Synchronization in complex networks with a modular structure

Partial synchronization and community switching in phase-oscillator networks and its analysis based on a bidirectional, weighted chain of three oscillators

Complex networks often possess communities defined based on network connectivity. When dynamics undergo in a network, one can also consider dynamical communities, i.e., a group of nodes displaying a similar dynamical process. We have investigated both analytically and numerically the development of a dynamical community structure, where the community is referred to as a group of nodes synchronized in frequency, in networks of phase oscillators. We first demonstrate that using a few example networks, the community structure changes when network connectivity or interaction strength is varied. In particular, we found that community switching, i.e., a portion of oscillators change the group to which they synchronize, occurs for a range of parameters. We then propose a three-oscillator model: a bidirectional, weighted chain of three Kuramoto phase oscillators, as a theoretical framework for understanding the community formation and its variation. Our analysis demonstrates that the model shows a variety of partially synchronized patterns: oscillators with similar natural frequencies tend to synchronize for weak coupling, while tightly connected oscillators tend to synchronize for strong coupling. We obtain approximate expressions for the critical coupling strengths by employing a perturbative approach in a weak coupling regime and a geometric approach in strong coupling regimes. Moreover, we elucidate the bifurcation types of transitions between different patterns. Our theory might be useful for understanding the development of partially synchronized patterns in a wider class of complex networks than community structured networks.

Maximizing synchronizability of networks with community structure based on node similarity

In reality, numerous networks have a community structure characterized by dense intra-community connections and sparse inter-community connections. In this article, strategies are proposed to enhance synchronizability of such networks by rewiring a certain number of inter-community links, where the research scope is complete synchronization on undirected and diffusively coupled dynamic networks. First, we explore the effect of adding links between unconnected nodes with different similarity levels on network synchronizability and find that preferentially adding links between nodes with lower similarity can improve network synchronizability more than that with higher similarity, where node similarity is measured by our improved Asymmetric Katz (AKatz) and Asymmetric Leicht-Holme-Newman (ALHNII) methods from the perspective of link prediction. Additional simulations demonstrate that the node similarity-based link-addition strategy is more effective in enhancing network synchronizability than the node centrality-based methods. Furthermore, we apply the node similarity-based link-addition or deletion strategy as the valid criteria to the rewiring process of inter-community links and then propose a Node Similarity-Based Rewiring Optimization (NSBRO) algorithm, where the optimization process is realized by a modified simulated annealing technique. Simulations show that our proposed method performs better in optimizing synchronization of such networks compared with other centrality-based heuristic methods. Finally, simulations on the Rössler system indicate that the network structure optimized by the NSBRO algorithm also leads to better synchronizability of coupled oscillators.

Random temporal connections promote network synchronization

We report a phenomenon of collective dynamics on discrete-time complex networks: a random temporal interaction matrix even of zero or/and small average is able to significantly enhance synchronization with probability one. According to current knowledge, there is no verifiably sufficient criterion for the phenomenon. We use the standard method of synchronization analytics and the theory of stochastic processes to establish a criterion, by which we rigorously and accurately depict how synchronization occurring with probability one is affected by the statistical characteristics of the random temporal connections such as the strength and topology of the connections as well as their probability distributions. We also illustrate the enhancement phenomenon using physical and biological complex dynamical networks.

Novel aperiodically intermittent stability criteria for Markovian switching stochastic delayed coupled systems

This paper concerns p th moment exponential stability of stochastic coupled systems with multiple time-varying delays, and Markovian switching topologies via intermittent control. Compared with previous research results, the mathematical model of this kind of stochastic coupled systems with multiple time-varying delays and Markovian switching topologies is studied for the first time. The intermittent control designed in this paper is aperiodical, which is more general in practice. Moreover, the restriction between control width and time delays is removed. By constructing a new differential inequality on delayed dynamical systems with Markovian switching topologies and combining the graph-theoretic approach with M-matrix theory, two sufficient criteria are derived to guarantee p th moment exponential stability of systems. Moreover, the exponential convergence rate has a close relationship with the maximum ratio of the rest width to the aperiodical time span (the sum of the control width and the rest width). Finally, we employ the theoretical results to study the exponential stability of stochastic coupled oscillators with multiple time-varying delays and Markovian switching topologies. Meanwhile, a numerical example is presented to illustrate the effectiveness and feasibility of the proposed results.

