Synchronization on small-world networks

Bifurcation behaviors of synchronized regions in logistic map networks with coupling delay

Network synchronized regions play an extremely important role in network synchronization according to the master stability function framework. This paper focuses on network synchronous state stability via studying the effects of nodal dynamics, coupling delay, and coupling way on synchronized regions in Logistic map networks. Theoretical and numerical investigations show that (1) network synchronization is closely associated with its nodal dynamics. Particularly, the synchronized region bifurcation points through which the synchronized region switches from one type to another are in good agreement with those of the uncoupled node system, and chaotic nodal dynamics can greatly impede network synchronization. (2) The coupling delay generally impairs the synchronizability of Logistic map networks, which is also dominated by the parity of delay for some nodal parameters. (3) A simple nonlinear coupling facilitates network synchronization more than the linear one does. The results found in this paper will help to intensify our understanding for the synchronous state stability in discrete-time networks with coupling delay.

Thermodynamic characterization of synchronization-optimized oscillator networks

We consider a canonical ensemble of synchronization-optimized networks of identical oscillators under external noise. By performing a Markov chain Monte Carlo simulation using the Kirchhoff index, i.e., the sum of the inverse eigenvalues of the Laplacian matrix (as a graph Hamiltonian of the network), we construct more than 1,000 different synchronization-optimized networks. We then show that the transition from star to core-periphery structure depends on the connectivity of the network, and is characterized by the node degree variance of the synchronization-optimized ensemble. We find that thermodynamic properties such as heat capacity show anomalies for sparse networks.

Analysis of epileptic seizures with complex network

Epilepsy is a disease of abnormal neural activities involving large area of brain networks. Until now the nature of functional brain network associated with epilepsy is still unclear. Recent researches indicate that the small world or scale-free attributes and the occurrence of highly clustered connection patterns could represent a general organizational principle in the human brain functional network. In this paper, we seek to find whether the small world or scale-free property of brain network is correlated with epilepsy seizure formation. A mass neural model was adopted to generate multiple channel EEG recordings based on regular, small world, random, and scale-free network models. Whether the connection patterns of cortical networks are directly associated with the epileptic seizures was investigated. The results showed that small world and scale-free cortical networks are highly correlated with the occurrence of epileptic seizures. In particular, the property of small world network is more significant during the epileptic seizures.

Robustness of cluster synchronous patterns in small-world networks with inter-cluster co-competition balance

All edges in the classical Watts and Strogatz's small-world network model are unweighted and cooperative (positive). By introducing competitive (negative) inter-cluster edges and assigning edge weights to mimic more realistic networks, this paper develops a modified model which possesses co-competitive weighted couplings and cluster structures while maintaining the common small-world network properties of small average shortest path lengths and large clustering coefficients. Based on theoretical analysis, it is proved that the new model with inter-cluster co-competition balance has an important dynamical property of robust cluster synchronous pattern formation. More precisely, clusters will neither merge nor split regardless of adding or deleting nodes and edges, under the condition of inter-cluster co-competition balance. Numerical simulations demonstrate the robustness of the model against the increase of the coupling strength and several topological variations.

Synchronization and stochastic resonance of the small-world neural network based on the CPG

According to biological knowledge, the central nervous system controls the central pattern generator (CPG) to drive the locomotion. The brain is a complex system consisting of different functions and different interconnections. The topological properties of the brain display features of small-world network. The synchronization and stochastic resonance have important roles in neural information transmission and processing. In order to study the synchronization and stochastic resonance of the brain based on the CPG, we establish the model which shows the relationship between the small-world neural network (SWNN) and the CPG. We analyze the synchronization of the SWNN when the amplitude and frequency of the CPG are changed and the effects on the CPG when the SWNN's parameters are changed. And we also study the stochastic resonance on the SWNN. The main findings include: (1) When the CPG is added into the SWNN, there exists parameters space of the CPG and the SWNN, which can make the synchronization of the SWNN optimum. (2) There exists an optimal noise level at which the resonance factor Q gets its peak value. And the correlation between the pacemaker frequency and the dynamical response of the network is resonantly dependent on the noise intensity. The results could have important implications for biological processes which are about interaction between the neural network and the CPG.

