDA; Macaulay2; CliqueTop; polymake; ergmReferences:
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