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.      

Weighted self-similar networks under preferential attachment

We introduce an abstract evolutionary formalism that generates weighted networks whose growth under stochastic preferential attachment triggers unrestricted weight rearrangements in existing links. The class of resulting algorithms for different parameter values includes the Barabási-Albert and Barrat-Barthélemy-Vespignani models as special cases. We solve the recursions that describe the average growth to derive exact solutions for the expected degree and strength distribution, the individual strength and weight development and the joint distribution of neighboring degrees. We find that the network exhibits a particular form of self-similarity, namely every sufficiently interconnected node has on average the same constitution of small-degree neighbors as any other node of large degree. Finally we suggest potential applications in several fields of interest.     

Synchronization in lattice-embedded scale-free networks

In this work, we study the effects of embedding a system of non-linear phase oscillators in a two-dimensional scale-free lattice. In order to analyze the effects of the embedding, we consider two different topologies. On the one hand, we consider a scale-free complex network where no constraint on the length of the links is taken into account. On the other hand, we use a method recently introduced for embedding scale-free networks in regular Euclidean lattices. In this case, the embedding is driven by a natural constraint of minimization of the total length of the links in the system. We analyze and compare the synchronization properties of a system of non-linear Kuramoto phase oscillators, when interactions between the oscillators take place in these networks. First, we analyze the behavior of the Kuramoto order parameter and show that the onset of synchronization is lower for non-constrained lattices. Then, we consider the behavior of the mean frequency of the oscillators as a function of the natural frequency for the two different networks and also for different values of the scale-free exponent. We show that, in contrast to non-embedded lattices that present a mean-field-like behavior characterized by the presence of a single cluster of synchronized oscillators, in embedded lattices the presence of a diversity of synchronized clusters at different mean frequencies can be observed. Finally, by considering the behavior of the mean frequency as a function of the degree, we study the role of hubs in the synchronization properties of the system.     

Relationship between degree-rank distributions and degree distributions of complex networks

Both the degree distribution and the degree-rank distribution, which is a relationship function between the degree and the rank of a vertex in the degree sequence obtained from sorting all vertices in decreasing order of degree, are important statistical properties to characterize complex networks. We derive an exact mathematical relationship between degree-rank distributions and degree distributions of complex networks. That is, for arbitrary complex networks, the degree-rank distribution can be derived from the degree distribution, and the reverse is true. Using the mathematical relationship, we study the degree-rank distributions of scale-free networks and exponential networks. We demonstrate that the degree-rank distributions of scale-free networks follow a power law only if scaling exponent λ>2. We also demonstrate that the degree-rank distributions of exponential networks follow a logarithmic law. The simulation results in the BA model and the exponential BA model verify our results.     

Synchronization in an array of coupled Boolean networks

This Letter presents an analytical study of synchronization in an array of coupled deterministic Boolean networks. A necessary and sufficient criterion for synchronization is established based on algebraic representations of logical dynamics in terms of the semi-tensor product of matrices. Some basic properties of a synchronized array of Boolean networks are then derived for the existence of transient states and the upper bound of the number of fixed points. Particularly, an interesting consequence indicates that a “large” mismatch between two coupled Boolean networks in the array may result in loss of synchrony in the entire system. Examples, including the Boolean model of coupled oscillations in the cell cycle, are given to illustrate the present results.     

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 W_s 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, W_s scales logarithmically with the system size.     

Synchronization in complex networks with switching topology

This Letter investigates synchronization issues of complex dynamical networks with switching topology. By constructing a common Lyapunov function, we show that local and global synchronization for a linearly coupled network with switching topology can be evaluated by the time average of second smallest eigenvalues corresponding to the Laplacians of switching topology. This result is quite powerful and can be further used to explore various switching cases for complex dynamical networks. Numerical simulations illustrate the effectiveness of the obtained results in the end.     

Synchronization of mobile chaotic agents on connected networks

This work studies the synchronization of a number of mobile agents on a substrate network. Each agent carries a chaotic map and randomly walks on a connected network. The collection of agents consists of another time-varying network derived from the substrate network. It is found that the synchronization conditions of this agent network depend on the average degree of the substrate network’s connectivity, the coupling strength between interacting agents, and the agent density in the network. Synchronization of the agent network on scale-free and ER networks is considered here, and it is found that the scale-free topology is more applicable to synchronize mobile chaotic agents. To get analytical insights, the star graph is taken and considered as a substrate network.     

Line graphs as social networks

It was demonstrated recently that the line graphs are clustered and assortative. These topological features are known to characterize some social networks [M.E.J. Newman, Y. Park, Why social networks are different from other types of networks, Phys. Rev. E 68 (2003) 036122]; it was argued that this similarity reveals their cliquey character. In the model proposed here, a social network is the line graph of an initial network of families, communities, interest groups, school classes and small companies. These groups play the role of nodes, and individuals are represented by links between these nodes. The picture is supported by the data on the LiveJournal network of about 8×10⁶ people.     

Synchronization in weighted complex networks: Heterogeneity and synchronizability

Synchronization in different types of weighted networks based on a scale-free weighted network model is investigated. It has been argued that heterogeneity suppresses synchronization in unweighted networks [T. Nishikawa, A.E. Motter, Y.C. Lai, F.C. Hoppensteadt, Phys. Rev. Lett. 91 (2003) 014101]. However, it is shown in this work that as the network becomes more heterogeneous, the synchronizability of Type I symmetrically weighted networks, and Type I and Type II asymmetrically weighted networks is enhanced, while the synchronizability of Type II symmetrically weighted networks is weakened.     

