Multi-agent systems are ubiquitous in Science, and they can be regarded as large systems of interacting particles with the ability to generate large-scale self-organized structures from simple local interactions rules between each agent and its neighbors. Since the size of the system is typically huge, an important question is to connect the microscopic and macroscopic scales in terms of mean-field limits, which is a fundamental problem in Physics and Mathematics closely related to Hilbert Sixth Problem. In most real-life applications, the communication between agents is not based on uniform all-to-all couplings, but on highly heterogeneous connections, and this makes agents distinguishable. However, the classical strategies based on mean-field limits are strongly based on the crucial assumption that agents are indistinguishable, and it therefore does not apply to our distinguishable setting, so that we need substantially new ideas.

 

In this talk I will present a recent work about the rigorous derivation of the mean-field limit for systems of non-exchangeable interacting diffusions on co-evolutionary networks. While previous research has primarily addressed continuum limits or systems with linear weight dynamics, our work overcomes these restrictions. The main challenge arises from the coupling between the network weight dynamics and the agents' states, which results in a non-Markovian dynamics where the system’s future depends on its entire history. Consequently, the mean-field limit is not described by a partial differential equation, but by a system of non-Markovian stochastic integrodifferential equations. A second difficulty stems from the non-linear weight dynamics, which requires a careful choice for the limiting network structure. Due to the limitations of the classical theory of graphons (Lovász and Szegedy, 2006) in handling non-linearities, we employ K-graphons (Lovász and Szegedy, 2010), also termed probability-graphons (Abraham, Delmas, and Weibel, 2025). This framework pro seems to provide a natural topology that is compatible with such non-linearities.