Factors that predict better synchronizability on complex networks

A generalized simplicial model and its application

Higher-order structures, consisting of more than two individuals, provide a new perspective to reveal the missed non-trivial characteristics under pairwise networks. Prior works have researched various higher-order networks, but research for evaluating the effects of higher-order structures on network functions is still scarce. In this paper, we propose a framework to quantify the effects of higher-order structures (e.g., 2-simplex) and vital functions of complex networks by comparing the original network with its simplicial model. We provide a simplicial model that can regulate the quantity of 2-simplices and simultaneously fix the degree sequence. Although the algorithm is proposed to control the quantity of 2-simplices, results indicate it can also indirectly control simplexes more than 2-order. Experiments on spreading dynamics, pinning control, network robustness, and community detection have shown that regulating the quantity of 2-simplices changes network performance significantly. In conclusion, the proposed framework is a general and effective tool for linking higher-order structures with network functions. It can be regarded as a reference object in other applications and can deepen our understanding of the correlation between micro-level network structures and global network functions.

Organization Synchronization in Response to Complex Project Delays: Network-Based Analysis

In response to frequent complex project delays, organization synchronization, a set of interactions, is a dynamic behavior that helps to restore the stability of complex projects after delays. However, few studies have figured out how organizations synchronize effectively in order to deal with delay issues. To solve this problem, this study first provides a preliminary list of CDFs and indices of organization interactions are also given. A total of 15 key CDFs and 10 interaction ways were refined according to a questionnaire survey. In addition, the complex network synchronization (CNS) theory was adopted to analyze the synchronizability and importance of nodes by comprehensively using multiple parameters. A complex metro project with 51 project organizations was used as a case study and we found that specific signal organizations synchronized through three effective interaction ways (meetings, discussion and study, and the Internet) to cope with six CDFs (safety accidents, prominent problems of land expropriation, unreasonable timelines by clients, improper construction designs, delayed payments, and high financial risks). This study contributes to defining organization synchronization, providing a feasible research framework for assessing network synchronizability and identifying signal organizations in complex projects, and guiding practitioners to effectively cope with delays by interactions between signal organizations.

Effect of topology upon relay synchronization in triplex neuronal networks

Relay synchronization in complex networks is characterized by the synchronization of remote parts of the network due to their interaction via a relay. In multilayer networks, distant layers that are not connected directly can synchronize due to signal propagation via relay layers. In this work, we investigate relay synchronization of partial synchronization patterns like chimera states in three-layer networks of interacting FitzHugh-Nagumo oscillators. We demonstrate that the phenomenon of relay synchronization is robust to topological random inhomogeneities of small-world type in the layer networks. We show that including randomness in the connectivity structure either of the remote network layers or of the relay layer increases the range of interlayer coupling strength where relay synchronization can be observed.

Enhancing network synchronizability by strengthening a single node

In improving the stability of complex dynamical systems, an outstanding problem is how to achieve the desired performance at a low cost. For engineering and biological complex systems whose performance and functionality rely on the synchronous motion of their units, an important question related to the performance-cost-balance problem is how to improve efficiently the system synchronizability when a small amount of additional coupling resource is available. Here, employing a complex network of coupled chaotic oscillators as the model, we address this question by introducing a small amount of coupling intensity to only a single oscillator and investigate how the improvement of the network synchronizability is dependent on the location of the target oscillator. Theoretical analysis shows that, to achieve the maximum network synchronizability, the target oscillator to be strengthened should be chosen according to the eigenvector of the most unstable mode. Based on the theoretical finding, we further propose a single-node-based scheme for improving synchronization: the eigenvector-centrality-based strengthening scheme. We describe in detail how to apply this scheme under different synchronization scenarios and justify its efficiency in various network models by numerical simulations. The performance of the new scheme is compared with the conventional ones based on betweenness, closeness, and degree centralities, and it is shown that the new scheme has a clear advantage over the conventional ones. Furthermore, by a brute-force search of the target oscillator over the network, it is verified numerically that the oscillator identified by the new scheme indeed gives the best synchronization performance.