Synchronization by uncorrelated noise: interacting rhythms in interconnected oscillator networks

Oscillators coupled in a network can synchronize with each other to yield a coherent population rhythm. How do multiple such rhythms interact with each other? Do these collective oscillations synchronize like individual oscillators? We show that this is not the case: for strong, inhibitory coupling rhythms can become synchronized by noise. In contrast to stochastic synchronization, this new mechanism synchronizes the rhythms even if the noisy inputs to different oscillators are completely uncorrelated. Key for the synchrony across networks is the reduced synchrony within the networks: it substantially increases the frequency range across which the networks can be entrained by other networks or by periodic pacemaker-like inputs. We demonstrate this type of robust synchronization for different classes of oscillators and network connectivities. The synchronization of different population rhythms is expected to be relevant for brain rhythms.

Accurate detection of hierarchical communities in complex networks based on nonlinear dynamical evolution

One of the most challenging problems in network science is to accurately detect communities at distinct hierarchical scales. Most existing methods are based on structural analysis and manipulation, which are NP-hard. We articulate an alternative, dynamical evolution-based approach to the problem. The basic principle is to computationally implement a nonlinear dynamical process on all nodes in the network with a general coupling scheme, creating a networked dynamical system. Under a proper system setting and with an adjustable control parameter, the community structure of the network would "come out" or emerge naturally from the dynamical evolution of the system. As the control parameter is systematically varied, the community hierarchies at different scales can be revealed. As a concrete example of this general principle, we exploit clustered synchronization as a dynamical mechanism through which the hierarchical community structure can be uncovered. In particular, for quite arbitrary choices of the nonlinear nodal dynamics and coupling scheme, decreasing the coupling parameter from the global synchronization regime, in which the dynamical states of all nodes are perfectly synchronized, can lead to a weaker type of synchronization organized as clusters. We demonstrate the existence of optimal choices of the coupling parameter for which the synchronization clusters encode accurate information about the hierarchical community structure of the network. We test and validate our method using a standard class of benchmark modular networks with two distinct hierarchies of communities and a number of empirical networks arising from the real world. Our method is computationally extremely efficient, eliminating completely the NP-hard difficulty associated with previous methods. The basic principle of exploiting dynamical evolution to uncover hidden community organizations at different scales represents a "game-change" type of approach to addressing the problem of community detection in complex networks.

Autapses promote synchronization in neuronal networks

Neurological disorders such as epileptic seizures are believed to be caused by neuronal synchrony. However, to ascertain the causal role of neuronal synchronization in such diseases through the traditional approach of electrophysiological data analysis remains a controversial, challenging, and outstanding problem. We offer an alternative principle to assess the physiological role of neuronal synchrony based on identifying structural anomalies in the underlying network and studying their impacts on the collective dynamics. In particular, we focus on autapses - time delayed self-feedback links that exist on a small fraction of neurons in the network, and investigate their impacts on network synchronization through a detailed stability analysis. Our main finding is that the proper placement of a small number of autapses in the network can promote synchronization significantly, providing the computational and theoretical bases for hypothesizing a high degree of synchrony in real neuronal networks with autapses. Our result that autapses, the shortest possible links in any network, can effectively modulate the collective dynamics provides also a viable strategy for optimal control of complex network dynamics at minimal cost.

