Integer Control Approximations for Graphon Dynamical Systems

Graphons generalize graphs and define a limit object of a converging graph sequence. The notion of graphons allows for a generic representation of coupled network dynamical systems. We are interested in approximating integer controls for graphon dynamical systems. To this end, we apply a decomposition approach comprised of a relaxation and a reconstruction step. We extend the sum-up rounding algorithm to operate on a finite partition of the continuous vertex set in a graphon setting and restore integer feasibility for a given relaxed control solution. Finally, we derive an $ L^{1} $ bound for the state variables for structurally similar graphs and approximated integer controls. We verify our claims by simulating the Kuramoto model for large networks.