Optimal transport and minimal trade problem, impacts on relational metrics and applications to large graphs and networks modularity

Abstract

This article presents a summary of the principal results found in MAR13. Starting with the seminal works on transportation theory of G. Monge and L. Kantorovich, while revisiting the works of Maurice Fréchet, we will introduce direct derivations of the optimal transport problem such as the so-called Alan Wilson's Entropy Model and the Minimal Trade Problem. We will show that optimal solutions of those models are mainly based in two dual principles: the independance and the indetermination structure between two categorical variables. Thanks to Mathematical Relational Analysis representation and the Antoine Caritat's (Condorcet) works on Relational Consensus, we will give an interesting interpretation to the indeterminaion structure and underline the duality Relationship between deviation to independence and deviation to indetermination structures. Finally, these results will lead us to the elaboration of a new criterion of modularization for large networks. © 2013 Springer-Verlag.