Adaptive approach to global synchronization of directed networks with fast switching topologies

Global synchronization of directed networks with switching topologies is investigated. It is found that if there exists at least one directed spanning tree in the network with the fixed time-average topology and the time-average topology is achieved sufficiently fast, the network will reach global synchronization for appreciate coupling strength. Furthermore, this appreciate coupling strength may be obtained by local adaptive approach. A sufficient condition about the global synchronization is given. Numerical simulations verify the effectiveness of the adaptive strategy.     

Synchronization of networks

We study the synchronization of coupled dynamical systems on networks. The dynamics is governed by a local nonlinear oscillator for each node of the network and interactions connecting different nodes via the links of the network. We consider existence and stability conditions for both single- and multi-cluster synchronization. For networks with time-varying topology we compare the synchronization properties of these networks with the corresponding time-average network. We find that if the different coupling matrices corresponding to the time-varying networks commute with each other then the stability of the synchronized state for both the time-varying and the time-average topologies are approximately the same. On the other hand, for non-commuting coupling matrices the stability of the synchronized state for the time-varying topology is in general better than the time-average topology.      

Synchronized state of coupled dynamics on time-varying networks

We consider synchronization properties of coupled dynamics on time-varying networks and the corresponding time-average network. We find that if the different Laplacians corresponding to the time-varying networks commute with each other then the stability of the synchronized state for both the time-varying and the time-average topologies are approximately the same. On the other hand for noncommuting Laplacians the stability of the synchronized state for the time-varying topology is in general better than the time-average topology.     

Optimal pinning synchronization on directed complex network

In this paper, pinning synchronization on directed network was considered. By analyzing, some general synchronization criteria on directed network were established. And then, we verified it on directed globally coupled network, directed scale-free network, and directed small-world network, respectively. The pinning nodes were selected, respectively, according to order of in-degrees and out-degrees. Through comparing and analyzing simulations, the optimal pinning scheme was found, and a practical principle was induced finally.     

Pinning synchronization of time-varying polytopic directed stochastic networks

In this Letter, pinning synchronization of a directed network with Markovian jump (DNMJ) and nonlinear perturbations is considered. By analyzing the structure of the network, a detailed pinning scheme is given to ensure the synchronization of all nodes in a DNMJ. This pinning scheme can overcome those difficulties on deciding which nodes needs to be pinned. This scheme can also identify the exact least number of pinned nodes for a DNMJ model. In addition, the time-varying polytopic directed network with Markovian jump is discussed. Finally, examples are provided to illustrate the effectiveness of the gained criteria.     

Synchronization in the network of chaotic microwave oscillators

Time scale synchronization in networks of chaotic microwave oscillators with the different topologies of the links between nodes has been studied. As a node element of the network the one-dimensional distributed model of the low-voltage vircator has been used. To characterize the degree of synchronization in the whole network the synchronization index has been introduced. The transition to the synchronous regime is shown to take place via cluster time scale synchronization. Meanwhile, the spectral structure of the output signals is complicated sufficiently which allows using such devices in a number of practical applications.     

Dynamics of delayed-coupled chaotic logistic maps: Influence of network topology,connectivity and delay times

We review our recent work on the synchronization of a network of delay-coupled maps, focusing on the interplay of the network topology and the delay times that take into account the finite velocity of propagation of interactions. We assume that the elements of the network are identical (N logistic maps in the regime where the individual maps, without coupling, evolve in a chaotic orbit) and that the coupling strengths are uniform throughout the network. We show that if the delay times are sufficiently heterogeneous, for adequate coupling strength the network synchronizes in a spatially homogeneous steady state, which is unstable for the individual maps without coupling. This synchronization behavior is referred to as ‘suppression of chaos by random delays’ and is in contrast with the synchronization when all the interaction delay times are homogeneous, because with homogeneous delays the network synchronizes in a state where the elements display in-phase time-periodic or chaotic oscillations. We analyze the influence of the network topology considering four different types of networks: two regular (a ring-type and a ring-type with a central node) and two random (free-scale Barabasi-Albert and small-world Newman-Watts). We find that when the delay times are sufficiently heterogeneous the synchronization behavior is largely independent of the network topology but depends on the network’s connectivity, i.e., on the average number of neighbors per node.      

Synchronization is enhanced in weighted complex networks

The propensity for synchronization of complex networks with directed and weighted links is considered. We show that a weighting procedure based upon the global structure of network pathways enhances complete synchronization of identical dynamical units in scale-free networks. Furthermore, we numerically show that very similar conditions hold also for phase synchronization of nonidentical chaotic oscillators.     

