Synchronization In Scale-free Networks Research Articles

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.

Synchronization in scale-free networks: The role of finite-size effects

Synchronization problems in complex networks are very often studied by researchers due to their many applications to various fields such as neurobiology, e-commerce and completion of tasks. In particular, scale-free networks with degree distribution , are widely used in research since they are ubiquitous in Nature and other real systems. In this paper we focus on the surface relaxation growth model in scale-free networks with , and study the scaling behavior of the fluctuations, in the steady state, with the system size N. We find a novel behavior of the fluctuations characterized by a crossover between two regimes at a value of that depends on λ: a logarithmic regime, found in previous research, and a constant regime. We propose a function that describes this crossover, which is in very good agreement with the simulations. We also find that, for a system size above , the fluctuations decrease with λ, which means that the synchronization of the system improves as λ increases. We explain this crossover analyzing the role of the network's heterogeneity produced by the system size N and the exponent of the degree distribution.

Effects of degree correlations on the explosive synchronization of scale-free networks.

We study the organization of finite-size, large ensembles of phase oscillators networking via scale-free topologies in the presence of a positive correlation between the oscillators' natural frequencies and the network's degrees. Under those circumstances, abrupt transitions to synchronization are known to occur in growing scale-free networks, while the transition has a completely different nature for static random configurations preserving the same structure-dynamics correlation. We show that the further presence of degree-degree correlations in the network structure has important consequences on the nature of the phase transition characterizing the passage from the phase-incoherent to the phase-coherent network state. While high levels of positive and negative mixing consistently induce a second-order phase transition, moderate values of assortative mixing, such as those ubiquitously characterizing social networks in the real world, greatly enhance the irreversible nature of explosive synchronization in scale-free networks. The latter effect corresponds to a maximization of the area and of the width of the hysteretic loop that differentiates the forward and backward transitions to synchronization.

Predictable nonwandering localization of covariant Lyapunov vectors and cluster synchronization in scale-free networks of chaotic maps.

Covariant Lyapunov vectors for scale-free networks of Hénon maps are highly localized. We revealed two mechanisms of the localization related to full and phase cluster synchronization of network nodes. In both cases the localization nodes remain unaltered in the course of the dynamics, i.e., the localization is nonwandering. Moreover, this is predictable: The localization nodes are found to have specific dynamical and topological properties and they can be found without computing of the covariant vectors. This is an example of explicit relations between the system topology, its phase-space dynamics, and the associated tangent-space dynamics of covariant Lyapunov vectors.

Hub-Induced Synchronization in Scale-Free Networks with Cluster Structure

A recent research indicated that the corticocortical connectivity network of the cat possesses cluster structure and that each cluster in the network is scale-free and has a most connected hub. Motivated by that research, we slightly modify the network model and derive sufficient conditions for cluster synchronization of the modified network based on Lyapunov function method. The obtained results indicate that cluster synchronization can be induced by the hubs of the scale-free networks. In our opinion, the concept of hub-induced synchronization provides a better understanding of cluster synchronization in scale-free networks. Numerical examples are provided to demonstrate the effectiveness of the theoretical results.

Reexamination of explosive synchronization in scale-free networks: The effect of disassortativity

Previous work [J. Gómez-Gardeñes, S. Gómez, A. Arenas, and Y. Moreno, Phys. Rev. Lett. 106, 128701 (2011)] has reported that explosive synchronization can be achieved in heterogeneous networks with a microscopic correlation between the structural and dynamical properties of the networks. This phenomenon, however, cannot be observed in all heterogeneous networks even if this structure-dynamics correlation is preserved. It is therefore of particular interest to identify the general topological factors that can induce the first order synchronization transition and to understand the underlying mechanisms. Here we investigate this issue using the scenario of the smooth transformation from homogeneous Erdös-Rènyi networks to heterogeneous Barabàsi-Albert networks. Specifically, we scrutinize how local and global properties of the network change during this process, and how these properties are associated with the emergence of explosive synchronization. We find that the local degree-degree correlation in the network contributes primarily to explosive synchronization, other than the global topological property or starlike subgraphs. We furthermore demonstrate that the degree of disassortative mixing also has a great effect in the presence of explosive synchronization.