Perturbation analysis and comparison of network synchronization methods

In many networked systems, synchronization is important and useful, and how to enhance synchronizability is an interesting problem. Based on the matrix perturbation theory, we analyze five methods of network synchronization enhancement, including the link removal, node removal, dividing hub node, pull control, and pinning control methods, and obtain explicit expressions for eigenvalue changes. By these comparisons, we find that, among all these methods, the pull control method is remarkable, as it can extend the synchronization (coupling strength) region from both the left and right sides, for any controlled node. Extensive simulation results are given to support the accuracy of the perturbation-based analysis.

Enhancing network cluster synchronization capability based on artificial immune algorithm

With the deeper study on complex networks, more and more attention has been paid to the research on the cluster synchronization phenomena based on complex networks. In the real world, synchronization phenomena or cluster synchronous behaviors occur frequently, some of which may result in larger negative impacts to the society, such as “cadmium rice event,” while others bring significant economic benefits to the society, such as the synchronization of the propaganda for “black Friday.” Therefore, research on cluster synchronism has great values for theoretical study and social applications. Currently, the study of cluster synchronicity is focused on the solution of the synchronization threshold and the analysis of the synchronization phenomenon, etc. However, the optimization to enhance the synchronous evolutionary effect is rarely presented in literatures. To overcome these limitations of current work, we explore the optimization of network structure with artificial immune algorithms under the condition of a constant network scale and finally realize the promotion of synchronous evolution effect in this paper. Moreover, the relevant research results are applied to real cases. On one hand, for the positive synchronous behaviors, the network structure with good synchronization capability is created to achieve better synchronization. On the other hand, the connection between nodes and edges in the network is cut off to avoid the occurrence of negative synchronous behaviors.

All-sense-all networks are suboptimal for sensorimotor synchronization

In human groups that seek to synchronize to a common steady beat, every member can typically perceive every other member. We question whether this naturally occurring all-sense-all condition is optimal for temporal coordination. We consider alternative configurations represented by directed graphs, in which individuals can only hear or see a subset of others. We identify a trade-off in the topology of such networks: While denser graphs provide stronger coupling, improving synchrony, density increases sensitivity to early taps, which produces rushing. Results from an experimental study with music conservatory students show that networks that combine short path length with low density match all-sense-all networks in synchrony while yielding a steadier beat. These findings suggest that professional teams in arts, sports, industry, and the military may improve temporal coordination by employing technology that strategically configures who can track whom.

Identifying nodal properties that are crucial for the dynamical robustness of multistable networks

We investigate the collective dynamics of bistable elements connected in different network topologies and estimate the network response to localized perturbations on different classes of nodes by introducing a variant of the concept of multinode basin stability. We show that perturbations on nodes with high closeness and betweeness centrality drastically reduces the capacity of the system to return to the original state. This effect is most pronounced for a star network, where perturbation of the single hub node can destroy the collective state, while the system manages to recover even when a majority of the peripheral nodes are strongly perturbed. This demonstrates the extreme effect of the centrality of the perturbed node on the stability of the network. Further, we exploit the difference in centrality distributions in random scale-free networks with m=1 and m=2 to probe which property most influences the collective dynamics in heterogeneous networks. Significantly, we find clear evidence that the betweeness centrality of the perturbed node is more crucial for dynamical robustness than closeness centrality or degree of the node. This allows us to decide which nodes to safeguard in order to maintain the collective state of a network against targeted localized attacks.

Source and sink nodes in absence seizures

As opposed to focal epilepsy, absence seizures do not exhibit a clear seizure onset zone or focus since its ictal activity rapidly engages both brain hemispheres. Yet recent graph theoretical analysis applied to absence seizures EEG suggests the cortical focal presence, an unexpected feature for this type of epilepsy. In this study, we explore the characteristics of absence seizure by classifying the nodes as to their source/sink natures via weighted directed graph analysis based on connectivity direction and strength estimation using information partial directed coherence (iPDC). By segmenting the EEG signals into relatively short 5-sec-long time windows we studied the evolution of coupling strengths from both sink and source nodes, and the network dynamics of absence seizures in eight patients.

