Presentation on YouTube.

Large networks are very common objects in engineering. One approach to modeling dynamical systems on large, dense networks is to use their associated graphon limit, which is a bounded function defined on the unit square [Lovasz, 2012]. In this talk, whose foundations were presented in [Dunyak, Caines, CDC 2022], we outline recent results extending classical stochastic linear systems theory to systems on very large graphs by utilizing their approximating graphons and Q-noise. This results in a stochastic differential equation in the space of square-integrable functions defined over the whole network. We demonstrate that a linear quadratic Gaussian (LQG) optimal control problem on a large network converges to a Q-noise LQG on a graphon. Then, when a graphon limit corresponds to a finite rank linear operator, the state of the system can be explicitly calculated. Finally, for a linear stochastic mean-field tracking game on a large graph, the Nash Equilibrium can be approximated by an optimal control problem on a graphon. The optimal inputs for each agent in the graphon can be solved for explicitly, giving a closed form solution.