Universal phase transition and scalings of phase oscillators with partial adaptive coupling

The self-adaptive mechanisms inherently in numerous real-world processes are fundamental for the emergence of collective dynamics. In this study, we establish a general framework elucidating universal phase transitions in adaptive networks of coupled phase oscillators, in which the interactions within a single layer include an adaptive strategy characterized by the feedback between the coupling strength and the magnitude of the order parameter, mediated through a power-law function with different correlation exponents. We analytically demonstrate that there exists a critical correlation exponent demarcating the first-order and second-order phase transitions of the system, with synchronization transition attributes exhibiting no dependence of the adaptive fractions on either side of this critical threshold. Nevertheless, we mathematically substantiate that, within the critical scenario characterized by the correlation exponent, the manifestation of either abrupt or continuous phase transition is contingent upon the product of the adaptive weights. Crucially, we reveal that the bifurcations governing various phase transitions-manifested by a spectrum of interrelated scaling laws of the order parameters-are comprehensively appreciated in terms of the perturbation theory. Our findings contribute profound insights into the dynamical nature of phase transitions and offer enhanced methodologies for the controlling and manipulating of synchronization transitions in adaptive networks.