Taming explosive growth through dynamic random links

We study the dynamics of a collection of nonlinearly coupled limit cycle oscillators relevant to a wide class of systems, ranging from neuronal populations to electrical circuits, over network topologies varying from a regular ring to a random network. We find that for sufficiently strong coupling strengths the trajectories of the system escape to infinity in the regular ring network. However when a fraction of the regular connections are dynamically randomized, the unbounded growth is suppressed and the system remains bounded. Further, we find a scaling relation between the critical fraction of random links necessary for successful prevention of explosive behavior and the network rewiring time-scale. These results suggest a mechanism by which blow-ups may be controlled in extended oscillator systems.

Finite-time and finite-size scaling of the Kuramoto oscillators

Phase transition in its strict sense can only be observed in an infinite system, for which equilibration takes an infinitely long time at criticality. In numerical simulations, we are often limited both by the finiteness of the system size and by the finiteness of the observation time scale. We propose that one can overcome this barrier by measuring the nonequilibrium temporal relaxation for finite systems and by applying the finite-time-finite-size scaling (FTFSS) which systematically uses two scaling variables, one temporal and the other spatial. The FTFSS method yields a smooth scaling surface, and the conventional finite-size scaling curves can be viewed as proper cross sections of the surface. The validity of our FTFSS method is tested for the synchronization transition of Kuramoto models in the globally coupled structure and in the small-world network structure. Our FTFSS method is also applied to the Monte Carlo dynamics of the globally coupled q-state clock model.

Nature of synchronization transitions in random networks of coupled oscillators

We consider a system of phase oscillators with random intrinsic frequencies coupled through sparse random networks and investigate how the connectivity disorder affects the nature of collective synchronization transitions. Various distribution types of intrinsic frequencies are considered: uniform, unimodal, and bimodal distribution. We employ a heterogeneous mean-field approximation based on the annealed networks and also perform numerical simulations on the quenched Erdös-Rényi networks. We find that the connectivity disorder drastically changes the nature of the synchronization transitions. In particular, the quenched randomness completely wipes away the diversity of the transition nature, and only a continuous transition appears with the same mean-field exponent for all types of frequency distributions. The physical origin of this unexpected result is discussed.

Impact of delays on the synchronization transitions of modular neuronal networks with hybrid synapses

The combined effects of the information transmission delay and the ratio of the electrical and chemical synapses on the synchronization transitions in the hybrid modular neuronal network are investigated in this paper. Numerical results show that the synchronization of neuron activities can be either promoted or destroyed as the information transmission delay increases, irrespective of the probability of electrical synapses in the hybrid-synaptic network. Interestingly, when the number of the electrical synapses exceeds a certain level, further increasing its proportion can obviously enhance the spatiotemporal synchronization transitions. Moreover, the coupling strength has a significant effect on the synchronization transition. The dominated type of the synapse always has a more profound effect on the emergency of the synchronous behaviors. Furthermore, the results of the modular neuronal network structures demonstrate that excessive partitioning of the modular network may result in the dramatic detriment of neuronal synchronization. Considering that information transmission delays are inevitable in intra- and inter-neuronal networks communication, the obtained results may have important implications for the exploration of the synchronization mechanism underlying several neural system diseases such as Parkinson's Disease.

