synchronizability of complex networks recent interest in the study of networks associated with complex systems has led to a better understanding of the factors and parameters relating the topology of the associated networks with the dynamics of the systems and in particular their synchronization in this paper by using known and new results from spectral graph theory we characterize relevant factors which affect the synchronization of complex networks

sharp results in the integral form john nirenberg inequality we consider the strong form of the john nirenberg inequality for the l 2 based bmo we construct explicit bellman functions for the inequality in the continuous and dyadic settings and obtain the sharp constant as well as the precise bound on the inequality s range of validity both previously unknown the results for the two cases are substantially different the paper not only gives another instance in the short list of such explicit calculations but also presents the bellman function method as a sequence of clear steps adaptable to a wide variety of applications

derivation of a hydrodynamic equation for ginzburg landau models in an external field the lattice approximation to a time dependent ginzburg landau equation is investigated in the presence of a small external field the evolution law conserves the spin but is not reversible a nonlinear diffusion equation of divergence type is obtained in the hydrodynamic limit the proof extends to certain stochastically perturbed hamiltonian systems

brief paper l2 norm performance index of synchronization and lqr control synthesis of complex networks in this paper the synchronizability problem of dynamical networks is addressed where better synchronizability means faster network synchronization with lower vibration the l 2 norm of the error vector e is taken as a performance index to measure this kind of synchronizability for the equilibrium synchronization case it is shown that there is a close relationship between the l 2 norm of the error vector e and the h 2 norm of the transfer function g of the linearized network about the equilibrium point consequently the effect of the network coupling topology on the h 2 norm of the transfer function g is analyzed finally an lqr controller is designed according to the lqr problem in modern control theory which can drive the whole network to its equilibrium point and meanwhile minimize the l 2 norm of the output of the linearized network

a new look at the john nirenberg and john stromberg theorems for bmo abstract we develop some techniques for studying various versions of the function space bmo special cases of one of our results give alternative proofs of the celebrated john nirenberg inequality and of related inequalities due to john and to wik our approach enables us to pose a simply formulated geometric question for which an affirmative answer would lead to a version of the john nirenberg inequality with dimension free constants

dynamic behaviors in directed networks motivated by the abundance of directed synaptic couplings in a real biological neuronal network we investigate the synchronization behavior of the hodgkin huxley model in a directed network we start from the standard model of the watts strogatz undirected network and then change undirected edges to directed arcs with a given probability still preserving the connectivity of the network a generalized clustering coefficient for directed networks is defined and used to investigate the interplay between the synchronization behavior and underlying structural properties of directed networks we observe that the directedness of complex networks plays an important role in emerging dynamical behaviors which is also confirmed by a numerical study of the sociological game theoretic voter model on directed networks

enhancing synchronizability of diffusively coupled dynamical networks a survey in this paper we review the literature on enhancing synchronizability of diffusively coupled dynamical networks with identical nodes the last decade has witnessed intensive investigations on the collective behavior over complex networks and synchronization of dynamical systems is the most common form of collective behavior for many applications it is desired that the synchronizability the ability of networks in synchronizing activity of their individual dynamical units is enhanced there are a number of methods for improving the synchronization properties of dynamical networks through structural perturbation in this paper we survey such methods including adding removing nodes and or edges rewiring the links and graph weighting these methods often try to enhance the synchronizability through minimizing the eigenratio of the laplacian matrix of the connection graph a synchronizability measure based on the master stability function formalism we also assess the performance of the methods by numerical simulations on a number of real world networks as well as those generated through models such as preferential attachment watts strogatz and erdos renyi

synchronous chaos in coupled oscillator systems we investigate the synchronization of chaotic oscillations in coupled oscillator systems both theoretically and in analog electronic circuits particular attention is paid to deriving and testing general conditions for the stability of synchronous chaotic behavior in cases where the coupled oscillator array possesses a shift invariant symmetry these cases include the well studied cases of nearest neighbor diffusive coupling and all to all or global coupling an approximate criterion is developed to predict the stability of synchronous chaotic oscillations in the strong coupling limit when the oscillators are coupled through a single coordinate scalar coupling this stability criterion is illustrated numerically in a set of coupled r ossler like oscillators synchronization experiments with coupled r ossler like oscillator circuits are also carried out to demonstrate the applicability of the theory to real systems

