Synchronization is enhanced in weighted complex networks

Synchronization enhancement subjected to adaptive blinking coupling

Synchronization holds a significant role, notably within chaotic systems, in various contexts where the coordinated behavior of systems plays a pivotal and indispensable role. Hence, many studies have been dedicated to investigating the underlying mechanism of synchronization of chaotic systems. Networks with time-varying coupling, particularly those with blinking coupling, have been proven essential. The reason is that such coupling schemes introduce dynamic variations that enhance adaptability and robustness, making them applicable in various real-world scenarios. This paper introduces a novel adaptive blinking coupling, wherein the coupling adapts dynamically based on the most influential variable exhibiting the most significant average disparity. To ensure an equitable selection of the most effective coupling at each time instance, the average difference of each variable is normalized to the synchronous solution's range. Due to this adaptive coupling selection, synchronization enhancement is expected to be observed. This hypothesis is assessed within networks of identical systems, encompassing Lorenz, Rössler, Chen, Hindmarsh-Rose, forced Duffing, and forced van der Pol systems. The results demonstrated a substantial improvement in synchronization when employing adaptive blinking coupling, particularly when applying the normalization process.

Optimal time-varying coupling function can enhance synchronization in complex networks

In this paper, we propose a time-varying coupling function that results in enhanced synchronization in complex networks of oscillators. The stability of synchronization can be analyzed by applying the master stability approach, which considers the largest Lyapunov exponent of the linearized variational equations as a function of the network eigenvalues as the master stability function. Here, it is assumed that the oscillators have diffusive single-variable coupling. All possible single-variable couplings are studied for each time interval, and the one with the smallest local Lyapunov exponent is selected. The obtained coupling function leads to a decrease in the critical coupling parameter, resulting in enhanced synchronization. Moreover, synchronization is achieved faster, and its robustness is increased. For illustration, the optimum coupling function is found for three networks of chaotic Rössler, Chen, and Chua systems, revealing enhanced synchronization.

Structure-function clustering in weighted brain networks

Functional networks, which typically describe patterns of activity taking place across the cerebral cortex, are widely studied in neuroscience. The dynamical features of these networks, and in particular their deviation from the relatively static structural network, are thought to be key to higher brain function. The interactions between such structural networks and emergent function, and the multimodal neuroimaging approaches and common analysis according to frequency band motivate a multilayer network approach. However, many such investigations rely on arbitrary threshold choices that convert dense, weighted networks to sparse, binary structures. Here, we generalise a measure of multiplex clustering to describe weighted multiplexes with arbitrarily-many layers. Moreover, we extend a recently-developed measure of structure-function clustering (that describes the disparity between anatomical connectivity and functional networks) to the weighted case. To demonstrate its utility we combine human connectome data with simulated neural activity and bifurcation analysis. Our results indicate that this new measure can extract neurologically relevant features not readily apparent in analogous single-layer analyses. In particular, we are able to deduce dynamical regimes under which multistable patterns of neural activity emerge. Importantly, these findings suggest a role for brain operation just beyond criticality to promote cognitive flexibility.

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.

Matryoshka and disjoint cluster synchronization of networks

The main motivation for this paper is to characterize network synchronizability for the case of cluster synchronization (CS), in an analogous fashion to Barahona and Pecora [Phys. Rev. Lett. 89, 054101 (2002)] for the case of complete synchronization. We find this problem to be substantially more complex than the original one. We distinguish between the two cases of networks with intertwined clusters and no intertwined clusters and between the two cases that the master stability function is negative either in a bounded range or in an unbounded range of its argument. Our proposed definition of cluster synchronizability is based on the synchronizability of each individual cluster within a network. We then attempt to generalize this definition to the entire network. For CS, the synchronous solution for each cluster may be stable, independent of the stability of the other clusters, which results in possibly different ranges in which each cluster synchronizes (isolated CS). For each pair of clusters, we distinguish between three different cases: Matryoshka cluster synchronization (when the range of the stability of the synchronous solution for one cluster is included in that of the other cluster), partially disjoint cluster synchronization (when the ranges of stability of the synchronous solutions partially overlap), and complete disjoint cluster synchronization (when the ranges of stability of the synchronous solutions do not overlap).