Explosive synchronization transitions in complex neural networks

It has been recently reported that explosive synchronization transitions can take place in networks of phase oscillators [Gómez-Gardeñes et al. Phys. Rev. Lett. 106, 128701 (2011)] and chaotic oscillators [Leyva et al. Phys. Rev. Lett. 108, 168702 (2012)]. Here, we investigate the effect of a microscopic correlation between the dynamics and the interacting topology of coupled FitzHugh-Nagumo oscillators on phase synchronization transition in Barabási-Albert (BA) scale-free networks and Erdös-Rényi (ER) random networks. We show that, if natural frequencies of the oscillations are positively correlated with node degrees and the width of the frequency distribution is larger than a threshold value, a strong hysteresis loop arises in the synchronization diagram of BA networks, indicating the evidence of an explosive transition towards synchronization of relaxation oscillators system. In contrast to the results in BA networks, in more homogeneous ER networks, the synchronization transition is always of continuous type regardless of the width of the frequency distribution. Moreover, we consider the effect of degree-mixing patterns on the nature of the synchronization transition, and find that the degree assortativity is unfavorable for the occurrence of such an explosive transition.

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.

Link-disorder fluctuation effects on synchronization in random networks

We consider one typical system of oscillators coupled through disordered link configurations in networks, i.e., a finite population of coupled phase oscillators with distributed intrinsic frequencies on a random network. We investigate the collective synchronization behavior, paying particular attention to link-disorder fluctuation effects on the synchronization transition and its finite-size scaling (FSS). Extensive numerical simulations as well as the mean-field analysis have been performed. We find that link-disorder fluctuations effectively induce uncorrelated random fluctuations in frequency, resulting in the FSS exponent ν[over ¯]=5/2, which is identical to that in the globally coupled case (no link disorder) with frequency-disorder fluctuations.

Interplay between collective behavior and spreading dynamics on complex networks

There are certain correlations between collective behavior and spreading dynamics on some real complex networks. Based on the dynamical characteristics and traditional physical models, we construct several new bidirectional network models of spreading phenomena. By theoretical and numerical analysis of these models, we find that the collective behavior can inhibit spreading behavior, but, conversely, this spreading behavior can accelerate collective behavior. The spread threshold of spreading network is obtained by using the Lyapunov function method. The results show that an effective spreading control method is to enhance the individual awareness to collective behavior. Many real-world complex networks can be thought of in terms of both collective behavior and spreading dynamics and therefore to better understand and control such complex networks systems, our work may provide a basic framework.

Network coordination and synchronization in a noisy environment with time delays

We study the effects of nonzero time delays in stochastic synchronization problems with linear couplings in complex networks. We consider two types of time delays: transmission delays between interacting nodes and local delays at each node (due to processing, cognitive, or execution delays). By investigating the underlying fluctuations for several delay schemes, we obtain the synchronizability threshold (phase boundary) and the scaling behavior of the width of the synchronization landscape, in some cases for arbitrary networks and in others for specific weighted networks. Numerical computations allow the behavior of these networks to be explored when direct analytical results are not available. We comment on the implications of these findings for simple locally or globally weighted network couplings and possible trade-offs present in such systems.

Determination of the critical coupling of explosive synchronization transitions in scale-free networks by mean-field approximations

An explosive synchronization can be observed in scale-free networks when Kuramoto oscillators have natural frequencies equal to their number of connections. The present paper reports on mean-field approximations to determine the critical coupling of such explosive synchronization. It has been verified that the equation obtained for the critical coupling has an inverse dependence on the network average degree. This expression differs from those whose frequency distributions are unimodal and even. In this case, the critical coupling depends on the ratio between the first and second statistical moments of the degree distribution. Numerical simulations were also conducted to verify our analytical results.

Noise-induced synchronization in small world networks of phase oscillators

A small-world (SW) network of similar phase oscillators, interacting according to the Kuramoto model, is studied numerically. It is shown that deterministic Kuramoto dynamics on SW networks has various stable stationary states. This can be attributed to the so-called defect patterns in an SW network, which it inherits from deformation of helical patterns in its regular parent. Turning on an uncorrelated random force causes vanishing of the defect patterns, hence increasing the synchronization among oscillators for moderate noise intensities. This phenomenon, called stochastic synchronization, is generally observed in some natural networks such as the brain neural network.

