ON THE SOCIOLOGICAL STUDY OF HIGHER-ORDER SYSTEMS AND COMPLEX NETWORKS
Abstract
Simplicial complexes and hypergraphs provide sociological theorists and researchers with frameworks for expressing and understanding the structure of higher-order systems and complex networks, which dyadic (i.e., pairwise) relationships alone are not able to represent easily. This article explores the use of simplicial complexes and hypergraphs to conceptualize and analyze higher-order systems and complex networks. Traditional social network analysis relies on dyadic ties to convey material or nonmaterial connections between social entities. However, many real-world networks involve higher-order or complex interactions that cannot be adequately captured by such simple graphs. Additionally, the concept of homology is introduced as a comprehensive way of generalizing from higher-order systems and complex networks salient topological features. Significant relational arrangements can be identified through the persistence of static or dynamic aspects of networks, which provides sociologists with needed means for handling increasingly complicated data that can quickly overwhelm the abilities of the analyst and the audience to make sense of and draw conclusions from them.
Body Text
Simplicial complexes are sets of simplices that generalize the notion of multiple relationships to arbitrary dimensions. In the context of network analysis, simplicial complexes allow analysts to model multi-way ties among nodes. The facets in simplicial complexes can help to visualize these grouped relationships more clearly. For instance, a co-authorship network may be represented as a simplicial complex where each respective subset under inclusion denotes the largest group of individuals who have collaborated on a paper.
Hypergraphs go a step further by extending the idea of edges in traditional graphs to hyperedges, which can connect any number of nodes. Frequently, but not always, they are depicted by a surrounding convex hull. This makes hypergraphs a powerful tool for representing higher-order relationships in networks. In a hypergraph, a hyperedge can signify a multi-way interaction, such as a three- or more-author paper collaboration involving several researchers.
By leveraging these advanced mathematical structures, researchers can gain deeper insights into the topology of higher-order systems and complex networks. For example, in the included figures, each representation describes the same set of relations held between actors. Only when presented with hypergraphs, such as in Figure 3, is an analyst able to discern the true pattern of paper co-authorship. In comparison, while simplicial complexes, such as in Figure 2, are an improvement over simple graphs, such as in Figure 1, they fall short of the sort of high-level and dense information carried by hypergraphs. However, simplicial complexes can ease geometric interpretations. This characteristic has proven useful in computational approaches for analyzing structural topology, both static and dynamic, including through varied applications of homology.
Figure 1. Simple graph of a four-person co-authorship network
Figure 2. Simplicial complex of a four-person co-authorship network
Figure 3. Hypergraph of a four-person co-authorship network
References:
Battiston, F. and G. Petri. 2022. Higher-Order Systems. Cham, Switzerland: Springer.
Landry, Nicholas W., Maxime Lucas, Iacopo Iacopini, Giovanni Petri, Alice Schwarze, Alice Patania, and Leo Torres. 2023. “XGI: A Python Package for Higher-Order Interaction Networks.” Journal of Open Source Software 8(85):5162. doi: 10.21105/joss.05162.