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.

翻译：社会系统的代数分析，或称代数社会网络分析，是指一系列旨在从有向图表示的社会系统中提取结构信息的方法。其中核心方法是确定给定系统中存在的角色与位置。对于仅涉及行动者间成对交互的社会系统，角色与位置分析已高度成熟——然而在当代社会网络分析中，能够考虑高阶交互的模型正日益普及。本文采用范畴论方法，探讨如何将角色与位置分析从图提升到能够容纳高阶交互的超图。我们运用通用余代数框架——一种源于计算机科学与逻辑的"系统理论"——来形式化角色与位置分析的核心概念，并将其扩展至包含图与超图的广泛结构类别。为验证定义的有效性，我们证明了一个极具普遍性的函子性定理，该定理在图的情形下可特化为关于位置分析与角色分析相容性的经典结论。