Emergent synchrony in oscillator networks with adaptive arbitrary-order interactions

Complex physical systems are often governed by interactions that extend beyond pairwise links, underscoring the need to establish a map between interpretable system parameters and emergent synchronisation phenomena in hyper-graphs. To achieve this, the current work formulates an adaptive Kuramoto model that incorporates hyperedges of arbitrary order and explores their effects on synchronisation. By deriving the exact order parameter dynamics in the thermodynamic limit, analytical expressions governing the collective dynamics are obtained. Subsequent numerics confirm the analytical predictions, in addition to capturing qualitatively different dynamical regimes and phase transitions. Further investigations based on numerically constructed order parameter distributions demonstrate how fluctuations due to finite system size can influence the long-term system dynamics by inducing spontaneous transitions. These results provide important insights and can have diverse applications, such as designing optimal surgical procedures for drug-resistant epilepsy in the human brain and identifying the sources of rumours in social networks.