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Stability of political structures modeled by simplicial complexes under mediation, splitting, and shellability

Duško Jojić, Franjo Šarčević

TL;DR

The paper develops a unified combinatorial-topological framework for stability of political structures modeled as simplicial complexes. By interpreting mediators as cones, substructure mediation, and agent splitting, it derives precise stability changes via $f$- and $h$-vectors, with shellability enabling transparent, nonnegative interpretations through the $h$-vector. Key results include explicit stability criteria for introducing single and multiple mediators, as well as splitting, plus a detailed treatment of shellable complexes and independence complexes of graphs. The work also introduces weighted stability, analyzes canonical examples (paths and crosspolytopes), and poses refinements to capture viability and finer distinctions beyond the basic stability measure. Overall, the paper provides a rigorous, modular toolkit for quantifying and comparing stability under structural modifications in agent-based models.

Abstract

Modeling political structures by simplicial complexes, we investigate whether introducing a mediator into a substructure increases or decreases the stability of the overall structure. We prove theorems that quantify the stability of a political structure when $n$ mediators are introduced, either one by one or simultaneously. We also examine how the stability is affected when a single agent is split into two. In addition, stability is expressed in terms of the $h$-vector, and special attention is given to a class of political structures modeled by shellable simplicial complexes. In the latter context, we analyze weighted political structures and examples of political structures modeled by independence complexes of graphs. This approach provides a rigorous, stepwise analysis of stability under different structural modifications, showing how the combinatorial and topological properties of the simplicial complex govern the structure's stability.

Stability of political structures modeled by simplicial complexes under mediation, splitting, and shellability

TL;DR

The paper develops a unified combinatorial-topological framework for stability of political structures modeled as simplicial complexes. By interpreting mediators as cones, substructure mediation, and agent splitting, it derives precise stability changes via - and -vectors, with shellability enabling transparent, nonnegative interpretations through the -vector. Key results include explicit stability criteria for introducing single and multiple mediators, as well as splitting, plus a detailed treatment of shellable complexes and independence complexes of graphs. The work also introduces weighted stability, analyzes canonical examples (paths and crosspolytopes), and poses refinements to capture viability and finer distinctions beyond the basic stability measure. Overall, the paper provides a rigorous, modular toolkit for quantifying and comparing stability under structural modifications in agent-based models.

Abstract

Modeling political structures by simplicial complexes, we investigate whether introducing a mediator into a substructure increases or decreases the stability of the overall structure. We prove theorems that quantify the stability of a political structure when mediators are introduced, either one by one or simultaneously. We also examine how the stability is affected when a single agent is split into two. In addition, stability is expressed in terms of the -vector, and special attention is given to a class of political structures modeled by shellable simplicial complexes. In the latter context, we analyze weighted political structures and examples of political structures modeled by independence complexes of graphs. This approach provides a rigorous, stepwise analysis of stability under different structural modifications, showing how the combinatorial and topological properties of the simplicial complex govern the structure's stability.

Paper Structure

This paper contains 8 sections, 9 theorems, 89 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Overview of some basic concepts and results
  3. Effects of introducing a mediator into the substructure of a political structure
  4. Effects of introducing many mediators into a political structure
  5. Splitting and stability
  6. Shellable political structures and their stability
  7. The independence complex of a graph as a political structure
  8. Further questions

Key Result

Proposition 2.11

Introducing a mediator into a political structure $P$ with $k+1$ vertices increases its stability, i.e. with equality holding if and only if $P$ is a $k$-simplex.

Figures (4)

  • Figure 1: An example of a political structure
  • Figure 2: Another example of a political structure
  • Figure 3: The political structure $P_{3,4;2}$
  • Figure 4: Political structures from Example \ref{['E:split']}

Theorems & Definitions (41)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • Example 2.6
  • Example 2.7
  • Example 2.8
  • Example 2.9
  • Definition 2.10
  • ...and 31 more