Modulating explosive synchronization with weighted pairwise and higher-order interactions

In this study, we investigate how weighted pairwise and higher-order interactions affect the onset of explosive synchronization in a higher-order Kuramoto network. We introduce a degree dependent weighting scheme for both dyadic and triadic couplings, controlled by two exponents that determine how strongly node connectivity affects the interaction strengths. Using extensive numerical simulations on Erdős–Rényi random networks with uniformly distributed natural frequencies, we examine how these weighting exponents interact with both the pairwise and higher-order coupling strengths. For the baseline case of linear weighting, in which the exponents are set to one, we find that stronger higher-order coupling reduces the level of pairwise coupling needed for the onset of synchronization and reduces the hysteresis between forward and backward continuations. When nonlinear weighting is introduced, the synchronization behavior becomes highly sensitive to the choice of exponents. In general, nonlinear weighting reduces the required pairwise coupling and weakens the hysteresis compared to linear weighting. When the higher-order weighting is kept linear and the pairwise exponent is varied, the hysteresis reaches its maximum for intermediate exponent values and decreases for both smaller and larger values. In contrast, when the pairwise weighting is linear while the higher-order exponent is varied, the hysteresis is largest for very small exponents and gradually diminishes as the exponent increases. Finally, we prove the existence of explosive transition and hysteresis area by theoretical analysis in the thermodynamic limit. Consequently, our results show that weighted higher-order interactions serve as an effective mechanism for tuning the abruptness of collective synchronization, allowing explosive transitions to be enhanced, weakened, or completely suppressed depending on how the weights are chosen.