Effects of hybrid synapses on the vibrational resonance in small-world neuronal networks

We investigate the effect of vibrational resonance in small-world neuronal networks with hybrid chemical and electrical synapses. It is shown that, irrespective of the probability of chemical synapses, an optimal amplitude of high-frequency component of the signal can optimize the dynamical response of neuron populations to the low-frequency component, which encodes the information. This effect of vibrational resonance of neuronal systems depends extensively on the network structure and parameters, which determine the ability of neuronal networks to enhance the outreach of localized subthreshold low-frequency signal. In particular, chemical synaptic coupling is more efficient than the electrical coupling for the transmission of local input signal due to its selective coupling. Moreover, there exists an optimal small-world topology characterized by an optimal value of rewiring probability, warranting the largest peak value of the system response. Considering that two-frequency signals are ubiquity in brain dynamics, we expect the presented results could have important implications for signal processing in neuronal systems.

Optimal entrainment of neural oscillator ensembles

In this paper, we derive the minimum-energy periodic control that entrains an ensemble of structurally similar neural oscillators to a desired frequency. The state-space representation of a nominal oscillator is reduced to a phase model by computing its limit cycle and phase response curve, from which the optimal control is derived by using formal averaging and the calculus of variations. We focus on the case of a 1:1 entrainment ratio and suggest a simple numerical method for approximating the optimal controls. The method is applied to asymptotically control the spiking frequency of neural oscillators modeled using the Hodgkin-Huxley equations. Simulations are used to illustrate the optimality of entrainment controls derived using phase models when applied to the original state-space system, which is crucial for using phase models in control synthesis for practical applications. This work addresses a fundamental problem in the field of neural dynamics and provides a theoretical contribution to the optimal frequency control of uncertain oscillating systems.

Synchronizability of small-world networks generated from ring networks with equal-distance edge additions

This paper investigates the impact of edge-adding number m and edge-adding distance d on both synchronizability and average path length of NW small-world networks generated from ring networks via random edge-adding. It is found that the synchronizability of the network as a function of the distance d is fluctuant and there exist some d that have almost no impact on the synchronizability and may only scarcely shorten the average path length of the network. Numerical simulations on a network of Lorenz oscillators confirm the above results. This phenomenon shows that the contributions of randomly added edges to both the synchronizability and the average path length are not uniform nor monotone in building an NW small-world network with equal-distance edge additions, implying that only if appropriately adding edges when building up the NW small-word network can help enhance the synchronizability and/or reduce the average path length of the resultant network. Finally, it is shown that this NW small-world network has worse synchronizability and longer average path length, when compared with the conventional NW small-world network, with random-distance edge additions. This may be due to the fact that with equal-distance edge additions, there is only one shortcut distance for better information exchange among nodes and for shortening the average path length, while with random-distance edge additions, there exist many different distances for doing so.

A general fractional-order dynamical network: synchronization behavior and state tuning

A general fractional-order dynamical network model for synchronization behavior is proposed. Different from previous integer-order dynamical networks, the model is made up of coupled units described by fractional differential equations, where the connections between individual units are nondiffusive and nonlinear. We show that the synchronous behavior of such a network cannot only occur, but also be dramatically different from the behavior of its constituent units. In particular, we find that simple behavior can emerge as synchronized dynamics although the isolated units evolve chaotically. Conversely, individually simple units can display chaotic attractors when the network synchronizes. We also present an easily checked criterion for synchronization depending only on the eigenvalues distribution of a decomposition matrix and the fractional orders. The analytic results are complemented with numerical simulations for two networks whose nodes are governed by fractional-order Lorenz dynamics and fractional-order Rössler dynamics, respectively.

