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Coalgebraic analysis of social systems

Nima Motamed, Nina Otter, Emily Roff

TL;DR

This paper uses 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.

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.

Coalgebraic analysis of social systems

TL;DR

This paper uses 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.

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.
Paper Structure (29 sections, 14 theorems, 68 equations, 8 figures)

This paper contains 29 sections, 14 theorems, 68 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Role analysis
  3. Positional analysis
  4. A coalgebraic framework
  5. Extending positional and role analysis
  6. The structure of the paper
  7. A reader's guide
  8. Positional and role analysis on graphs
  9. Positional analysis
  10. Role analysis
  11. Compatibility of positional and role analysis
  12. Social systems as coalgebras
  13. Social systems as hypergraphs
  14. Graphs and hypergraphs as coalgebras
  15. Positional analysis via bisimulation
  16. ...and 14 more sections

Key Result

Proposition 2.14

Let $G = (A, (R_i)_{i=1}^k)$ be a multirelational graph and let $E \subseteq A \times A$ be an outward-regular equivalence on $(A,R_i)$ for all $i \in \{1,\ldots,k\}$. Then there is a role reduction defined by $\phi(R) = {R}/{E}$. The same holds if $E$ is an inward-regular equivalence on $(A,R_i)$ for all $i \in \{1,\ldots,k\}$.

Figures (8)

  • Figure 1: Left, a multirelational network containing a 'parent' relation $P$, 'sister' relation $S$, and 'brother' relation $B$. Center, a role reduction combining $S$ and $B$ into the 'sibling' relation $\overline{S}$. Right, a blockmodel by 'generations'.
  • Figure 2: A graph $G$ and its blockmodels with respect to two equivalence relations, of which only $E_2$ is a regular equivalence. See \ref{['eg:graph-blockmodels']}.
  • Figure 3: A role reduction in which the 'sister' and 'brother' relations $S$ and $B$ are identified within the 'sibling' relation $\overline{S}$. See \ref{['eg:parent-sibling']}.
  • Figure 4: A blockmodel of $\overline{F}$ (\ref{['fig:parent-sibling']}) into 'generations', inducing a role reduction that identifies 'parent' with 'sibling's parent'. See \ref{['eg:generations']}.
  • Figure 5: Three models of the same co-authorship system. See \ref{['subsec:higher-order-relations']}.
  • ...and 3 more figures

Theorems & Definitions (71)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9
  • Example 2.10
  • ...and 61 more