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This paper deals with the derivation of the mean-field limit for multi-agent systems on a large class of sparse graphs. More specifically, the case of non-exchangeable multi-agent systems consisting of non-identical agents is addressed. The analysis does not only involve PDEs and stochastic analysis but also graph theory through a new concept of limits of sparse graphs (extended graphons) that reflect the structure of the connectivities in the network and has critical effects on the collective dynamics. In this article some of the main restrictive hypothesis in the previous literature on the connectivities between the agents (dense graphs) and the cooperation between them (symmetric interactions) are removed.

In this note, we propose a new relative entropy combination of the methods developed by P.-E. Jabin and Z. Wang [Inventiones (2018)] and by S. Serfaty [Proc. Int. Cong. of Math, (2018) and references therein] to treat more general kernels in mean field limit theory. This new relative entropy may be understood as introducing appropriate weights in the relative entropy developed by P.-E. Jabin and Z. Wang (in the spirit of what has been recently developed by D. Bresch and P.-E. Jabin [Annals of Maths (2018)]) to cancel the more singular terms involving the divergence of the flow. As an example, a full rigorous derivation (with quantitative estimates) of the Patlak-Keller-Segel model in some subcritical regimes is obtained. Our new relative entropy allows to treat singular potentials which combine large smooth part, small attractive singular part and large repulsive singular part. Résumé: Limite champ moyen pour des noyaux plus généraux. Dans cette note, on propose une nouvelle entropie relative combinant les méthodes développées par P.-E. Jabin et Z. Wang [Inventiones (2018)] et par S. Serfaty [see review in Proc. Int. Con og Math (2018) et références] pour traiter des noyaux plus généraux en thorie de la limite champ moyen. Cette nouvelle entropie relative consiste en l'introduction d'une famille de poids appropriés dans l'entropie relative développée par P.-E. Jabin et Z. Wang (dans le même esprit du travail récent de D. Bresch et P.-E. Jabin [Annals of Maths (2018)]) pour compenser les termes les plus singuliers qui font intervenir la divergence du champ de vitesse. Comme exemple, une preuve avec estimation quantitative de la limite champ moyen vers le modèle de Patlak-Keller-Segel en régime sous-critique est obtenue. Notre méthode permet de couvrir dses potentiels singuliers qui peuvent combiner une partie réguliere, une petite partie singulière attractive et une grande partie singulire répulsive.

We consider large systems of particles interacting through rough but bounded interaction kernels. We are able to control the relative entropy between the N -particle distribution and the expected limit which solves the corresponding Vlasov system. This implies the Mean Field limit to the Vlasov system together with Propagation of Chaos through the strong convergence of all the marginals. The method works at the level of the Liouville equation and relies on precise combinatorics results.