Design of oscillator networks with enhanced synchronization tolerance against noise

Can synchronization properties of a network of identical oscillators in the presence of noise be improved through appropriate rewiring of its connections? What are the optimal network architectures for a given total number of connections? We address these questions by running the optimization process, using the stochastic Markov Chain Monte Carlo method with replica exchange, to design networks of phase oscillators with increased tolerance against noise. As we find, the synchronization of a network, characterized by the Kuramoto order parameter, can be increased up to 40%, as compared to that of the randomly generated networks, when the optimization is applied. Large ensembles of optimized networks are obtained, and their statistical properties are investigated.

Onset of synchronization in complex networks of noisy oscillators

We study networks of noisy phase oscillators whose nodes are characterized by random degrees counting the number of their connections. Both these degrees and the natural frequencies of the oscillators are distributed according to a given probability density. Replacing the randomly connected network by an all-to-all coupled network with weighted edges allows us to formulate the dynamics of a single oscillator coupled to the mean field and to derive the corresponding Fokker-Planck equation. From the latter we calculate the critical coupling strength for the onset of synchronization as a function of the noise intensity, the frequency distribution, and the first two moments of the degree distribution. Our approach is applied to a dense small-world network model, for which we calculate the degree distribution. Numerical simulations prove the validity of the replacement. We also test the applicability to more sparsely connected networks and formulate homogeneity and absence of correlations in the degree distribution as limiting factors of our approach.

Effects of long-range links on metastable states in a dynamic interaction network

We introduce a model for random-walking agents on a two-dimensional periodic lattice, where the dynamic interaction network is defined using local short-range interactions and E randomly added long-range interactions. With periodic states for agents and an interaction rule of repeated averaging, we numerically find two types of metastable states at low- and high-E limits, respectively, along with consensus states. If we apply this model to opinion dynamics, metastable states can be interpreted as sustainable diversities in our societies, and our results then imply that, while diversities decrease and eventually disappear with more long-range interactions, other types of diverse states can appear under this rule when networks are almost fully connected.

Accuracy of mean-field theory for dynamics on real-world networks

Mean-field analysis is an important tool for understanding dynamics on complex networks. However, surprisingly little attention has been paid to the question of whether mean-field predictions are accurate, and this is particularly true for real-world networks with clustering and modular structure. In this paper, we compare mean-field predictions to numerical simulation results for dynamical processes running on 21 real-world networks and demonstrate that the accuracy of such theory depends not only on the mean degree of the networks but also on the mean first-neighbor degree. We show that mean-field theory can give (unexpectedly) accurate results for certain dynamics on disassortative real-world networks even when the mean degree is as low as 4.

Rewiring-induced chaos in pulse-coupled neural networks

The dependence of the dynamics of pulse-coupled neural networks on random rewiring of excitatory and inhibitory connections is examined. When both excitatory and inhibitory connections are rewired, periodic synchronization emerges with a Hopf-like bifurcation and a subsequent period-doubling bifurcation; chaotic synchronization is also observed. When only excitatory connections are rewired, periodic synchronization emerges with a saddle node-like bifurcation, and chaotic synchronization is also observed. This result suggests that randomness in the system does not necessarily contaminate the system, and sometimes it even introduces rich dynamics to the system such as chaos.

Chaotic phase synchronization in a modular neuronal network of small-world subnetworks

We investigate the onset of chaotic phase synchronization of bursting oscillators in a modular neuronal network of small-world subnetworks. A transition to mutual phase synchronization takes place on the bursting time scale of coupled oscillators, while on the spiking time scale, they behave asynchronously. It is shown that this bursting synchronization transition can be induced not only by the variations of inter- and intra-coupling strengths but also by changing the probability of random links between different subnetworks. We also analyze the effect of external chaotic phase synchronization of bursting behavior in this clustered network by an external time-periodic signal applied to a single neuron. Simulation results demonstrate a frequency locking tongue in the driving parameter plane, where bursting synchronization is maintained, even with the external driving. The width of this synchronization region increases with the signal amplitude and the number of driven neurons but decreases rapidly with the network size. Considering that the synchronization of bursting neurons is thought to play a key role in some pathological conditions, the presented results could have important implications for the role of externally applied driving signal in controlling bursting activity in neuronal ensembles.