Explosive synchronization enhanced by time-delayed coupling

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.

Synchronization in scale free networks with degree correlation

In this paper we study a model of synchronization process on scale free networks with degree–degree correlations. This model was already studied on this kind of networks without correlations by Pastore y Piontti et al. [A.L. Pastore y Piontti, P.A. Macri, L.A. Braunstein, Phys. Rev. E 76 (2007) 046117]. Here, we study the effects of the degree–degree correlation on the behavior of the load fluctuations Ws in the steady state. We found that for assortative networks there exist a specific correlation where the system is optimally synchronized. In addition, we found that close to this optimal value the fluctuations does not depend on the system size and therefore the system becomes fully scalable. This result could be very important for some technological applications. On the other hand, far from the optimal correlation, Ws scales logarithmically with the system size.

Hub synchronization in scale-free networks

Heterogeneity in the degree distribution is known to suppress global synchronization in complex networks of symmetrically coupled oscillators. Scale-free networks display a great deal of heterogeneity, containing a few nodes, termed hubs, that are highly connected, while most nodes receive only a few connections. Here, we show that a group of synchronized nodes may appear in scale-free networks: hubs undergo a transition to synchronization while the other nodes remain unsynchronized. This general phenomenon can occur even in the absence of global synchronization. Our results suggest that scale-free networks may have evolved to complement various levels of synchronization.

Generalized synchronization in complex dynamical networks via adaptive couplings

Abstract This paper investigates generalized synchronization of three typical classes of complex dynamical networks: scale-free networks, small-world networks, and interpolating networks. The proposed synchronization strategy is to adjust adaptively a node’s coupling strength based on the node’s local generalized synchronization information. By taking the auxiliary-system approach and using the Lyapunov function method, we prove that for any given initial coupling strengths, the generalized synchronization can take place in complex networks consisting of nonidentical dynamical systems. It is demonstrated that the coupling strengths are affected by topologies of the networks. Furthermore, it is found that there are hierarchical features in the processes of generalized synchronization in scale-free networks because of their highly heterogeneous distributions of connection degree. Finally, we discuss in detail how a network’s degree of heterogeneity affects its generalization synchronization behavior.

Paths to globally generalized synchronization in scale-free networks

We apply the auxiliary-system approach to study paths to globally generalized synchronization in scale-free networks of identical chaotic oscillators, including Hénon maps, logistic maps, and Lorentz oscillators. As the coupling strength epsilon between nodes of the network is increased, transitions from partially to globally generalized synchronization and intermittent behaviors near the synchronization thresholds, are found. The generalized synchronization starts from the hubs of the network and then spreads throughout the whole network with the increase of epsilon . Our result is useful for understanding the synchronization process in complex networks.

Chaotic phase synchronization in scale-free networks of bursting neurons

There is experimental evidence that the neuronal network in some areas of the brain cortex presents the scale-free property, i.e., the neuron connectivity is distributed according to a power law, such that neurons are more likely to couple with other already well-connected ones. From the information processing point of view, it is relevant that neuron bursting activity be synchronized in some weak sense. A coherent output of coupled neurons in a network can be described through the chaotic phase synchronization of their bursting activity. We investigated this phenomenon using a two-dimensional map to describe neurons with spiking-bursting activity in a scale-free network, in particular the dependence of the chaotic phase synchronization on the coupling properties of the network as well as its synchronization with an externally applied time-periodic signal.

SYNCHRONIZATION AND GRAPH TOPOLOGY

This paper clarifies the relation between synchronization and graph topology. Applying the Connection Graph Stability method developed by Belykh et al. [2004a] to the study of synchronization in networks of coupled oscillators, we show which graph properties matter for synchronization. In particular, while we explicitly link the stability of synchronization with the average path length for a wide class of coupling graphs, we prove by a simple argument that the average path length is not always the crucial quantity for synchronization. We also show that synchronization in scale-free networks can be described by means of regular networks with a star-like coupling structure. Finally, by considering an example of coupled Hindmarsh–Rose neuron models, we demonstrate how global stability of synchronization depends on the parameters of the individual oscillator.