Synchronization is a phenomenon representing the emergence of collective behavior in natural and synthetic complex systems. Synchronization is commonly found to follow a smooth, second order phase transition as shown generally by Kuramoto. However, in some cases there is a first-order, or discontinuous, synchronization transition, i.e. an abrupt and irreversible phase transition with hysteresis to the synchronized state of coupled oscillators, that is called explosive synchronization. There are three cases in which transition to synchronized state occurs explosively. The first condition is where the network topology is scale-free and there exists a positive correlation between the natural frequencies of the oscillators and their degrees. The second case that happens is in all to all network, random network and uncorrelated scale free network, in which there is a positive correlation between coupling strengths of the oscillators and the absolute value of their natural frequencies. A similar condition occurs in random network in which there is a positive correlation between coupling strengths of the oscillators and the absolute value of difference in natural frequencies of two interacting node. Our intention is to iect the interplay between the function and the structure of network, in emergence of collective behaviors such as explosive synchronization.
