Explosive synchronization of multi-layer complex networks based on star connection between layers with delay

Explosive synchronization(ES)is a kind of first-order jump phenomenon that exists in physical and biological systems.In recent years,researchers have focused on ES between single-layer and multi-layer networks.Most research on complex networks with delay has focused on single-layer or double-layer networks,multi-layer networks are seldom explored.In this paper,we propose a Kuramoto model of frequency weights in multi-layer complex networks with delay and star connections between layers.Through theoretical analysis and numerical verification,the factors affecting the backward critical coupling strength are analyzed.The results show that the interaction between layers and the average node degree has a direct effect on the backward critical coupling strength of each layer network.The location of the delay,the size of the delay,the number of network layers,the number of nodes,and the network topology are revealed to have no direct impact on the backward critical coupling strength of the network.Delay is introduced to explore the influence of delay and other related parameters on ES.

Explosive synchronization:From synthetic to real-world networks

Synchronization is a widespread phenomenon in both synthetic and real-world networks.This collective behavior of simple and complex systems has been attracting much research during the last decades.Two different routes to synchrony are defined in networks;first-order,characterized as explosive,and second-order,characterized as continuous transition.Although pioneer researches explained that the transition type is a generic feature in the networks,recent studies proposed some frameworks in which different phase and even chaotic oscillators exhibit explosive synchronization.The relationship between the structural properties of the network and the dynamical features of the oscillators is mainly proclaimed because some of these frameworks show abrupt transitions.Despite different theoretical analyses about the appearance of the firstorder transition,studies are limited to the mean-field theory,which cannot be generalized to all networks.There are different real-world and man-made networks whose properties can be characterized in terms of explosive synchronization,e.g.,the transition from unconsciousness to wakefulness in the brain and spontaneous synchronization of power-grid networks.In this review article,explosive synchronization is discussed from two main aspects.First,pioneer articles are categorized from the dynamical-structural framework point of view.Then,articles that considered different oscillators in the explosive synchronization frameworks are studied.In this article,the main focus is on the explosive synchronization in networks with chaotic and neuronal oscillators.Also,efforts have been made to consider the recent articles which proposed new frameworks of explosive synchronization.

Explosive synchronization of complex networks with different chaotic oscillators

Recent studies have shown that explosive synchronization transitions can be observed in networks of phase oscillators [Goemez-Gardenes J, Goemez S, Arenas A and Moreno Y 2011 Phys. Rev. Lett. 106 128701] and chaotic oscillators [Leyva I, Sevilla-Escoboza R, Buldu J M, Sendifia-Nadal I, Goemez-Gardefies J, Arenas A, Moreno Y, Goemez S, Jaimes-Reaitegui R and Boccaletti S 2012 Phys. Rev. Lett. 108 168702]. Here, we study the effect of different chaotic dynamics on the synchronization transitions in small world networks and scale free networks. The continuous transition is discovered for R6ssler systems in both of the above complex networks. However, explosive transitions take place for the coupled Lorenz systems, and the main reason is the abrupt change of dynamics before achieving complete synchronization. Our results show that the explosive synchronization transitions are accompanied by the change of system dynamics.

Explosive synchronization in a mobile network in the presence of a positive feedback mechanism

Synchronization is a process that describes the coherent dynamics of a large ensemble of interacting units.The study of explosive synchronization transition attracts considerable attention.Here,I report the explosive transition within the framework of a mobile network,while each oscillator is controlled by global-order parameters of the system.Using numerical simulation,I find that the explosive synchronization(ES)transition behavior can be controlled by simply adjusting the fraction of controlled oscillators.The influences of some parameters on explosive synchronization are studied.Moreover,due to the presence of the positive feedback mechanism,I prevent the occurrence of the synchronization of continuous-phase transition and make phase transition of the system a first-order phase transition accompanied by a hysteresis loop.

Explosive synchronization of multi-layer complex networks based on inter-layer star network connection

Explosive synchronization(ES)is a first-order transition phenomenon that is ubiquitous in various physical and biological systems.In recent years,researchers have focused on explosive synchronization in a single-layer network,but few in multi-layer networks.This paper proposes a frequency-weighted Kuramoto model in multi-layer complex networks with star connection between layers and analyzes the factors affecting the backward critical coupling strength by both theoretical analysis and numerical validation.Our results show that the backward critical coupling strength of each layer network is influenced by the inter-layer interaction strength and the average degree.The number of network layers,the number of nodes,and the network topology can not directly affect the synchronization of the network.Enhancing the inter-layer interaction strength can prevent the emergence of explosive synchronization and increasing the average degree can promote the generation of explosive synchronization.

Explosive synchronization enhances selectivity： Example of the cochlea

Acoustical signal transduction in the cochlea is an active process that involves nonlinear amplification and spontaneous otoacoustic emissions. Signal transduction involves individual subunits composed of globally coupled hair cells, which can be modeled as oscillators close to a Hopf bifurcation. The coupling may induce a transition toward synchronization, which in turn leads to a strong nonlinear response. In the model studied here, the synchronization transition of the subunit is discontinuous （explosive） in the absence of an external stimulus. We show that, in the presence of an external stimulus and for a coupling strength slightly lower than the critical value leading to explosive synchronization, the response of the subunit has better frequency selectivity and a larger signal-to-noise ratio. From physiological observations that subunits are themselves coupled together, we further propose a model of the complete cochlea, accounting for the ensemble of frequencies that the organ is able to detect.

Effects of frustration on explosive synchronization

In this study, we consider 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 frustration is included in the system. This assumption can enhance or delay the explosive transition to synchronization. Interestingly, a de-synchronization phenomenon occurs and the type of phase transition is also changed. Furthermore, we provide an analytical treatment based on a star graph, which resembles that obtained in scale-free networks. Finally, a self-consistent approach is implemented to study the de-synchronization regime. Our findings have important implications for controlling synchronization in complex networks because frustration is a controllable parameter in experiments and a discontinuous abrupt phase transition is always dangerous in engineering in the real world.

Understanding the mechanisms of brain functions from the angle of synchronization and complex network

The human brain is the most complicated and fascinated system and executes various important brain functions, but its underlying mechanism is a long-standing problem. In recent years, based on the progress of complex network science, much attention has been paid to this problem and many important results have been achieved, thus it is the time to make a summary to help further studies. For this purpose, we here make a brief but comprehensive review on those results from the aspect of brain networks, i.e., from the angle of synchronization and complex network. First, we briefly discuss the main features of human brain and its cognitive functions through synchronization. Then, we discuss how to construct both the anatomical and functional brain networks, including the pathological brain networks such as epilepsy and Alzheimer’s diseases. Next, we discuss the approaches of studying brain networks. After that, we discuss the current progress of understanding the mechanisms of brain functions, including the aspects of chimera state, remote synchronization, explosive synchronization, intelligence quotient, and remote propagation. Finally, we make a brief discussion on the envision of future study.