Synchronization of an ensemble of oscillators is an emergent phenomenon present in several complex systems, ranging from social and physical to biological and technological systems. The most successful approach to describe how coherent behavior emerges in these complex systems is given by the paradigmatic Kuramoto model. This model has been traditionally studied in complete graphs. However, besides being intrinsically dynamical, complex systems present very heterogeneous structure, which can be represented as complex networks. This report is dedicated to review main contributions in the field of synchronization in networks of Kuramoto oscillators. In particular, we provide an overview of the impact of network patterns on the local and global dynamics of coupled phase oscillators. We cover many relevant topics, which encompass a description of the most used analytical approaches and the analysis of several numerical results. Furthermore, we discuss recent developments on variations of the Kuramoto model in networks, including the presence of noise and inertia. The rich potential for applications is discussed for special fields in engineering, neuroscience, physics and Earth science. Finally, we conclude by discussing problems that remain open after the last decade of intensive research on the Kuramoto model and point out some promising directions for future research.

The emergence of explosive synchronization has been reported as an abrupt transition in complex networks of first-order Kuramoto oscillators. In this Letter, we demonstrate that the nodes in a second-order Kuramoto model, perform a cascade of transitions toward a synchronous macroscopic state, which is a novel phenomenon that we call cluster explosive synchronization. We provide a rigorous analytical treatment using a mean-field analysis in uncorrelated networks. Our findings are in good agreement with numerical simulations and fundamentally deepen the understanding of microscopic mechanisms toward synchronization.PACS numbers: 89.75. Hc,89.75.Kd,05.45.Xt In the past few years, much research effort has been devoted to investigating the influence of network organization on dynamical processes, such as random walks [1], congestion [2,3], epidemic spreading [3] and synchronization [4,5]. Regarding synchronization of coupled oscillators, it has been demonstrated that the emergence of collective behavior in these structures depends on the patterns of connectivity of the underlying network. For instance, through a mean-field analysis, it has been found that Kuramoto oscillators display a second-order phase transition to the synchronous state with a critical coupling strength that depends on the network topology [5].Recently, discontinuous transitions to phase synchronization have been observed in SF networks [6][7][8][9]. This phenomenon, called explosive synchronization, was proved to be caused exclusively by a microscopic correlation between the network topology and the intrinsic dynamics of each oscillator. More specifically, Gómez-Gardeñez et al.[6] considered the natural frequencies positively correlated with the degree distribution of the network, defining the natural frequency of each oscillator as equal to its number of connections.In this Letter, we substantially extend a first-order Kuramoto model used in [6] to a second-order Kuramoto model [10-13] that we modify in order to analyze global synchronization, considering the natural frequency of each node proportional to its degree [14][15][16]. In this model, we find a discontinuous phase transition in which small degree nodes join the synchronous component simultaneously, whereas other nodes synchronize successively according to their degrees (in contrast to [6] where all the nodes join the synchronous component abruptly): This is a novel phenomenon which we call cluster explosive synchronization. By developing a mean-field theory we derive self-consistent equations that produce the lower and upper critical coupling strength associated to a hysteretic behavior of the synchronization for uncorrelated networks. The analytical results are in good agreement with numerical simulations. Moreover, we show that decreasing the network average frequency and increasing the coupling strength are the key factors that lead to cluster explosive synchronization.The second-order Kuramoto model consists of the following set of equations [10-13]:where θ i is the phase of th...

This paper deals with the emergence of explosive synchronization in scale-free networks by considering the Kuramoto model of coupled phase oscillators. The natural frequencies of oscillators are assumed to be correlated with their degrees, and a time delay is included in the system. This assumption allows enhancing the explosive transition to reach a synchronous state. We provide an analytical treatment developed in a star graph, which reproduces results obtained in scale-free networks. Our findings have important implications in understanding the synchronization of complex networks since the time delay is present in most real-world complex systems due to the finite speed of the signal transmission over a distance.