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Political structures and the topology of simplicial complexes

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  • Mock, Andrea
  • Volić, Ismar

Abstract

We use the topology of simplicial complexes to model political structures following Abdou and Keiding (2019). Simplicial complexes are a natural tool to encode interactions in political structures since a simplex can be used to represent a subset of compatible agents. We translate the wedge, cone, and suspension operations into the language of political structures and show how these constructions correspond to merging structures and introducing mediators. We introduce the notions of the viability of an agent and the stability of a political system and examine their interplay with the simplicial complex topology, casting their interactions in category-theoretic language whenever possible. We also introduce a refinement of the model by assigning weights to simplices corresponding to the number of issues the agents agree on. In addition, homology of simplicial complexes is used to detect non-viabilities, certain cycles of incompatible agents, and the (non)presence of mediators. Finally, we extend some results from Abdou and Keiding (2019), bringing viability and stability into the language of friendly delegations and using homology to examine the existence of R-compromises and D-compromises.

Suggested Citation

  • Mock, Andrea & Volić, Ismar, 2021. "Political structures and the topology of simplicial complexes," Mathematical Social Sciences, Elsevier, vol. 114(C), pages 39-57.
    • Handle: RePEc:eee:matsoc:v:114:y:2021:i:c:p:39-57
    DOI: 10.1016/j.mathsocsci.2021.09.004
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    References listed on IDEAS

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    1. Iacopo Iacopini & Giovanni Petri & Alain Barrat & Vito Latora, 2019. "Simplicial models of social contagion," Nature Communications, Nature, vol. 10(1), pages 1-9, December.
    2. Abdou, Joseph M. & Keiding, Hans, 2019. "A qualitative theory of conflict resolution and political compromise," Mathematical Social Sciences, Elsevier, vol. 98(C), pages 15-25.
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    Cited by:

    1. Rajsbaum, Sergio & Raventós-Pujol, Armajac, 2022. "A Combinatorial Topology Approach to Arrow's Impossibility Theorem," MPRA Paper 112004, University Library of Munich, Germany.

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