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

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.