What are higher-order networks? (2104.11329v3)

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.

• The paper provides a comprehensive survey of higher-order networks, emphasizing interactions beyond dyadic relationships.
• The paper reviews definitions and methodologies, including hypergraphs and simplicial complexes, for capturing multi-way interactions.
• The paper highlights implications for structural analysis and network dynamics, suggesting advanced modeling for complex systems.

Higher-Order Networks: An Exploration of Definitions, Implications, and Applications

The paper "What are higher-order networks?" by Bick, Gross, Harrington, and Schaub provides a comprehensive survey of the emerging field of higher-order network analysis. This field extends traditional graph-based approaches by considering interactions that surpass pairwise relationships, offering a richer representation framework crucial for various complex systems. The paper meticulously reviews recent developments and offers insights into structural, statistical, and dynamical perspectives of higher-order networks.

Definition and Relevance

The authors define higher-order networks as those that feature interactions involving more than two nodes, contrasting with the traditional dyadic interaction paradigm of graphs. These include structures such as hypergraphs and simplicial complexes, which are increasingly recognized for their ability to capture complex interactions in systems ranging from sociology to biology. By moving beyond pairwise interactions, higher-order networks can more accurately model systems where multi-way interactions are significant, such as in socio-economic activities, biochemical reactions, and structural balance in social networks.

Major Themes

The paper addresses three core questions:

1. What are higher-order networks? They explore various mathematical constructs that embody higher-order interactions, such as hypergraphs and simplicial complexes, and discuss their theoretical foundations.
2. Why paper higher-order networks? They suggest that these networks provide a more detailed understanding of relational data where interactions are inherently non-dyadic. The representation of non-linear, multi-way relationships offers advantages in applications requiring complex interaction modeling.
3. How can higher-order networks be used?
   • Structural Analysis (Topology and Geometry): The authors discuss the utility of higher-order networks in uncovering the geometric and topological properties of data. Persistent homology and simplicial complexes are tools of choice for characterizing data "shapes" and addressing questions about connectivity and loops in datasets.

• Relational Data Analysis: When data sets include higher-order interactions, traditional graph-based representations can fall short. The authors review probabilistic models, including exponential random graph models and configuration models adapted to hypergraphs and simplicial complexes, to analyze non-dyadic relational data effectively.
• Dynamical Systems: The exploration expands network dynamical systems to include higher-order interactions, discussing implications for synchronization and stability, particularly in coupled oscillator networks. This extension to hypergraphs and simplicial complexes allows for modeling complex dependencies in dynamics, which are not captured in pairwise interaction frameworks.

Implications and Future Directions

The discussion in the paper postulates several implications of adopting higher-order network frameworks. These models provide a broader and potentially more accurate depiction of intricate systems, holding promise for advancements in machine learning, network theory, and data science. The interdisciplinary approach in network dynamics, linking topological and geometrical methods with statistical analysis, encourages collaborative research across fields.

The survey recognizes that while higher-order networks offer a powerful modeling framework, they are also computationally and theoretically complex, necessitating further research. Future work is likely to explore scalable algorithms and methods of inference and optimization tailored to these models, enabling their application in real-world systems with sizable data sets.

In conclusion, the paper advocates for the integration of higher-order network perspectives in the modeling and understanding of complex systems. By encompassing a richer set of interactions, these models hold the potential to significantly enhance our comprehension and manipulation of the intricate webs of relations that define modern scientific challenges.