Concurrent enhancement of percolation and synchronization in adaptive networks

Co-evolutionary adaptive mechanisms are not only ubiquitous in nature, but also beneficial for the functioning of a variety of systems. We here consider an adaptive network of oscillators with a stochastic, fitness-based, rule of connectivity, and show that it self-organizes from fragmented and incoherent states to connected and synchronized ones. The synchronization and percolation are associated to abrupt transitions, and they are concurrently (and significantly) enhanced as compared to the non-adaptive case. Finally we provide evidence that only partial adaptation is sufficient to determine these enhancements. Our study, therefore, indicates that inclusion of simple adaptive mechanisms can efficiently describe some emergent features of networked systems’ collective behaviors, and suggests also self-organized ways to control synchronization and percolation in natural and social systems.

Impact of a leader on cluster synchronization

We study the mechanisms of frequency-synchronized cluster formation in coupled nonidentical oscillators and investigate the impact of presence of a leader on the cluster synchronization. We find that the introduction of a leader, a node having large parameter mismatch, induces a profound change in the cluster pattern as well as in the mechanism of the cluster formation. The emergence of a leader generates a transition from the driven to the mixed cluster state. The frequency mismatch turns out to be responsible for this transition. Additionally, for a chaotic evolution, the driven mechanism stands as a primary mechanism for the cluster formation, whereas for a periodic evolution the self-organization mechanism becomes equally responsible.

Control for a synchronization-desynchronization switch

How to freely enhance or suppress synchronization of networked dynamical systems is of great importance in many disciplines. A unified precise control method for a synchronization-desynchronization switch, called the pull-push control method, is suggested. Namely, synchronization can be achieved when the original systems are desynchronous by pulling (or protecting) one node or a certain subset of nodes, whereas desynchronization can be accomplished when the systems are already synchronous by pushing (or kicking) one node or a certain subset of nodes. With this method, the controlled nodes should be chosen by the generalized eigenvector centrality of the critical synchronization mode of the Laplacian matrix. Compared with existing control methods for synchronization, it displays high efficiency, flexibility, and precision as well.

Synchronization of interconnected networks: the role of connector nodes

In this Letter we identify the general rules that determine the synchronization properties of interconnected networks. We study analytically, numerically, and experimentally how the degree of the nodes through which two networks are connected influences the ability of the whole system to synchronize. We show that connecting the high-degree (low-degree) nodes of each network turns out to be the most (least) effective strategy to achieve synchronization. We find the functional relation between synchronizability and size for a given network of networks, and report the existence of the optimal connector link weights for the different interconnection strategies. Finally, we perform an electronic experiment with two coupled star networks and conclude that the analytical results are indeed valid in the presence of noise and parameter mismatches.

Emergence and scaling of synchronization in moving-agent networks with restrictive interactions

In fields such as robotics and sensor networks, synchronization among mobile and dynamic agents is a basic task. We articulate an effective strategy to achieve synchronization in dynamic networks of moving chaotic agents. Our counterintuitive idea is to restrict agents' ability to interact with each other, which can be implemented by designating a finite number of fixed zones in the space, in which agents are allowed to interact with each other but agents outside the zones are deprived of the ability of mutual interaction. Our setting is thus different from the one used in existing works on synchronization of mobile agents where each agent is associated with an interacting zone that moves with the agent. We find, through a mathematical analysis, that an optimal interval exists in the interaction probability, where stable synchronization emerges. An inverse square-root scaling law is uncovered which relates the interval with the system size, i.e., the total number of moving agents. Extensive numerical support for physical spaces of one, two, and three dimensions is provided.

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.

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-Rényi.

