Mean-Field Approximation of Dynamics on Networks

Many real-world phenomena can be modelled as dynamical processes on networks, a prominent example being the spread of infectious diseases such as COVID-19. Mean-field approximations are a widely used tool to analyse such dynamical processes on networks, but these are typically derived using plausible probabilistic reasoning, introducing uncontrolled errors that may lead to invalid mathematical conclusions. In this paper we present a rigorous approach to derive mean-field approximations from the exact description of Markov chain dynamics on networks through a process of averaging called approximate lumping. We consider a general class of Markov chain dynamics on networks in which each vertex can adopt a finite number of ``vertex-states'' (e.g. susceptible, infected, recovered etc.), and transition rates depend on the number of neighbours of each type. Our approximate lumping is based on counting the number of each type of vertex-state in subsets of vertices, and this results in a density dependent population process. In the large graph limit, this reduces to a low dimensional system of ordinary differential equations, special cases of which are well known mean-field approximations. Our approach provides a general framework for the derivation of mean-field approximations of dynamics on networks that unifies previously disconnected approaches and highlights the sources of error.