betweenness centrality correlation in social networks scale free sf networks exhibiting a power law degree distribution can be grouped into the assortative dissortative and neutral networks according to the behavior of the degree degree correlation coefficient here we investigate the betweenness centrality bc correlation for each type of sf networks while the bc bc correlation coefficients behave similarly to the degree degree correlation coefficients for the dissortative and neutral networks the bc correlation is nontrivial for the assortative ones found mainly in social networks the mean bc of neighbors of a vertex with bc g i is almost independent of g i implying that each person is surrounded by almost the same influential environments of people no matter how influential the person may be

consensus in a heterogeneous influence network we consider a dynamical network model in which a number of agents all move on the plane with the same constant absolute velocity at each time step each agent s direction is updated as the average of its direction plus the directions of other agents who can influence it the influencing capability of each agent is represented by its influencing radius which is randomly chosen according to a power law distribution with a scaling exponent between 2 and ensuremath infty as the value of the scaling exponent decreases the radius distribution becomes more heterogeneous and the network becomes much easier to achieve direction consensus among agents due to the leading roles played by a few hub agents furthermore almost all agents will finally move in the same desired direction in a strong heterogeneous influence network if and only if a small fraction of hub agents can be controlled to move in the desired direction these results also reflect the robust yet fragile feature of a heterogeneous influence network

three structural properties reflecting the synchronizability of complex networks during the process of adding links we find that the synchronizability of the classical barabasi albert ba scale free or watts strogatz ws small world networks can be statistically quantified by three essentially structural quantities of these networks i e the eccentricity variance of the degree distribution and clustering coefficients the results indicate that both the eccentricity and clustering coefficient are positively linearly correlated with synchronizability while the variance is negatively linearly correlated moreover the efficiency of some particular strategies of adding links to change the synchronizability is also investigated this information can be used to guide us to design corresponding strategies of structure evolving processes to manipulate the synchronizability of a given network

linked by loops network structure and switch integration in complex dynamical systems abstract simple nonlinear dynamical systems with multiple stable stationary states are often taken as models for switchlike biological systems this paper considers the interaction of multiple such simple multistable systems when they are embedded together into a larger dynamical supersystem attention is focused on the network structure of the resulting set of coupled differential equations and the consequences of this structure on the propensity of the embedded switches to act independently versus cooperatively specifically it is argued that both larger average and larger variance of the node degree distribution lead to increased switch independence given the frequency of empirical observations of high variance degree distributions e g power law in biological networks it is suggested that the results presented here may aid in identifying switch integrating subnetworks as comparatively homogenous low degree substructures potential applications to ecological problems such as the relationship of stability and complexity are also briefly discussed

theory of locally concave functions and its applications to sharp estimates of integral functionals we prove a duality theorem the computation of certain bellman functions is usually based on as a byproduct we obtain sharp results about the norms of monotonic rearrangements the main novelty of our approach is a special class of martingales and an extremal problem on this class which is dual to the minimization problem for locally concave functions

decoupling process for better synchronizability on scale free networks we propose a decoupling process performed in scale free networks to enhance the synchronizability of the network together with preserving the scale free structure simulation results show that the decoupling process can effectively promote the network synchronizability which is measured in terms of eigenratio of the coupling matrix moreover we investigate the correlation between some important structural properties and the collective synchronization and find that the maximum vertex betweenness seems to be the most strongly correlated with the synchronizability among the major structural features considered we explain the effect of the decoupling process from a viewpoint of coupling information transmission our work provides some evidence that the dynamics of synchronization is related to that of information or vehicle traffic because of the low cost in modifying the coupling network the decoupling process may have potential applications

clustering and the synchronization of oscillator networks by manipulating the clustering coefficient of a network without changing its degree distribution we examine the effect of clustering on the synchronization of phase oscillators on networks with poisson and scale free degree distributions for both types of networks increased clustering hinders global synchronization as the network splits into dynamical clusters that oscillate at different frequencies surprisingly in scale free networks clustering promotes the synchronization of the most connected nodes hubs even though it inhibits global synchronization as a result they show an additional advanced transition instead of a single synchronization threshold this cluster enhanced synchronization of hubs may be relevant to the brain that is scale free and highly clustered

