Optimal Filtering and the Separation Principle on Very Large Networks: a Graphon Q-noise Analysis

Estimation of stochastic systems on very large networks is intractable computationally. Graphon theory provides limit objects for infinite sequences of graphs by mapping adjacency matrices to the unit square, enabling the modelling of dynamical systems on arbitrarily large graphs via functional analytic methods. In previous work (Dunyak and Caines, 2022, 2023, 2024), Q-noise was used to extend stochastic systems on large graphs to stochastic systems in Hilbert spaces on graphons; in this paper the linear system state estimation problem on large networks and their graphon limits are analysed, and a Separation Principle of control and estimation is introduced. Convergence of finite network linear system state estimates, together with the corresponding Kalman filter systems to their graph limit counterparts, is established. A computational example of this convergence is illustrated on a standard graphon example.