Pulsed chaos synchronization in networks with adaptive couplings

Networks of chaotic units with static couplings can synchronize to a common chaotic trajectory. The effect of dynamic adaptive couplings on the cooperative behavior of chaotic networks is investigated. The couplings adjust to the activities of its two units by two competing mechanisms: An exponential decrease of the coupling strength is compensated for by an increase due to desynchronized activity. This mechanism prevents the network from reaching a steady state. Numerical simulations of a coupled map lattice show chaotic trajectories of desynchronized units interrupted by pulses of mutually synchronized clusters. These pulses occur on all scales, sometimes extending to the entire network. Clusters of synchronized units can be triggered by a small group of synchronized units.

Synchronization in interdependent networks

We explore the synchronization behavior in interdependent systems, where the one-dimensional (1D) network (the intranetwork coupling strength J(I)) is ferromagnetically intercoupled (the strength J) to the Watts-Strogatz (WS) small-world network (the intranetwork coupling strength J(II)). In the absence of the internetwork coupling (J=0), the former network is well known not to exhibit the synchronized phase at any finite coupling strength, whereas the latter displays the mean-field transition. Through an analytic approach based on the mean-field approximation, it is found that for the weakly coupled 1D network (J(I)≪1) the increase of J suppresses synchrony, because the nonsynchronized 1D network becomes a heavier burden for the synchronization process of the WS network. As the coupling in the 1D network becomes stronger, it is revealed by the renormalization group (RG) argument that the synchronization is enhanced as J(I) is increased, implying that the more enhanced partial synchronization in the 1D network makes the burden lighter. Extensive numerical simulations confirm these expected behaviors, while exhibiting a reentrant behavior in the intermediate range of J(I). The nonmonotonic change of the critical value of J(II) is also compared with the result from the numerical RG calculation.

Coarse graining for synchronization in directed networks

Coarse-graining model is a promising way to analyze and visualize large-scale networks. The coarse-grained networks are required to preserve statistical properties as well as the dynamic behaviors of the initial networks. Some methods have been proposed and found effective in undirected networks, while the study on coarse-graining directed networks lacks of consideration. In this paper we proposed a path-based coarse-graining (PCG) method to coarse grain the directed networks. Performing the linear stability analysis of synchronization and numerical simulation of the Kuramoto model on four kinds of directed networks, including tree networks and variants of Barabási-Albert networks, Watts-Strogatz networks, and Erdös-Rényi networks, we find our method can effectively preserve the network synchronizability.

Enhancing synchronization by directionality in complex networks

We propose a method called the residual edge-betweenness gradient (REBG) to enhance the synchronizability of networks by assigning the link direction while keeping the topology and link weights unchanged. Direction assignment has been shown to improve the synchronizability of undirected networks in general, but we find that in some cases incommunicable components emerge and networks fail to synchronize. We show that the REBG method improves the residual degree gradient (RDG) method by effectively avoiding the synchronization failure. Further experiments show that the REBG method enhances the synchronizability in networks with a community structure compared with the RDG method.

Rewiring dynamical networks with prescribed degree distribution for enhancing synchronizability

In this paper, we present an algorithm for enhancing synchronizability of dynamical networks with prescribed degree distribution. The algorithm takes an unweighted and undirected network as input and outputs a network with the same node-degree distribution and enhanced synchronization properties. The rewirings are based on the properties of the Laplacian of the connection graph, i.e., the eigenvectors corresponding to the second smallest and the largest eigenvalues of the Laplacian. A term proportional to the eigenvectors is adopted to choose potential edges for rewiring, provided that the node-degree distribution is preserved. The algorithm can be implemented on networks of any sizes as long as their eigenvalues and eigenvectors can be calculated with standard algorithms. The effectiveness of the proposed algorithm in enhancing the network synchronizability is revealed by numerical simulation on a number of sample networks including scale-free, Watts-Strogatz, and Erdős-Rényi graphs. Furthermore, a number of network's structural parameters such as node betweenness centrality, edge betweenness centrality, average path length, clustering coefficient, and degree assortativity are tracked as a function of optimization steps.