Maximizing synchronizability of duplex networks

We study the synchronizability of duplex networks formed by two randomly generated network layers with different patterns of interlayer node connections. According to the master stability function, we use the smallest nonzero eigenvalue and the eigenratio between the largest and the second smallest eigenvalues of supra-Laplacian matrices to characterize synchronizability on various duplexes. We find that the interlayer linking weight and linking fraction have a profound impact on synchronizability of duplex networks. The increasingly large inter-layer coupling weight is found to cause either decreasing or constant synchronizability for different classes of network dynamics. In addition, negative node degree correlation across interlayer links outperforms positive degree correlation when most interlayer links are present. The reverse is true when a few interlayer links are present. The numerical results and understanding based on these representative duplex networks are illustrative and instructive for building insights into maximizing synchronizability of more realistic multiplex networks.

Mesoscale Architecture Shapes Initiation and Richness of Spontaneous Network Activity

Spontaneous activity in the absence of external input, including propagating waves of activity, is a robust feature of neuronal networks in vivo and in vitro The neurophysiological and anatomical requirements for initiation and persistence of such activity, however, are poorly understood, as is their role in the function of neuronal networks. Computational network studies indicate that clustered connectivity may foster the generation, maintenance, and richness of spontaneous activity. Since this mesoscale architecture cannot be systematically modified in intact tissue, testing these predictions is impracticable in vivo Here, we investigate how the mesoscale structure shapes spontaneous activity in generic networks of rat cortical neurons in vitro In these networks, neurons spontaneously arrange into local clusters with high neurite density and form fasciculating long-range axons. We modified this structure by modulation of protein kinase C, an enzyme regulating neurite growth and cell migration. Inhibition of protein kinase C reduced neuronal aggregation and fasciculation of axons, i.e., promoted uniform architecture. Conversely, activation of protein kinase C promoted aggregation of neurons into clusters, local connectivity, and bundling of long-range axons. Supporting predictions from theory, clustered networks were more spontaneously active and generated diverse activity patterns. Neurons within clusters received stronger synaptic inputs and displayed increased membrane potential fluctuations. Intensified clustering promoted the initiation of synchronous bursting events but entailed incomplete network recruitment. Moderately clustered networks appear optimal for initiation and propagation of diverse patterns of activity. Our findings support a crucial role of the mesoscale architectures in the regulation of spontaneous activity dynamics.SIGNIFICANCE STATEMENT Computational studies predict richer and persisting spatiotemporal patterns of spontaneous activity in neuronal networks with neuron clustering. To test this, we created networks of varying architecture in vitro Supporting these predictions, the generation and spatiotemporal patterns of propagation were most variable in networks with intermediate clustering and lowest in uniform networks. Grid-like clustering, on the other hand, facilitated spontaneous activity but led to degenerating patterns of propagation. Neurons outside clusters had weaker synaptic input than neurons within clusters, in which increased membrane potential fluctuations facilitated the initiation of synchronized spike activity. Our results thus show that the intermediate level organization of neuronal networks strongly influences the dynamics of their activity.

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.

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.

Filippov systems and quasi-synchronization control for switched networks

This paper is concerned with the quasi-synchronization issue of linearly coupled networks with discontinuous nonlinear functions in each isolated node. Under the framework of Filippov systems, the existence and boundedness of solutions for such complex networks can be guaranteed by the matrix measure approach. A design method is presented for the synchronization controllers of coupled networks with non-identical discontinuous systems. Moreover, a sufficient condition is derived to ensure the quasi-synchronization of switched coupled complex networks with discontinuous isolated nodes, which could be controlled by some designed linear controllers. The obtained results extend the previous work on the synchronization issue of coupled complex networks with Lipschitz continuous conditions. Numerical simulations on the coupled chaotic systems are given to demonstrate the effectiveness of the theoretical results.

