On the synchronization of ReLU version of dense networks

In this article, we analyze a Kuramoto-type system in which the classical sine coupling is replaced by the Rectified Linear Unit (ReLU) function max{0,sin(x)}. Both identical and non-identical oscillators are studied on connected graphs, and non-identical oscillators are also examined on two-group social networks. For the identical oscillators, our results show that simple connectivity of the network suffices for synchronization in both the classical and ReLU-modified models. For the non-identical oscillators, since the connection between oscillators is not all-to-all and the coupling function is the ReLU sine function, this causes the complexity of the synchronization analysis. To overcome this subtlety, we design a special Ansatz to proceed the proof by contradiction. In such a setting, by assuming the oscillators will not achieve a frequency synchronization asymptotically and analyzing the integrability of the difference of the second derivatives of phase functions for carefully chosen N pairs of connected oscillators, we show there should be at least N+1 distinct oscillators in this ReLU system, which violates the fact that the total number of the oscillators is N. Moreover, we show that the synchronized frequency for the ReLU model is the maximum among all the natural frequencies; while it is well-known that the synchronized frequency in the classical Kuramoto model is the mean of all the natural frequencies. In the social-network setting, characterized by intra-group competition and inter-group attraction, we derive explicit conditions that guarantee asymptotic synchronization. Numerical experiments support the theory and show that the classical model converges faster on equivalent topologies. Overall, our results provide a theoretical foundation for understanding synchronization in systems with asymmetric competitive–cooperative interactions.