Induced synchronization of a mobile agent network by phase locking

We investigate synchronization issues of a set of mobile agents in plane, where each agent carries an identical chaotic oscillator and interacts with its neighbors through a blinking coupling mechanism. We discuss the effect of blinking pattern on synchronization of the related network. In particular, we show that phase locking of the blinking behavior can dramatically enhance synchronizability of the mobile agent network, while the random blinking pattern works little. Also, we show that by assessing the convergence time, the mobile agent networks with different blinking periods and duty ratios share a common idle duration which is independent of both the blinking period and the corresponding duty ratio.

Dynamics and directionality in complex networks

We investigate how we can improve the synchronizability of complex networks simply by changing the link direction while conserving the local link weights and topology. Performing the linear stability analysis of synchronization and numerical simulation of the Kuramoto model in the directed networks, we find that while a random assignment of link directions generally weakens the degree of synchronization, a properly organized directionality can systematically enhance the network synchronization. In this respect, we suggest a simple method of changing the link direction according to the larger residual degree starting from small residual degree nodes. This result provides plausible applications to control the synchronizability of systems in various fields.

Stripe domain coarsening in geographical small-world networks on a Euclidean lattice

We study phase ordering dynamics of spatially periodic striped patterns on the small-world network that is derived from a two-dimensional regular lattice with distance-dependent random connections. It is demonstrated numerically that addition of spatial disorder in the form of shortcuts makes the growth of domains much slower or even frozen at late times.

Three structural properties reflecting the synchronizability of complex networks

During the process of adding links, we find that the synchronizability of the classical Barabási-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.

Experimental approach to the study of complex network synchronization using a single oscillator

We propose an experimental setup based on a single oscillator for studying large networks formed by identical unidirectionally coupled systems. A chaotic wave form generated by the oscillator is stored in a computer to adjust the signal according to the desired network configuration to feed it again into the same oscillator. No previous theoretical knowledge about the oscillator dynamics is needed. To visualize network synchronization we introduce a network synchronization bifurcation diagram that should prove to be an effective tool for analysis, design, and optimization of complex networks.

Efficient rewirings for enhancing synchronizability of dynamical networks

In this paper, we present an algorithm for optimizing synchronizability of complex dynamical networks. Starting with an undirected and unweighted network, we end up with an undirected and unweighted network with the same number of nodes and edges having enhanced synchronizability. To this end, based on some network properties, rewirings, i.e., eliminating an edge and creating a new edge elsewhere, are performed iteratively avoiding always self-loops and multiple edges between the same nodes. We show that the method is able to enhance the synchronizability of networks of any size and topological properties in a small number of steps that scales with the network size. For numerical simulations, an optimization algorithm based on simulated annealing is used. Also, the evolution of different topological properties of the network such as distribution of node degree, node and edge betweenness centrality is tracked with the iteration steps. We use networks such as scale-free, Strogatz-Watts and random to start with and we show that regardless of the initial network, the final optimized network becomes homogeneous. In other words, in the network with high synchronizability, parameters, such as, degree, shortest distance, node, and edge betweenness centralities are almost homogeneously distributed. Also, parameters, such as, maximum node and edge betweenness centralities are small for the rewired network. Although we take the eigenratio of the Laplacian as the target function for optimization, we show that it is also possible to choose other appropriate target functions exhibiting almost the same performance. Furthermore, we show that even if the network is optimized taking into account another interpretation of synchronizability, i.e., synchronization cost, the optimal network has the same synchronization properties. Indeed, in networks with optimized synchronizability, different interpretations of synchronizability coincide. The optimized networks are Ramanujan graphs, and thus, this rewiring algorithm could be used to produce Ramanujan graphs of any size and average degree.

Evolving functional network properties and synchronizability during human epileptic seizures

We assess electrical brain dynamics before, during, and after 100 human epileptic seizures with different anatomical onset locations by statistical and spectral properties of functionally defined networks. We observe a concave-like temporal evolution of characteristic path length and cluster coefficient indicative of a movement from a more random toward a more regular and then back toward a more random functional topology. Surprisingly, synchronizability was significantly decreased during the seizure state but increased already prior to seizure end. Our findings underline the high relevance of studying complex systems from the viewpoint of complex networks, which may help to gain deeper insights into the complicated dynamics underlying epileptic seizures.