Synchronization in asymmetrically coupled networks with node balance

We study global stability of synchronization in asymmetrically connected networks of limit-cycle or chaotic oscillators. We extend the connection graph stability method to directed graphs with node balance, the property that all nodes in the network have equal input and output weight sums. We obtain the same upper bound for synchronization in asymmetrically connected networks as in the network with a symmetrized matrix, provided that the condition of node balance is satisfied. In terms of graphs, the symmetrization operation amounts to replacing each directed edge by an undirected edge of half the coupling strength. It should be stressed that without node balance this property in general does not hold.     

Adaptive generalized matrix projective lag synchronization between two different complex networks with non-identical nodes and different dimensions

<正>The adaptive generalized matrix projective lag synchronization between two different complex networks with non-identical nodes and different dimensions is investigated in this paper.Based on Lyapunov stability theory and Barbalat’s lemma,generalized matrix projective lag synchronization criteria are derived by using the adaptive control method.Furthermore,each network can be undirected or directed,connected or disconnected,and nodes in either network may have identical or different dynamics.The proposed strategy is applicable to almost all kinds of complex networks.In addition,numerical simulation results are presented to illustrate the effectiveness of this method,showing that the synchronization speed is sensitively influenced by the adaptive law strength,the network size,and the network topological structure.     

Synchronization Dynamics in Non-Normal Networks: The Trade-Off for Optimality

Synchronization is an important behavior that characterizes many natural and human made systems that are composed by several interacting units. It can be found in a broad spectrum of applications, ranging from neuroscience to power-grids, to mention a few. Such systems synchronize because of the complex set of coupling they exhibit, with the latter being modeled by complex networks. The dynamical behavior of the system and the topology of the underlying network are strongly intertwined, raising the question of the optimal architecture that makes synchronization robust. The Master Stability Function (MSF) has been proposed and extensively studied as a generic framework for tackling synchronization problems. Using this method, it has been shown that, for a class of models, synchronization in strongly directed networks is robust to external perturbations. Recent findings indicate that many real-world networks are strongly directed, being potential candidates for optimal synchronization. Moreover, many empirical networks are also strongly non-normal. Inspired by this latter fact in this work, we address the role of the non-normality in the synchronization dynamics by pointing out that standard techniques, such as the MSF, may fail to predict the stability of synchronized states. We demonstrate that, due to a transient growth that is induced by the structure’s non-normality, the system might lose synchronization, contrary to the spectral prediction. These results lead to a trade-off between non-normality and directedness that should be properly considered when designing an optimal network, enhancing the robustness of synchronization.     

Complex transitions to synchronization in delay-coupled networks of logistic maps

A network of delay-coupled logistic maps exhibits two different synchronization regimes, depending on the distribution of the coupling delay times. When the delays are homogeneous throughout the network, the network synchronizes to a time-dependent state [F.M. Atay, J. Jost, A. Wende, Phys. Rev. Lett. 92, 144101 (2004)], which may be periodic or chaotic depending on the delay; when the delays are sufficiently heterogeneous, the synchronization proceeds to a steady-state, which is unstable for the uncoupled map [C. Masoller, A.C. Marti, Phys. Rev. Lett. 94, 134102 (2005)]. Here we characterize the transition from time-dependent to steady-state synchronization as the width of the delay distribution increases. We also compare the two transitions to synchronization as the coupling strength increases. We use transition probabilities calculated via symbolic analysis and ordinal patterns. We find that, as the coupling strength increases, before the onset of steady-state synchronization the network splits into two clusters which are in anti-phase relation with each other. On the other hand, with increasing delay heterogeneity, no cluster formation is seen at the onset of steady-state synchronization; however, a rather complex unsynchronized state is detected, revealed by a diversity of transition probabilities in the network nodes.     

Synchronization transition of identical phase oscillators in a directed small-world network

We numerically study a directed small-world network consisting of attractively coupled, identical phase oscillators. While complete synchronization is always stable, it is not always reachable from random initial conditions. Depending on the shortcut density and on the asymmetry of the phase coupling function, there exists a regime of persistent chaotic dynamics. By increasing the density of shortcuts or decreasing the asymmetry of the phase coupling function, we observe a discontinuous transition in the ability of the system to synchronize. Using a control technique, we identify the bifurcation scenario of the order parameter. We also discuss the relation between dynamics and topology and remark on the similarity of the synchronization transition to directed percolation.     

Synchronization in a power-driven moving agent network

Motivated by the fact that couplings between individual units of many real-world complex systems are relevant to energy, we propose a power-driven moving agent network model as a simple representation. The presented network exhibits a directed and time-varying topological structure, where each agent associated with a chaotic oscillator is depicted as a random walker in a planar space, and interactions among agents are established via communication by assigning different emission powers to them. To investigate the effect of power distribution, synchronization is further explored for the power-driven moving agent network. Under the constraint of fast-switching, we theoretically show that synchronization of the agent network is determined by the power density which is independent of both the power distribution and the size of network. Several numerical simulations are given to validate the acquired results.     