Adaptive mechanism between dynamical synchronization and epidemic behavior on complex networks

Many realistic epidemic networks display statistically synchronous behavior which we will refer to as epidemic synchronization. However, to the best of our knowledge, there has been no theoretical study of epidemic synchronization. In fact, in many cases, synchronization and epidemic behavior can arise simultaneously and interplay adaptively. In this paper, we first construct mathematical models of epidemic synchronization, based on traditional dynamical models on complex networks, by applying the adaptive mechanisms observed in real networks. Then, we study the relationship between the epidemic rate and synchronization stability of these models and, in particular, obtain the conditions of local and global stability for epidemic synchronization. Finally, we perform numerical analysis to verify our theoretical results. This work is the first to draw a theoretical bridge between epidemic transmission and synchronization dynamics and will be beneficial to the study of control and the analysis of the epidemics on complex networks.

Competition between intra-community and inter-community synchronization and relevance in brain cortical networks

In this paper the effects of inter-community links on the synchronization performance of community networks, especially on the competition between individual community and the whole network, are studied in detail. The study is organized from two aspects: the number or portion of inter-community links and the connection strategy of inter-community links between different communities. A critical point is found in the competition of global network and individual communities. Increasing the number of inter-community links will enhance the global synchronizability but degrade the synchronization performance of individual community before this point. After that the individual community will synchronize better and better as part of the whole network because the community structure is not so prominent. The critical point represents a balance region where the individual community is maximally independent while the information transmission remains effective between different communities. Among various connection strategies, connecting nodes belonging to different communities randomly rather than connecting nodes with larger degrees are the most efficient way to enhance global synchronization of the network. However, the dynamical modularity is the reverse case. A preferential connection scheme linking most of the hubs from the communities will allow rather efficient global synchronization while maintaining strong dynamical clustering of the communities. Interestingly, the observations are found to be relevant in a realistic network of cat cortex. The synchronization state is just at the critical point, which shows a reasonable combination of segregated function in individual communities and coordination among them. Our work sheds light on principles underlying the emergence of modular architectures in real network systems and provides guidance for the manipulation of synchronization in community networks.

Burst synchronization transitions in a neuronal network of subnetworks

In this paper, the transitions of burst synchronization are explored in a neuronal network consisting of subnetworks. The studied network is composed of electrically coupled bursting Hindmarsh-Rose neurons. Numerical results show that two types of burst synchronization transitions can be induced not only by the variations of intra- and intercoupling strengths but also by changing the probability of random links between different subnetworks and the number of subnetworks. Furthermore, we find that the underlying mechanisms for these two bursting synchronization transitions are different: one is due to the change of spike numbers per burst, while the other is caused by the change of the bursting type. Considering that changes in the coupling strengths and neuronal connections are closely interlaced with brain plasticity, the presented results could have important implications for the role of the brain plasticity in some functional behavior that are associated with synchronization.

Mixed outer synchronization of coupled complex networks with time-varying coupling delay

In this paper, the problem of outer synchronization between two complex networks with the same topological structure and time-varying coupling delay is investigated. In particular, we introduce a new type of outer synchronization behavior, i.e., mixed outer synchronization (MOS), in which different state variables of the corresponding nodes can evolve into complete synchronization, antisynchronization, and even amplitude death simultaneously for an appropriate choice of the scaling matrix. A novel nonfragile linear state feedback controller is designed to realize the MOS between two networks and proved analytically by using Lyapunov-Krasovskii stability theory. Finally, numerical simulations are provided to demonstrate the feasibility and efficacy of our proposed control approach.

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.

Cluster synchronization in community networks with nonidentical nodes

In this paper dynamical networks with community structure and nonidentical nodes and with identical local dynamics for all individual nodes in each community are considered. The cluster synchronization of these networks with or without time delay is studied by using some feedback control schemes. Several sufficient conditions for achieving cluster synchronization are obtained analytically and are further verified numerically by some examples with chaotic or nonchaotic nodes. In addition, an essential relation between synchronization dynamics and local dynamics is found by detailed analysis of dynamical networks without delay through the stage detection of cluster synchronization.

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.