on synchronizability and heterogeneity in unweighted networks abstract synchronization in classes of continuous time dynamical unweighted networks with different topologies is investigated a synchronization optimal network model based on rewiring of links is proposed compared with other networks it exhibits a stronger synchronizability we presented link density and investigated the correlation of synchronizability link density and heterogeneity in degree distribution in this work it will be shown that synchronizability of type i networks is independent of heterogeneity in the degree distribution when the link density is smaller than 0 02 synchronizability and link density are proportional when the link density is larger than 0 025 heterogeneity decides the curve slope the synchronizability of type ii networks is drastically enhanced by enhancing the link density when the link density is smaller than 0 025 however synchronizability increases weakly by enhancing the link density when the link density is larger than 0 025

synchronizability of dynamical scale free networks subject to random errors in this paper the robustness of network synchronizability against random deletion of nodes i e errors in dynamical scale free networks was studied to this end two measures of network synchronizability namely the eigenratio of the laplacian and the order parameter quantifying the degree of phase synchrony were adopted and the synchronizability robustness on preferential attachment scale free graphs was investigated the findings revealed that as the network size decreases the robustness of its synchronizability against random removal of nodes declines i e the more the number of randomly removed nodes from the network the worse its synchronizability we also showed that this dependence of the synchronizability on the network size is different with that in the growing scale free networks the profile of a number of network properties such as clustering coefficient efficiency assortativity and eccentricity as a function of the network size was investigated in these two cases growing scale free networks and those with randomly removed nodes the results showed that these processes are also different in terms of these metrics

lamellar pattern formation in small world media abstract numerical and analytical techniques are used to investigate the effects of quenched disorder of small world networks on the phase ordering dynamics of lamellar patterns as modeled by the swift hohenberg equation morphologies for small and large values of the network randomness are quite different it is found that addition of shortcuts to an underlying regular lattice makes the growth of domains evolving from random initial conditions much slower at late times as the randomness increases the evolution is eventually frozen

network synchronization spectral versus statistical properties we consider synchronization of weighted networks possibly with asymmetrical connections focusing on causal relations rather than the observed correlations we show that the synchronizability of networks cannot be directly inferred from their statistical properties small local changes in the network structure can sensitively affect the eigenvalues relevant for synchronization while the gross statistical network properties remain essentially unchanged consequently commonly used statistical properties including the degree distribution degree homogeneity average degree average distance degree correlation and clustering coefficient can fail to characterize the synchronizability of networks in terms of causal relations despite the observed correlations c 2006 elsevier b v all rights reserved

synchronization of kuramoto oscillators in small world networks abstract we study the synchronization of kuramoto oscillators on small world networks firstly we analyze the impact of coupling strength rewiring probability and average degree of networks on the phase synchronization in weak coupled system then by increasing the coupling strength we find an interesting phenomenon that when coupling strength is much larger than the synchronization threshold there is another threshold which can decrease the synchronization performance of the system finally in order to enhance the synchronization performance of strong coupled system we adopt a simple way to delete some nodes randomly to construct disordered environment numerical results show that a certain degree of disorder can promote the synchronization performance of strong coupled small world networks

structure properties and synchronizability of cobweb like networks abstract the cobweb like network was constructed and its structure properties and synchronizability were studied by numerical simulation and probability theory for cobweb like networks the behaviors of structure properties with the adding probability are similar to those for the small world network however the properties of cobweb like networks also distinctly depend on the spoke ring ratio r with increasing r the average path length decreases and the average clustering coefficient increases simultaneously the connectivity distribution becomes heterogeneous moreover in the case of bounded synchronized region the synchronizability shows complicated behaviors with r and is enhanced favorably at r 2 28 with homogeneous load distribution the reason is that homogeneous load distribution has an advantage over the communication between oscillators which efficiently leads to global synchronization this work could be useful for design and kinetic property research of cobweb like networks

