| Abstract: |
| Traditional models of interacting particle systems often assume a fixed network of connections, which simplifies the analysis but fails to capture many real-world phenomena. Indeed, interacting particles where the network structure and particle states co-evolve in mutual influence, are increasingly recognised as essential in diverse fields. For instance, they appear in neuroscience, where learning is encoded through the strengthening and weakening of synaptic connections. In this talk I will present the rigorous mean-field limit for systems of non-exchangeable interacting diffusions on co-evolutionary networks. The main challenge arises from the coupling between the network 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 (Lov\`asz and Szegedy, 2006), in our mean-field limit we employ for the first time K-graphons (Lov\`asz and Szegedy, 2010), also termed probability-graphons (Abraham, Delmas, and Weibel, 2025), as they provide a natural framework compatible with non-linear structures. |
|