Coalgebraic analysis of social systems

2026-02-26

AI Summary

This work proposes a formal generalization of role and positional analysis in social systems from traditional graph-based models—limited to pairwise interactions—to hypergraph-based models that accommodate higher-order interactions. Leveraging category theory and a universal coalgebraic framework, the authors rigorously formalize the core notions of roles and positions and systematically extend them to a unified structure encompassing both graphs and hypergraphs. The resulting framework not only subsumes classical graph models as special cases but also establishes theoretical coherence and validity through functoriality theorems. This advancement provides a mathematically rigorous foundation for analyzing higher-order structures in social networks, thereby enabling more nuanced and expressive modeling of complex relational data beyond dyadic ties.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Cognitive Modeling & Cognitive Systems: Social Cognition And Interaction
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Social Networks and Social Media: Social media analysis through the lenses of networks
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

The algebraic analysis of social systems, or algebraic social network analysis, refers to a collection of methods designed to extract information about the structure of a social system represented as a directed graph. Central among these are methods to determine the roles that exist within a given system, and the positions. The analysis of roles and positions is highly developed for social systems that involve only pairwise interactions among actors - however, in contemporary social network analysis it is increasingly common to use models that can take into account higher-order interactions as well. In this paper we take a category-theoretic approach to the question of how to lift role and positional analysis from graphs to hypergraphs, which can accommodate higher-order interactions. We use the framework of universal coalgebra - a 'theory of systems' with origins in computer science and logic - to formalize the main concepts of role and positional analysis and extend them to a large class of structures that includes both graphs and hypergraphs. As evidence for the validity of our definitions, we prove a very general functoriality theorem that specializes, in the case of graphs, to a folkloric observation about the compatibility of positional and role analysis.

Problem

• role analysis
• positional analysis
• hypergraphs
• higher-order interactions
• social systems

Innovation

• universal coalgebra
• hypergraphs
• role analysis
• positional analysis
• higher-order interactions