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Epidemics, voting behaviour and cascading failures in power grids are examples of natural, social and technological phenomena that can be modelled as dynamical processes on networks. The study of such important complex systems requires approximation, but the assumptions that underpin the standard mean-field approaches are routinely violated by dynamics on real-world networks, leading to uncontrolled errors and even controversial results. Consequently, determining the approximation precision has been recognised as a key challenge. We present a micro-scale foundation for mean-field approximation of a wide range of dynamics on networks that facilitates quantification of approximation error, elucidating its connection to network structure and model dynamics. We show that our coarse-graining approach minimises approximation error and we obtain an upper bound on this uncertainty. We illustrate our approach using epidemic dynamics on real-world networks.

Micro-scale foundation with error quantification for the approximation of dynamics on networks