Differential analysis of dynamic and static networks

Network is a popular concept for complex and irregular structures used in various areas of research: contact tracing in epidemiology, polymers in chemistry, the brain in neuroscience, communication infrastructures in computer science, and many more. What is the single mathematical object that captures the essence of this concept? Is it possible to operate with dynamic networks with the same versatility as is done, for example, with density functions in mathematical physics? How to detect with certainty the presence of heterogeneous network models in noisy data? To answer these questions, we will develop a methodology that views dynamic and static networks as elements of a functional space. We will define infinitesimal perturbations of networks and -- for the first time -- introduce abstract differential equations governing network evolution. Establishing this link will enable studying network-like structures with mathematical analysis, enabling structural optimisation, optimal control, and statistical inference of networks. Once achieved, this vision will aid the transition from documenting networks observed in the real world to casting predictions, performing a rational design, and help to explain the mechanisms behind the prevalence of particular network structures in nature. We will demonstrate the latter through collaborations with academic and industrial partners from neuroscience, materials science, and social policies, wherein this project will aid development of new models, better materials and adaptive policies. The proposal builds upon my expertise in random graphs, which will allow us to model networks with structural heterogeneities without a significant loss of detailing. This project is supported by preparatory studies and informed by my extensive collaborations with stakeholders using networks in science.