factors that predict better synchronizability on complex networks while shorter characteristic path length has in general been believed to enhance synchronizability of a coupled oscillator system on a complex network the suppressing tendency of the heterogeneity of the degree distribution even for shorter characteristic path length has also been reported to see this we investigate the effects of various factors such as the degree characteristic path length heterogeneity and betweenness centrality on synchronization and find a consistent trend between the synchronization and the betweenness centrality the betweenness centrality is thus proposed as a good indicator for synchronizability

degree based attacks are not optimal for desynchronization in general networks suppressing harmful synchronization in complex networked systems is receiving increasing interest in various contexts including epileptic seizure and the internet and traffic congestion it is traditionally believed that by disrupting the most important nodes i e those nodes of high degrees synchronization can be effectively mitigated however this approach can perform poorly in more general situations such as homogeneous random or small world networks in this article we investigate how topological properties such as the heterogeneity of the network can have an impact on its desynchronization in particular we propose a topology aware scheme that chooses topologically sufficiently diversified contrarian nodes by taking into account the patterns of connectivity among them hence a maximal number of nodes in the network can be influenced leading to a more global impact and greater disruption to synchronization our scheme demonstrates significantly improved performance on various networked systems including homogeneous networks small world and even scale free networks

a mathematical analysis of the effects of hebbian learning rules on the dynamics and structure of discrete time random recurrent neural networks we present a mathematical analysis of the effects of hebbian learning in random recurrent neural networks with a generic hebbian learning rule including passive forgetting and different timescales for neuronal activity and learning dynamics previous numerical work has reported that hebbian learning drives the system from chaos to a steady state through a sequence of bifurcations here we interpret these results mathematically and show that these effects involving a complex coupling between neuronal dynamics and synaptic graph structure can be analyzed using jacobian matrices which introduce both a structural and a dynamical point of view on neural network evolution furthermore we show that sensitivity to a learned pattern is maximal when the largest lyapunov exponent is close to 0 we discuss how neural networks may take advantage of this regime of high functional interest

a novel definition of generalized synchronization on networks and a numerical simulation example this paper develops a novel definition of generalized synchronization on complex networks consisting of systems evolving in a chaotic or regular fashion with two usual methods for detecting generalized synchronization two criteria for generalized synchronization on networks are advanced some complex dynamical behaviors are discussed briefly on the basis of numerical simulations of a real network example

improving consensus and synchronizability of networks of coupled systems via adding links abstract the problem of improving the consensus and synchronizability for weighted networks of dynamic systems via adding links among them is investigated in this paper a weighted network entropy is introduced to measure the homogeneity of the weighted networks the relationship between the homogeneity and the consensus and synchronizability is analyzed a solution scheme is proposed to improve the consensus and synchronizability of the weighted networks through maximizing their homogeneity via adding as few links as possible weighted networks created from the barrat model are tested by using the proposed scheme and two commonly used link adding approaches are selected to compare with this scheme computer simulation results show that the proposed link adding scheme can enhance the consensus and the synchronizability of the weighted networks effectively and efficiently

synchronization criterions and pinning control of general complex networks with time delay the problems of synchronization and pinning control for general time delay complex dynamical networks are investigated in this paper less conservative criterions for both continuous time and discrete time complex dynamical networks with time delay are obtained pinning control strategies are respectively designed to make these complex dynamical networks synchronized moreover the problems of designing controllers are converted into solving optimal problems of a series of linear matrix inequalities which reduces the computation complexity finally numerical simulations verify the effectiveness of our methodology

relaxation of synchronization on complex networks point of information transfer or the dynamics of system recovery from a perturbation we measure the relaxation time that is required to establish global synchronization by varying the structural properties of the networks it is found that the relaxation time in a strong coupling regime kkc logarithmically increases with network size n which is attributed to the initial random phase fluctuation given by on 1 2 after elimination of the initial phase fluctuation the relaxation time is found to be independent of the system size this implies that the local interaction that depends on the structural connectivity is irrelevant in the relaxation dynamics of the synchronization in the strong coupling regime the relaxation dynamics is analytically derived in a form independent of the system size and it exhibits good consistency with numerical simulations as an application we also explore the recovery dynamics of the oscillators when perturbations enter the system