Inferring connectivity of interacting phase oscillators

The question as to how network topology properties influence network dynamical behavior has been extensively investigated. Here we treat the inverse problem, i.e., how to infer network connection topology from the dynamic evolution, and suggest a control based topology identification method. This method includes two steps: (i) driving the network to a steady state and (ii) inferring all elements of the connectivity matrix by exploiting information obtained from the observed steady state response of each node. We adopt different strategies for model-dependent (i.e., each local phase dynamics and coupling functions are known) and model-free (i.e., each local phase dynamics and coupling functions are unknown) cases and give detailed conditions for both cases under which network topology can be identified correctly. The influence of noise on topology identification is discussed as well. All proposed approaches are motivated and illustrated with networks of phase oscillators. We argue that these topology identification methods can be extended to general dynamical networks and are not restricted to only networks of phase oscillators.

Dynamical complexity in small-world networks of spiking neurons

A computer model is described which is used to assess the dynamical complexity of a class of networks of spiking neurons with small-world properties. Networks are constructed by forming an initially segregated set of highly intraconnected clusters and then applying a probabilistic rewiring method reminiscent of the Watts-Strogatz procedure to make intercluster connections. Causal density, which counts the number of independent significant interactions among a system's components, is used to assess dynamical complexity. This measure was chosen because it employs lagged observations, and is therefore more sensitive to temporally smeared evidence of segregation and integration than its alternatives. The results broadly support the hypothesis that small-world topology promotes dynamical complexity, but reveal a narrow parameter range within which this occurs for the network topology under investigation, and suggest an inverse correlation with phase synchrony inside this range.

Localizations on complex networks

We study the structural characteristics of complex networks using the representative eigenvectors of the adjacent matrix. The probability distribution function of the components of the representative eigenvectors are proposed to describe the localization on networks where the Euclidean distance is invalid. Several quantities are used to describe the localization properties of the representative states, such as the participation ratio, the structural entropy, and the probability distribution function of the nearest neighbor level spacings for spectra of complex networks. Whole-cell networks in the real world and the Watts-Strogatz small-world and Barabasi-Albert scale-free networks are considered. The networks have nontrivial localization properties due to the nontrivial topological structures. It is found that the ascending-order-ranked series of the occurrence probabilities at the nodes behave generally multifractally. This characteristic can be used as a structural measure of complex networks.

Transition to global synchronization in clustered networks

A clustered network is characterized by a number of distinct sparsely linked subnetworks (clusters), each with dense internal connections. Such networks are relevant to biological, social, and certain technological networked systems. For a clustered network the occurrence of global synchronization, in which nodes from different clusters are synchronized, is of interest. We consider Kuramoto-type dynamics and obtain an analytic formula relating the critical coupling strength required for global synchronization to the probabilities of intracluster and intercluster connections, and provide numerical verification. Our work also provides direct support for a previous spectral-analysis-based result concerning the role of random intercluster links in enhancing the synchronizability of a clustered network.

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.

Optimization of synchronization in complex clustered networks

There has been mounting evidence that many types of biological or technological networks possess a clustered structure. As many system functions depend on synchronization, it is important to investigate the synchronizability of complex clustered networks. Here we focus on one fundamental question: Under what condition can the network synchronizability be optimized? In particular, since the two basic parameters characterizing a complex clustered network are the probabilities of intercluster and intracluster connections, we investigate, in the corresponding two-dimensional parameter plane, regions where the network can be best synchronized. Our study yields a quite surprising finding: a complex clustered network is most synchronizable when the two probabilities match each other approximately. Mismatch, for instance caused by an overwhelming increase in the number of intracluster links, can counterintuitively suppress or even destroy synchronization, even though such an increase tends to reduce the average network distance. This phenomenon provides possible principles for optimal synchronization on complex clustered networks. We provide extensive numerical evidence and an analytic theory to establish the generality of this phenomenon.

