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Dimension-reduction of dynamics on real-world networks with symmetry
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences  (IF2.704),  Pub Date : 2021-07-21, DOI: 10.1098/rspa.2021.0026
Jonathan A. Ward

We derive explicit formulae to quantify the Markov chain state-space compression, or lumping, that can be achieved in a broad range of dynamical processes on real-world networks, including models of epidemics and voting behaviour, by exploiting redundancies due to symmetries. These formulae are applied in a large-scale study of such symmetry-induced lumping in real-world networks, from which we identify specific networks for which lumping enables exact analysis that could not have been done on the full state-space. For most networks, lumping gives a state-space compression ratio of up to ${10}^{7}$, but the largest compression ratio identified is nearly ${10}^{12}$. Many of the highest compression ratios occur in animal social networks. We also present examples of types of symmetry found in real-world networks that have not been previously reported.