Network synchronizability analysis: a graph-theoretic approach

This paper addresses the fundamental problem of complex network synchronizability from a graph-theoretic approach. First, the existing results are briefly reviewed. Then, the relationships between the network synchronizability and network structural parameters (e.g., average distance, degree distribution, and node betweenness centrality) are discussed. The effects of the complementary graph of a given network and some graph operations on the network synchronizability are discussed. A basic theory based on subgraphs and complementary graphs for estimating the network synchronizability is established. Several examples are given to show that adding new edges to a network can either increase or decrease the network synchronizability. To that end, some new results on the estimations of the synchronizability of coalescences are reported. Moreover, a necessary and sufficient condition for a network and its complementary network to have the same synchronizability is derived. Finally, some examples on Chua circuit networks are presented for illustration.

Sequence nets

We study a class of networks generated by sequences of letters taken from a finite alphabet consisting of m letters (corresponding to m types of nodes) and a fixed set of connectivity rules. Recently, it was shown how a binary alphabet might generate threshold nets in a similar fashion [A. Hagberg, Phys. Rev. E 74, 056116 (2006)]. Just like threshold nets, sequence nets in general possess a modular structure reminiscent of everyday-life nets and are easy to handle analytically (i.e., calculate degree distribution, shortest paths, betweenness centrality, etc.). Exploiting symmetry, we make a full classification of two- and three-letter sequence nets, discovering two classes of two-letter sequence nets. These sequence nets retain many of the desirable analytical properties of threshold nets while yielding richer possibilities for the modeling of everyday-life complex networks more faithfully.

Relaxation of synchronization on complex networks

We study collective synchronization in a large number of coupled oscillators on various complex networks. In particular, we focus on the relaxation dynamics of the synchronization, which is important from the viewpoint of information transfer or the dynamics of system recovery from a perturbation. We measure the relaxation time tau 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 (K>Kc) logarithmically increases with network size N , which is attributed to the initial random phase fluctuation given by O(N-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.

Synchronizability of network ensembles with prescribed statistical properties

It has been shown that synchronizability of a network is determined by the local structure rather than the global properties. With the same global properties, networks may have very different synchronizability. In this paper, we numerically studied, through the spectral properties, the synchronizability of ensembles of networks with prescribed statistical properties. Given a degree sequence, it is found that the eigenvalues and eigenratios characterizing network synchronizability have well-defined distributions, and statistically, the networks with extremely poor synchronizability are rare. Moreover, we compared the synchronizability of three network ensembles that have the same nodes and average degree. Our work reveals that the synchronizability of a network can be significantly affected by the local pattern of connections, and the homogeneity of degree can greatly enhance network synchronizability for networks of a random nature.

Geographical effect on small-world network synchronization

We investigate the geographical effect on the synchronization of small-world oscillator networks. We construct small-world geographical networks by randomly adding links to one- and two-dimensional regular lattices, and we find that the synchronizability is a nonmonotonic function of both the coupling strength and the geographical distance of randomly added shortcuts. Our findings demonstrate that the geographical effect plays an important role in network synchronization, which may shed some light on the study of collective dynamics of complex networks.

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.

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.

Entrainment transition in populations of random frequency oscillators

The entrainment transition of coupled random frequency oscillators is revisited. The Kuramoto model (global coupling) is shown to exhibit unusual sample-dependent finite-size effects leading to a correlation size exponent nu=5/2. Simulations of locally coupled oscillators in d dimensions reveal two types of frequency entrainment: mean-field behavior at d>4 and aggregation of compact synchronized domains in three and four dimensions. In the latter case, scaling arguments yield a correlation length exponent nu=2/(d-2), in good agreement with numerical results.