Relaying phase synchrony in chaotic oscillator chains

We study the manner in which the effect of an external drive is transmitted through mutually coupled response systems by examining the phase synchrony between the drive and the response. Two different coupling schemes are used. Homogeneous couplings are via the same variables while heterogeneous couplings are through different variables. With the latter scenario, synchronization regimes are truncated with an increasing number of mutually coupled oscillators in contrast to homogeneous coupling schemes. Our results are illustrated for systems of coupled chaotic Rössler oscillators.

Dynamics and pattern formation in large systems of spatially-coupled oscillators with finite response times

We consider systems of many spatially distributed phase oscillators that interact with their neighbors. Each oscillator is allowed to have a different natural frequency, as well as a different response time to the signals it receives from other oscillators in its neighborhood. Using the ansatz of Ott and Antonsen [Chaos 18, 037113 (2008)] and adopting a strategy similar to that employed in the recent work of Laing [Physica D 238, 1569 (2009)], we reduce the microscopic dynamics of these systems to a macroscopic partial-differential-equation description. Using this macroscopic formulation, we numerically find that finite oscillator response time leads to interesting spatiotemporal dynamical behaviors including propagating fronts, spots, target patterns, chimerae, spiral waves, etc., and we study interactions and evolutionary behaviors of these spatiotemporal patterns.

Fast and robust image segmentation by small-world neural oscillator networks

Inspired by the temporal correlation theory of brain functions, researchers have presented a number of neural oscillator networks to implement visual scene segmentation problems. Recently, it is shown that many biological neural networks are typical small-world networks. In this paper, we propose and investigate two small-world models derived from the well-known LEGION (locally excitatory and globally inhibitory oscillator network) model. To form a small-world network, we add a proper proportion of unidirectional shortcuts (random long-range connections) to the original LEGION model. With local connections and shortcuts, the neural oscillators can not only communicate with neighbors but also exchange phase information with remote partners. Model 1 introduces excitatory shortcuts to enhance the synchronization within an oscillator group representing the same object. Model 2 goes further to replace the global inhibitor with a sparse set of inhibitory shortcuts. Simulation results indicate that the proposed small-world models could achieve synchronization faster than the original LEGION model and are more likely to bind disconnected image regions belonging together. In addition, we argue that these two models are more biologically plausible.

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.

Chaotic phase synchronization in small-world networks of bursting neurons

We investigate the chaotic phase synchronization in a system of coupled bursting neurons in small-world networks. A transition to mutual phase synchronization takes place on the bursting time scale of coupled oscillators, while on the spiking time scale, they behave asynchronously. It is shown that phase synchronization is largely facilitated by a large fraction of shortcuts, but saturates when it exceeds a critical value. We also study the external chaotic phase synchronization of bursting oscillators in the small-world network by a periodic driving signal applied to a single neuron. It is demonstrated that there exists an optimal small-world topology, resulting in the largest peak value of frequency locking interval in the parameter plane, where bursting synchronization is maintained, even with the external driving. The width of this interval increases with the driving amplitude, but decrease rapidly with the network size. We infer that the externally applied driving parameters outside the frequency locking region can effectively suppress pathologically synchronized rhythms of bursting neurons in the brain.

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.

