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，或称代数社交网络分析，指一系列旨在提取以有向图表示的社会系统结构信息的方法。其中核心方法包括确定给定系统中存在的角色与位置。仅涉及参与者二元交互的社会系统中，角色与位置分析已高度发展——然而在现代社交网络分析中，越来越多采用能够兼顾高阶交互的模型。本文采用范畴论方法，研究如何将角色与位置分析从图推广到可容纳高阶交互的超图。我们利用泛余代数框架——这一源自计算机科学与逻辑学的"系统理论"——形式化角色与位置分析的核心概念，并将其扩展到包括图与超图在内的广泛结构类别。为验证定义的有效性，我们证明了一个极具一般性的函子性定理，该定理在图的特例中可归结为关于位置分析与角色分析兼容性的经验观察。