Complex network synchronizability: analysis and control

In this paper, the investigation is first motivated by showing two examples of simple regular symmetrical graphs, which have the same structural parameters, such as average distance, degree distribution, and node betweenness centrality, but have very different synchronizabilities. For a given network with identical node dynamics, it is further shown that two key factors influencing the network synchronizability are the network inner linking matrix and the eigenvalues of the network topological matrix. Several examples are then provided to show that adding new edges to a network can either increase or decrease the network synchronizability. In searching for conditions under which the network synchronizability may be increased by adding edges, it is found that for networks with disconnected complementary graphs, adding edges never decreases their synchronizability. Moreover, it is found that an unbounded synchronized region is always easier to analyze than a bounded synchronized region. Therefore to effectively enhance the network synchronizability, a design method is finally presented for the inner linking matrix of rank 1 such that the resultant network has an unbounded synchronized region, for the case where the synchronous state is an equilibrium point of the network.

Enhanced synchronizability via age-based coupling

In this Brief Report, we study the synchronization of growing scale-free networks. An asymmetrical age-based coupling method is proposed with only one free parameter alpha . Although the coupling matrix is asymmetric, our coupling method could guarantee that all the eigenvalues are non-negative reals. The eigenratio R will approach 1 in the large limit of alpha .

Nature-inspired interconnects for self-assembled large-scale network-on-chip designs

Future nanoscale electronics built up from an Avogadro number of components need efficient, highly scalable, and robust means of communication in order to be competitive with traditional silicon approaches. In recent years, the networks-on-chip (NoC) paradigm emerged as a promising solution to interconnect challenges in silicon-based electronics. Current NoC architectures are either highly regular or fully customized, both of which represent implausible assumptions for emerging bottom-up self-assembled molecular electronics that are generally assumed to have a high degree of irregularity and imperfection. Here, we pragmatically and experimentally investigate important design tradeoffs and properties of an irregular, abstract, yet physically plausible three-dimensional (3D) small-world interconnect fabric that is inspired by modern network-on-chip paradigms. We vary the framework's key parameters, such as the connectivity, number of switch nodes, and distribution of long- versus short-range connections, and measure the network's relevant communication characteristics. We further explore the robustness against link failures and the ability and efficiency to solve a simple toy problem, the synchronization task. The results confirm that (1) computation in irregular assemblies is a promising and disruptive computing paradigm for self-assembled nanoscale electronics and (2) that 3D small-world interconnect fabrics with a power-law decaying distribution of shortcut lengths are physically plausible and have major advantages over local two-dimensional and 3D regular topologies.

Synchronizability determined by coupling strengths and topology on complex networks

We investigate in depth the synchronization of coupled oscillators on top of complex networks with different degrees of heterogeneity within the context of the Kuramoto model. In a previous paper [Phys. Rev. Lett. 98, 034101 (2007)], we unveiled how for fixed coupling strengths local patterns of synchronization emerge differently in homogeneous and heterogeneous complex networks. Here, we provide more evidence on this phenomenon, extending the previous work to networks that interpolate between homogeneous and heterogeneous topologies. We also introduce details of the path towards synchronization for the evolution of clustering in the synchronized patterns. Finally, we investigate the synchronization of networks with modular structure and conclude that, in these cases, local synchronization is first attained at the most internal level of organization of modules, progressively evolving to the outer levels as the coupling constant is increased. The present work introduces parameters that are proved to be useful for the characterization of synchronization phenomena in complex networks.

Paths to synchronization on complex networks

The understanding of emergent collective phenomena in natural and social systems has driven the interest of scientists from different disciplines during decades. Among these phenomena, the synchronization of a set of interacting individuals or units has been intensively studied because of its ubiquity in the natural world. In this Letter, we show how for fixed coupling strengths local patterns of synchronization emerge differently in homogeneous and heterogeneous complex networks, driving the process towards a certain global synchronization degree following different paths. The dependence of the dynamics on the coupling strength and on the topology is unveiled. This study provides a new perspective and tools to understand this emerging phenomena.