analyzing and controlling the network synchronization regions abstract in this paper the commonly concerned issue of synchronization regions of complex dynamical networks is investigated for the case when the synchronous state is an equilibrium point some simple sufficient conditions for a network to have or have no unbounded synchronization regions of the form 1 are established where 1 is a constant in addition a sufficient condition for the existence of a bounded synchronization region of the form 2 3 is derived where 2 and 3 are constants by using the parameter dependent lyapunov function method furthermore some effective controller design methods are presented that can change the synchronization regions thereby managing the synchronizability of the network finally some numerical examples are given to show that a dynamical network may have disconnected synchronization regions particularly it may have the coexistence of unbounded and bounded synchronization regions in the form of 1 2 3

a note on bmo and its application abstract in this note at first we will point out a fact which is implicitly contained in the original paper of john and nirenberg comm pure appl math 14 1961 415 426 if a bmo r n function f satisfies lim a f a f d x obviously if the value of left term is finite it must be zero then there holds 1 f t c f n ln 2 1 t where f t is the nondecreasing rearrangement of f as a consequence we get the following inequality 2 f p c n p f r r p f bmo 1 r p 1 r p by this inequality the bilinear estimates in bmo in math z 235 2000 173 194 can be easily improved these bilinear estimates are crucial for the main results in that paper

effects of degree correlation on the controllability of networks abstract in this paper we investigate the effects of degree correlation on the controllability of undirected networks with different degree mixing patterns based on simulation analysis considering random pinning max degree pinning and mix degree pinning some relevance factors on controllability such as the control gain and the number of pinned nodes are discussed in detail it is found that disassortative mixing enhances the network controllability contrast to assortative mixing to which the network controllability is sensible furthermore too large or too low value of the feedback gain can reduce the controllability from the viewpoint of practical application mix degree pinning strategy is suggested in assortative network while in disassortative network max degree pinning scheme is better than random pinning

enhancing robustness and synchronizability of networks homogenizing their degree distribution a new family of networks called entangled has recently been proposed in the literature these networks have optimal properties in terms of synchronization robustness against errors and attacks and efficient communication they are built with an algorithm which uses modified simulated annealing to enhance a well known measure of networks ability to reach synchronization among nodes in this work we suggest that a class of networks similar to entangled networks can be produced by changing some of the connections in a given network or by just adding a few connections we call this class of networks weak entangled although entangled networks can be considered as a subset of weak entangled networks we show that both classes share similar properties especially with respect to synchronization and robustness and that they have similar structural properties

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 h enon maps logistic maps and lorentz oscillators as the coupling strength ensuremath 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 ensuremath epsilon our result is useful for understanding the synchronization process in complex networks

finite size scaling of synchronized oscillation on complex networks the onset of synchronization in a system of random frequency oscillators coupled through a random network is investigated using a mean field approximation we characterize sample to sample fluctuations for networks of finite size and derive the corresponding scaling properties in the critical region for scale free networks with the degree distribution p k approximately k gamma at large k we found that the finite size exponent nu takes on the value 5 2 when gamma 5 the same as in the globally coupled kuramoto model for highly heterogeneous networks 3 gamma 5 nu and the order parameter exponent beta depend on gamma the analytical expressions for these exponents obtained from the mean field theory are shown to be in excellent agreement with data from extensive numerical simulations

synchronization processes in complex networks we present an extended analysis based on the dynamics towards synchronization of a system of coupled oscillators of the hierarchy of communities in complex networks in the synchronization process different structures corresponding to well defined communities of nodes appear in a hierarchical way the analysis also provides a useful connection between synchronization dynamics complex networks topology and spectral graph analysis

sharp multiplicative inequalities with bmo i abstract we find the best possible constant c in the inequality l r c l p p r bmo 1 p r where 2 r and p r we employ the bellman function technique to solve this problem in the case of an interval and then transfer our results to the circle and the line

enhanced synchronizability by structural perturbations in this brief report we investigate the collective synchronization of a system of coupled oscillators on a barab asi albert scale free network we propose an approach of structural perturbations aiming at those nodes with maximal betweenness this method can markedly enhance the network synchronizability and is easy to realize the simulation results show that the eigenratio will sharply decrease by one half when only 0 6 of those hub nodes occur under three division processes when the network size n 2000 in addition the present study also provides numerical evidence that the maximal betweenness plays a major role in network synchronization