Consensus formation on a simplicial complex of opinions

• Opinion of an agent is considered as an unordered set of judgments.
• Opinion space is mapped onto a simplicial complex.
• Concepts of combinatorial algebraic topology are used in the analysis.
• Maximal number of realized opinions associated to agents are monitored.
• Dimension of opinion is monitored as a function of the initial number of judgments.

Geometric realization of an opinion is considered as a simplex and the opinion space of a group of individuals is a simplicial complex whose topological features are monitored in the process of opinion formation. The agents are physically located at the nodes of a scale-free and a random network. Social interactions include all concepts of social dynamics present in the mainstream models, augmented by four additional interaction mechanisms which depend on the local properties of opinions and their overlapping properties. The results pertaining to the formation of consensus are of particular interest. An analogy with quantum mechanical pure states is established through the application of the high-dimensional combinatorial Laplacian.