Comment on Consensus formation on a simplicial complex of opinions (arXiv:1212.1940, Physica A, Vol. 397 (1), pp. 111-120, 2014)

TL;DR

This paper critiques a previous work claiming an analogy between opinion formation on simplicial complexes and quantum states, demonstrating the analogy's mathematical and conceptual errors and clarifying the correct spectral analysis of these structures.

Contribution

It provides a rigorous correction to prior claims, clarifying the proper spectral analysis of simplicial complexes and their relation to quantum density matrices.

Findings

The previous analogy with quantum states is mathematically incorrect.

Correct spectral analysis of simplicial complexes is essential for topological characterization.

The paper emphasizes the importance of accurate mathematical formalism in topological opinion models.

Abstract

In the commented paper, the authors declare an analogy with the quantum mechanical pure states established through the application of the high-dimensional combinatorial Laplacian considering that the simplicial complex is in the pure state when it is formed by collection of the pure states. In their work, besides giving a completely erroneous analogy for the pure quantum mechanical state, which contradicts to very basic postulates of quantum mechanic, the authors clearly fail to provide and explain mathematical formalism behind their claims. Their intention is to rigorously prove that the existence of opinion space in all considered cases consists of the pure quantum states using incorrect normalization constant which does not produce trace equal to one. In this comment we prove out that their claims are erroneous. In order to show incorrectness of Maletic & Rajkovic model and conclusions we calculated and examined connections between different combinatorial structures of q- dimensional simplicial complexes and their Laplacian spectra and analyzed resulting symmetric and positive semidefinite matrices in the context of the density matrix of a quantum mechanical system. Importance of presented research is to further develop accurate quantitative topological-based characterization method for convertibility and distillation of density matrix.