Functional clustering in hippocampal cultures: relating network structure and dynamics

In this work we investigate the relationship between gross anatomic structural network properties, neuronal dynamics and the resultant functional structure in dissociated rat hippocampal cultures. Specifically, we studied cultures as they developed under two conditions: the first supporting glial cell growth (high glial group), and the second one inhibiting it (low glial group). We then compared structural network properties and the spatio-temporal activity patterns of the neurons. Differences in dynamics between the two groups could be linked to the impact of the glial network on the neuronal network as the cultures developed. We also implemented a recently developed algorithm called the functional clustering algorithm (FCA) to obtain the resulting functional network structure. We show that this new algorithm is useful for capturing changes in functional network structure as the networks evolve over time. The FCA detects changes in functional structure that are consistent with expected dynamical differences due to the impact of the glial network. Cultures in the high glial group show an increase in global synchronization as the cultures age, while those in the low glial group remain locally synchronized. We additionally use the FCA to quantify the amount of synchronization present in the cultures and show that the total level of synchronization in the high glial group is stronger than in the low glial group. These results indicate an interdependence between the glial and neuronal networks present in dissociated cultures.

Complexity versus modularity and heterogeneity in oscillatory networks: combining segregation and integration in neural systems

Normal functioning in many realistic complex dynamical systems, such as neural networks, requires coherence and synchronization for collective actions of network components. However, strong synchronization of the whole network is often related to pathological situations. A regime in between enabling both segregation in subsystems and integration as a whole is thus desirable. Here, we characterize this regime by complexity of synchronization patterns and study its relationship to heterogeneous and modular architecture in complex network of oscillators. We show that these networks possess a broad range of high complexity associated with the formation of dynamical clusters and the coordination between the clusters. In realistic networks of C. elegans and cat cortex, the complexity is reduced when the original network is rewired in various ways, reflecting that the neural systems are organized to provide a combination of segregation and integration with the coexistence of various complex network features, especially modularity and heterogeneity. Our work can stimulate further studies on structure-function relationships in neural systems through the inquiry of the specific functional roles of the intermediate dynamical regime.

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.

Inferring the physical connectivity of complex networks from their functional dynamics

Background

Biological networks, such as protein-protein interactions, metabolic, signalling, transcription-regulatory networks and neural synapses, are representations of large-scale dynamic systems. The relationship between the network structure and functions remains one of the central problems in current multidisciplinary research. Significant progress has been made toward understanding the implication of topological features for the network dynamics and functions, especially in biological networks. Given observations of a network system's behaviours or measurements of its functional dynamics, what can we conclude of the details of physical connectivity of the underlying structure?

Results

We modelled the network system by employing a scale-free network of coupled phase oscillators. Pairwise phase coherence (PPC) was calculated for all the pairs of oscillators to present functional dynamics induced by the system. At the regime of global incoherence, we observed a Significant pairwise synchronization only between two nodes that are physically connected. Right after the onset of global synchronization, disconnected nodes begin to oscillate in a correlated fashion and the PPC of two nodes, either connected or disconnected, depends on their degrees.

Based on the observation of PPCs, we built a weighted network of synchronization (WNS), an all-to-all functionally connected network where each link is weighted by the PPC of two oscillators at the ends of the link. In the regime of strong coupling, we observed a Significant similarity in the organization of WNSs induced by systems sharing the same substrate network but different configurations of initial phases and intrinsic frequencies of oscillators.

We reconstruct physical network from the WNS by choosing the links whose weights are higher than a given threshold. We observed an optimal reconstruction just before the onset of global synchronization.

Finally, we correlated the topology of the background network to the observed change of the functional activities in the system.

Conclusions

The results presented in this study indicate a strong relationship between the structure and dynamics of complex network systems. As coupling strength increases, synchronization emerges among hub nodes and recruits small-degree nodes. The results show that the onset of global synchronization in the system hinders the reconstruction of an underlying complex structure. Our analysis helps to clarify how the synchronization is achieved in systems of different network topologies.

Design of easily synchronizable oscillator networks using the Monte Carlo optimization method

Starting with an initial random network of oscillators with a heterogeneous frequency distribution, its autonomous synchronization ability can be largely improved by appropriately rewiring the links between the elements. Ensembles of synchronization-optimized networks with different connectivities are generated and their statistical properties are studied.