Alternating synchronizability of complex clustered networks with regular local structure

Small network distance and homogeneous degree distribution have been found to be critical to efficient network synchronization. In this paper, we investigate the synchronizability of clustered networks with regular subnetworks and report a counterintuitive phenomenon: As the density of intracluster links is increased, the network exhibits strong and weak synchronizability in an alternating manner. A theory based on analyzing the eigenvalues and eigenvectors of the coupling matrix is provided to explain this phenomenon. The relevance of the network model to tissue organization for intercellular communication in biological systems is discussed. An implication is that, in order to achieve synchronization, local coupling density in the network needs to be tuned properly.

Optimization of synchronization in gradient clustered networks

We consider complex clustered networks with a gradient structure, where the sizes of the clusters are distributed unevenly. Such networks describe actual networks in biophysical systems and in technological applications more closely than the previous models. Theoretical analysis predicts that the network synchronizability can be optimized by the strength of the gradient field, but only when the gradient field points from large to small clusters. A remarkable finding is that, if the gradient field is sufficiently strong, synchronizability of the network is mainly determined by the properties of the subnetworks in the two largest clusters. These results are verified by numerical eigenvalue analysis and by direct simulation of synchronization dynamics on coupled-oscillator networks.

Network synchronization of groups

In this paper we study synchronized motions in complex networks in which there are distinct groups of nodes where the dynamical systems on each node within a group are the same but are different for nodes in different groups. Both continuous time and discrete time systems are considered. We initially focus on the case where two groups are present and the network has bipartite topology (i.e., links exist between nodes in different groups but not between nodes in the same group). We also show that group synchronous motions are compatible with more general network topologies, where there are also connections within the groups.

Effects of the network structural properties on its controllability

In a recent paper, it has been suggested that the controllability of a diffusively coupled complex network, subject to localized feedback loops at some of its vertices, can be assessed by means of a Master Stability Function approach, where the network controllability is defined in terms of the spectral properties of an appropriate Laplacian matrix. Following that approach, a comparison study is reported here among different network topologies in terms of their controllability. The effects of heterogeneity in the degree distribution, as well as of degree correlation and community structure, are discussed.

Frustration, drift, and antiphase coupling in a neural array

Synchronization among neurons is critical for many processes in the nervous system, ranging from the processing of sensory information to the onset of pathological conditions such as epilepsy. Here, we study synchronization in an array of neurons, each modeled by a set of nonlinear ordinary differential equations. We find that an array of 20x20 coupled neurons undergoes a series of alternating low and high synchronization states, as measured by phase-locking and frequency entrainment, as the coupling constant is tuned. The role of long-range connections in inducing "small-world networks" has recently been of great interest in many physical and biological problems. Since long-range connections do exist in the brain, we investigated the role of such connections in our neural array. Introducing a biologically realistic percentage of long-range connections has no significant effect on synchronization. We find that it is rather the type of coupling and the total number of connections that determine the synchronization state of the array. We also show that some coupling conditions can lead to frustration in the system, resulting from an inability to simultaneously satisfy conflicting phase requirements. This frustration leads to a drift in the overall behavior of the network, which may offer an explanation for transitions between different types of neural oscillations observed experimentally.

Desynchronization waves in small-world networks

A regular array of oscillators with random coupling exhibits a transition from synchronized motion to desynchronized but ordered waves as a global coupling parameter is increased, due to the spread of localized instability of eigenvectors of the Laplacian matrix. We find that shortcuts, which make a regular network small-world, can destroy ordered desynchronization wave patterns. Wave patterns in a small-world network are usually destroyed gradually as the degree of regularity in the network deteriorates. No ordered wave patterns are observed in scale-free and random networks. The formation of ordered wave patterns in a coupled oscillator network can be explained by considering the time evolution of phase in each oscillator. We derive a general type of the Kardar-Parisi-Zhang equation for phase evolution in a prototype oscillator network. The equation demonstrates well the ordered desynchronized wave patterns found in the network with and without shortcuts. Our results provide a qualitative justification for the requirement of certain degree of regularity in the network for ordered wave patterns to arise.