The role of degree-weighted couplings in the synchronous onset of Kuramoto oscillator networks

This paper investigates the role of asymmetrical degree-dependent weighted couplings in synchronization of a network of Kuramoto oscillators, where the conditions of coupling criticality for the onset of phase synchronization in degree-weighted complex networks are arrived at. The numerical simulations visualize that for networks having power-law or exponential degree distributions, asymmetrical degree-weighted couplings (with increasing weighting exponent β) increases the critical coupling to achieve the onset of phase synchronization in the networks.     

Scale-free properties of information flux networks in genetic algorithms

In this study, we present empirical analysis of statistical properties of mating networks in genetic algorithms (GAs). Under the framework of GAs, we study a class of interaction network model—information flux network (IFN), which describes the information flow among generations during evolution process. The IFNs are found to be scale-free when the selection operator uses a preferential strategy rather than a random. The topology structure of IFN is remarkably affected by operations used in genetic algorithms. The experimental results suggest that the scaling exponent of the power-law degree distribution is shown to decrease when crossover rate increases, but increase when mutation rate increases, and the reason may be that high crossover rate leads to more edges that are shared between nodes and high mutation rate leads to many individuals in a generation possessing low fitness. The magnitude of the out-degree exponent is always more than the in-degree exponent for the systems tested. These results may provide a new viewpoint with which to view GAs and guide the dissemination process of genetic information throughout a population.     

Synchronization and information transmission in spatio-temporal networks of deformable units

We study the relationship between synchronization and the rate with which information is exchanged between nodes in a spatio-temporal network that describes the dynamics of classical particles under a substrate Remoissenet-Peyrard potential. We also show how phase and complete synchronization can be detected in this network. The difficulty in detecting phase synchronization in such a network appears due to the highly non-coherent character of the particle dynamics which unables a proper definition of the phase dynamics. The difficulty in detecting complete synchronization appears due to the spatio character of the potential which results in an asymptotic state highly dependent on the initial state.      

Lobby index in networks

We propose a new node centrality measure in networks, the lobby index, which is inspired by Hirsch’s h-index. It is shown that in scale-free networks with exponent α the distribution of the l-index has power tail with exponent α(α+1). Properties of the l-index and extensions are discussed.     

Synchronization of linearly coupled networks of deterministic ratchets

This Letter focuses on the synchronization in a class of dynamical complex networks with each node being a deterministic ratchet. In virtue of the technique derived from pendulum-like nonlinear analytic theory and Kalman–Yakubovich–Popov (KYP) lemma, simple linear matrix inequality (LMI) formulations are established to guarantee the stable synchronization of such networks. An interesting conclusion is reached that the stability of synchronization in the coupled whole N-dimensional networks can be converted into that of the simplest 2-dimensional space.     

Noisy scale-free networks

The impact of observational noise on the analysis of scale-free networks is studied. Various noise sources are modeled as random link removal, random link exchange and random link addition. Emphasis is on the resulting modifications for the node-degree distribution and for a functional ranking based on betweenness centrality. The implications for estimated gene-expressed networks for childhood acute lymphoblastic leukemia are discussed.     

Tuning degree distributions: Departing from scale-free networks

Scale-free networks are characterized by a degree distribution with power-law behavior. Although scale-free networks have been shown to arise in many areas, ranging from the World Wide Web to transportation or social networks, degree distributions of other observed networks often differ from the power-law type. Data based investigations require modifications of the typical scale-free network.We present an algorithm that generates networks in which the shape of the degree distribution is tunable by modifying the preferential attachment step of the Barabási-Albert construction algorithm. The shape of the distribution is represented by dispersion measures such as the variance and the skewness, both of which are highly correlated with the maximal degree of the network and, therefore, adequately represents the influence of superspreaders or hubs. By combining our algorithm with work of Holme and Kim, we show how to generate networks with a variety of degree distributions and clustering coefficients.     

Synchronization in complex delayed dynamical networks with impulsive effects

The present paper is mainly concerned with the issues of synchronization dynamics of complex delayed dynamical networks with impulsive effects. A general model of complex delayed dynamical networks with impulsive effects is formulated, which can well describe practical architectures of more realistic complex networks related to impulsive effects. Based on impulsive stability theory on delayed dynamical systems, some simple but less conservative criterion are derived for global synchronization of such dynamical network. It is shown that synchronization of the networks is heavily dependent on impulsive effects of connecting configuration in the networks. Furthermore, the theoretical results are applied to a typical SF network composing of impulsive coupled chaotic delayed Hopfield neural network nodes, and are also illustrated by numerical simulations.     

Scaling and correlations in three bus-transport networks of China

We report the statistical properties of three bus-transport networks (BTN) in three different cities of China. These networks are composed of a set of bus lines and stations serviced by these. Network properties, including the degree distribution, clustering and average path length are studied in different definitions of network topology. We explore scaling laws and correlations that may govern intrinsic features of such networks. Besides, we create a weighted network representation for BTN with lines mapped to nodes and number of common stations to weights between lines. In such a representation, the distributions of degree, strength and weight are investigated. A linear behavior between strength and degree s(k)∼k$s(k)\sim k$ is also observed.