Mediation, splitting and shellability in political structures modeled by simplicial complexes

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.