Linear Stochastic Processes on Networks and Low Rank Graph Limits

The modelling of stochastic linear systems in large complex networks is intractable computationally and may be impossible due to data-collection costs or privacy concerns. Graphon theory provides an approach to overcome these issues by providing potentially simple limit objects for infinite sequences of graphs, permitting one to approximate arbitrarily large networks by infinite dimensional operators. Graphon system theory is extended here to stochastic systems by the use of Q-noise, a generalization of Wiener processes in finite dimensional spaces to processes in function spaces. The theory is developed for low rank systems as a special case.