Necessary condition for frequency synchronization in network structures

We present the necessary condition for complete frequency synchronization of phase-coupled oscillators in network structures. The surface area of a set of sites is defined as the number of links between the sites within the set and those outside the set. The necessary condition is that the surface area of any set of cN (0 < c < 1) oscillators in the N-oscillator system must exceed square root of N in the limit N --> infinity. We also provide the necessary condition for macroscopic frequency synchronization. Thus, we identify networks in which one or both of the above mentioned types of synchronization do not occur.

Quasiperiodic forcing of coupled chaotic systems

We study the manner in which the effect of quasiperiodic modulation is transmitted in a coupled nonlinear dynamical system. A system of Rössler oscillators is considered, one of which is subject to driving, and the dynamics of other oscillators which are, in effect, indirectly forced is observed. Strange nonchaotic dynamics is known to arise only in quasiperiodically driven systems, and thus the transmitted effect is apparent when such motion is seen in subsystems that are not directly modulated. We also find instances of imperfect phase synchronization with forcing, where the system transits from one phase synchronized state to another, with arbitrary phase slips. The stability of phase synchrony for arbitrary initial conditions with identical forcing is observed as a general property of strange nonchaotic motion.

Roles of inhibitory neurons in rewiring-induced synchronization in pulse-coupled neural networks

The roles of inhibitory neurons in synchronous firing are examined in a network of excitatory and inhibitory neurons with Watts and Strogatz's rewiring. By examining the persistence of the synchronous firing that exists in the random network, it was found that there is a probability of rewiring at which a transition between the synchronous state and the asynchronous state takes place, and the dynamics of the inhibitory neurons play an important role in determining this probability.

Structural properties of the synchronized cluster on complex networks

We investigate how the largest synchronized connected component (LSCC) is formed and evolves to achieve a global synchronization on complex networks using Kuramoto model. In this study we use two different networks, Erdösi-Rényi network and Barabási-Albert network. From the finite-size scaling analysis, we find that the scaling exponents for the percolation order parameter and mean cluster size on both networks agree with the mean-field percolation theory, beta=gamma=1. We also find that the finite-size scaling exponent, nu, also agrees with the mean-field percolation result, nu=3. Moreover, we also show that the cluster size distributions are identical with the mean-field percolation distribution on both networks. Combining with the analysis for the merging clusters, we directly show that the LSCC on both networks evolves by merging clusters of various sizes.

Generalized synchronization of complex dynamical networks via impulsive control

This paper investigates the generalized synchronization (GS) of two typical complex dynamical networks, small-world networks and scale-free networks, in terms of impulsive control strategy. By applying the auxiliary-system approach to networks, we demonstrate theoretically that for any given coupling strength, GS can take place in complex dynamical networks consisting of nonidentical systems. Particularly, for Barabasi-Albert scale-free networks, we look into the relations between GS error and topological parameter m, which denotes the number of edges linking to a new node at each time step, and find out that GS speeds up with increasing m. And for Newman-Watts small-world networks, the time needed to achieve GS decreases as the probability of adding random edges increases. We further reveal how node dynamics affects GS speed on both small-world and scale-free networks. Finally, we analyze how the development of GS depends on impulsive control gains. Some abnormal but interesting phenomena regarding the GS process are also found in simulations.

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.

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.

The development of generalized synchronization on complex networks

In this paper, we numerically investigate the development of generalized synchronization (GS) on typical complex networks, such as scale-free networks, small-world networks, random networks, and modular networks. By adopting the auxiliary-system approach to networks, we observe that GS generally takes place in oscillator networks with both heterogeneous and homogeneous degree distributions, regardless of whether the coupled chaotic oscillators are identical or nonidentical. We show that several factors, such as the network topology, the local dynamics, and the specific coupling strategies, can affect the development of GS on complex networks.