Dynamics of High Dimensional Complex Networks

Stochastic ordinary differential equations on random graphs provide a general framework for modelling a range of important applications, including electrical power grids, communication or social networks, machine learning algorithms, epidemiology disease spread, neuroscience, and material models in statistical physics. This project will develop new characterizations of how interactions among large networks of individual particles or agents lead to important larger scale macroscopic phenomena. Examples range from the billions of neurons in the brain that can form macroscopic oscillations that are observable through electrocardiograms (ECG), to individual populations that may incur large outbreaks in disease spread. The project will formulate new classes of macroscopic models that incorporate the microscopic effects of network delays, and local connectivity. It will also involve the training of graduate and undergraduate students in this research area. This project aims to rigorously establish new macroscopic autonomous McKean-Vlasov equations (Fokker Planck type, hydrodynamic limits) for large systems of stochastically interacting particles on random graphs. A key mathematical contribution will be the introduction of new non-standard empirical measures that encode sufficient information regarding the microscopic state of the system. The structure of these measures are a key novelty that enables the derivation, and convergence to, autonomous McKean-Vlasov equations. The project will also establish new formulas for rare events through rigorous large deviations principles. These formulas will characterize how microscopic fluctuations can lead to large scale macroscopic changes. The results will be developed for practically important classes of random graphs (relevant for the visual cortex model and social models) that allow for state-dependent edge connections, and hypergraphs that arise in many-agent interaction models. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.