Global Synchronization of Directed Networks with Fast Switching Topologies

Global synchronization of a class of directed dynamical networks with switching topologies is investigated. It is found that if there is a directed spanning tree in the fixed time-average of network topology and the time-average is achieved sufficiently fast, then the network will reach global synchronization for sufficiently large coupling strength.     

Speed of complex network synchronization

Synchrony is one of the most common dynamical states emerging on networks. The speed of convergence towards synchrony provides a fundamental collective time scale for synchronizing systems. Here we study the asymptotic synchronization times for directed networks with topologies ranging from completely ordered, grid-like, to completely disordered, random, including intermediate, partially disordered topologies. We extend the approach of master stability functions to quantify synchronization times. We find that the synchronization times strongly and systematically depend on the network topology. In particular, at fixed in-degree, stronger topological randomness induces faster synchronization, whereas at fixed path length, synchronization is slowest for intermediate randomness in the small-world regime. Randomly rewiring real-world neural, social and transport networks confirms this picture.     

Physical-layer distributed synchronization in wireless networks and applications

Physical-layer (pulse-coupled) techniques for distributed synchronization in wireless networks are attracting significant attention for their efficiency and scalability. In this paper, the model of pulse-coupled discrete Phase Locked Loops is reviewed and further investigated in two directions. At first, we extend the characterization of (frequency or phase) synchronous states and convergence conditions from homogeneous networks, where all the nodes have the same power constraints, to more general heterogeneous networks. Towards this goal, we build on recent results on algebraic graph theory for generally non-bidirectional graphs, and derive: (i) necessary and sufficient conditions for global synchronization of the network; (ii) closed-form expressions for the asymptotic values of frequency and phases, as a function of the network topology. In the second part of the paper, an application of pulse-coupled synchronization is considered, namely data collection in a sensor network. The energy efficiency of two medium access protocols for data collection from a set of randomly located sensors to an access point is compared: (i) basic ALOHA (which does not require time synchronization among the sensors); (ii) slotted ALOHA, where time synchronization is achieved via pulse-coupled clocks. Analysis shows that the energy spent for maintaining synchronization in slotted ALOHA pays off in terms of total energy consumption with respect to basic ALOHA provided that the number of sensors is sufficiently small. Moreover, the energy gain is proved to depend explicitly on the system load (in terms of packets /s), hardware and topology of the network.     

When weak inhibition synchronizes strongly desynchronizing networks of bursting neurons

We show that weak common inhibition applied to a network of bursting neurons with strong desynchronizing connections can induce burst and complete synchronization. We demonstrate that the weak synchronizing inhibition from the same pacemaker neuron can win out over much stronger desynchronizing connections within the network, provided that the neuron's duty cycle is sufficiently long. We also gain insight into how the changes in burst duty cycles can trigger unexpected clusters of synchrony in bursting networks.     

Consensus on de Bruijn graphs

We study the consensus dynamics with or without time-delays on directed and undirected de Bruijn graphs. Our results show that consensus on an undirected de Bruijn graph has a lower converging speed and larger time-delay tolerance in comparison with that on an undirected scale-free network. Although there is not much difference between the eigenvalue ratios of the two undirected networks, we found that their dynamical properties are remarkably different; consequently, it is seemingly more informative to consider the second smallest and the largest eigenvalues separately rather than considering their ratio in the study of synchronization of a coupled oscillators network. Moreover, our study on directed de Bruijn graphs reveals that properly setting directions on edges can improve the converging speed and time-delay tolerance simultaneously.     

From lag synchronization to pattern formation in networked dynamics

The relationship between lag synchronization and pattern formation is investigated in this article by taking networked dynamics as spatio–temporal models. Firstly some results in two-dimensional open flow models with unidirectional coupling term are presented, and sufficient conditions for globally asymptotically stable lag synchronization in discrete and continuous cases are revealed by analytic method to elucidate the wave pattern evoked by the lag synchronization in such models. Then, an ad hoc network topology, a so-called “tier-network”, is introduced, which is an acyclic digraph with one center node and all the directed pathways from any other node to the center node are equal in length. And the dynamics in tier-networks is investigated. Some similarly sufficient conditions for globally asymptotically stable lag synchronization is obtained by analytic method, and the wave pattern in tier-networks evoked by lag synchronization is illustrated. The above results are supported by the numerical simulation in discrete case.