Abstract
Cluster synchronisation is a key phenomenon observed in networks of coupled dynamical units. Its presence has been linked to symmetry and, more generally, to equability of the underlying pattern of interactions between dynamical units. However, it is not known under which conditions equitability-induced synchronisation is the only cluster synchronisation that can occur on a particular system. Here, we reveal a natural linear independent condition such that equitability becomes necessary, and sufficient, for the existence of cluster synchronised solutions on a very general dynamical system which allows multiplex or higher-order, arbitrarily weighted interactions. Our results explain the ubiquity of explosive synchronisation, as opposed to cluster synchronisation, in multiplex and higher-order networks: equitability imposes additional constraints that must be simultaneously satisfied on the same set of nodes. The outcomes of this work have significant implications for the design of complex dynamical systems of coupled dynamical units with arbitrary cluster synchronisation patterns and coupling functions.
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Acknowledgements
G.C.-A. is partially funded by the URJC under fellowship PREDOC-21-026-2164 and the INCIBE/URJC Agreement M3386/2024/0031/001 within the framework of the Recovery, Transformation and Resilience Plan funds of the European Union (Next Generation EU). C.I.d.G. acknowledges funding from the Bulgarian Ministry of Education and Science, under Project No. BG-RRP-2.004-0006-C02. R.J.S.-G. has been partially supported by the Erlangen Hub: Mathematical Foundations of Intelligence grant EP/Y028872/1.
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R.J.S.-G., S.B., and C.I.d.G. conceived the project. The theoretical framework was developed by R.J.S.-G. and K.K. The numerical simulations were done by G.C.-A., K.K., and C.I.d.G. All authors wrote and reviewed the final manuscript.
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Kovalenko, K., Contreras-Aso, G., del Genio, C.I. et al. Equitability and explosive synchronisation in multiplex and higher-order networks. Commun Phys (2026). https://doi.org/10.1038/s42005-026-02543-5
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DOI: https://doi.org/10.1038/s42005-026-02543-5