Looking beyond community structure leads to the discovery of dynamical communities in weighted networks

A fundamental question is whether groups of nodes of a complex network can possibly display long-term cluster-synchronized behavior. While this question has been addressed for the restricted classes of unweighted and labeled graphs, it remains an open problem for the more general class of weighted networks. The emergence of coordinated motion of nodes in natural and technological networks is directly related to the network structure through the concept of an equitable partition, which determines which nodes can show long-term synchronized behavior and which nodes cannot. We provide a method to detect the presence of nearly equitable partitions in weighted networks, based on minimal information about the network structure. With this approach we are able to discover the presence of dynamical communities in both synthetic and real technological, biological, and social networks, to a statistically significant level. We show that our approach based on dynamical communities is better at predicting the emergence of synchronized behavior than existing methods to detect community structure.

Stability of synchronization in simplicial complexes

Various systems in physics, biology, social sciences and engineering have been successfully modeled as networks of coupled dynamical systems, where the links describe pairwise interactions. This is, however, too strong a limitation, as recent studies have revealed that higher-order many-body interactions are present in social groups, ecosystems and in the human brain, and they actually affect the emergent dynamics of all these systems. Here, we introduce a general framework to study coupled dynamical systems accounting for the precise microscopic structure of their interactions at any possible order. We show that complete synchronization exists as an invariant solution, and give the necessary condition for it to be observed as a stable state. Moreover, in some relevant instances, such a necessary condition takes the form of a Master Stability Function. This generalizes the existing results valid for pairwise interactions to the case of complex systems with the most general possible architecture.

Assessing synchronizability provided by coupling variable from the algebraic structure of dynamical systems

Synchronization is a very generic phenomenon which can be encountered in a large variety of coupled dynamical systems. Being able to synchronize chaotic systems is strongly dependent on the nature of their coupling. Few attempts to explain such a dependency using observability and/or controllability were not fully satisfactory and synchronizability yet remained unexplained. Synchronizability can be defined as the range of coupling parameter values for which two nearly identical systems are fully synchronized. Our objective is here to investigate whether synchronizability can be related to the main rotation necessarily required for structuring any type of attractor, that is, whether synchronizability is significantly improved when the coupling variable is one of the variables involved in the main rotation. We thus propose a semianalytic procedure from a single isolated system to discard the worst variable for fully synchronizing two (nearly) identical copies of that system.

Global robustness versus local vulnerabilities in complex synchronous networks

In complex network-coupled dynamical systems, two questions of central importance are how to identify the most vulnerable components and how to devise a network making the overall system more robust to external perturbations. To address these two questions, we investigate the response of complex networks of coupled oscillators to local perturbations. We quantify the magnitude of the resulting excursion away from the unperturbed synchronous state through quadratic performance measures in the angle or frequency deviations. We find that the most fragile oscillators in a given network are identified by centralities constructed from network resistance distances. Further defining the global robustness of the system from the average response over ensembles of homogeneously distributed perturbations, we find that it is given by a family of topological indices known as generalized Kirchhoff indices. Both resistance centralities and Kirchhoff indices are obtained from a spectral decomposition of the stability matrix of the unperturbed dynamics and can be expressed in terms of resistance distances. We investigate the properties of these topological indices in small-world and regular networks. In the case of oscillators with homogeneous inertia and damping coefficients, we find that inertia only has small effects on robustness of coupled oscillators. Numerical results illustrate the validity of the theory.

Resonance induced by coupling diversity in globally coupled bistable oscillators

We investigate the collective response of an ensemble of bistable oscillators to an external periodic signal, where the coupling strength between oscillators is diverse. We find that there exists an optimal level of coupling diversity, at which the collective response of the system can be largely improved, i.e., resonance induced by coupling diversity. We also observe that the system splits into three oscillation clusters when this resonance happens. We finally propose a reduced model based on the three oscillation clusters, which can well predict the collective response of the system.

Competitive networks using chaotic circuits with hierarchical structure

Coupled oscillatory systems are good models that are able to describe a variety of higher dimensional nonlinear phenomena. Coupled chaotic circuits produce many kinds of interesting synchronization phenomena. In recent years, research studies on complex networks related to the synchronization of coupled oscillators have attracted much attention. In the real world, there are a variety of different network structures. We focus on the competitive interaction network that includes conflict between two networks. Here, we propose a new paradigm for this competitive interaction network using coupled chaotic circuits.

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.

Synchronization in network geometries with finite spectral dimension

Recently there is a surge of interest in network geometry and topology. Here we show that the spectral dimension plays a fundamental role in establishing a clear relation between the topological and geometrical properties of a network and its dynamics. Specifically we explore the role of the spectral dimension in determining the synchronization properties of the Kuramoto model. We show that the synchronized phase can only be thermodynamically stable for spectral dimensions above four and that phase entrainment of the oscillators can only be found for spectral dimensions greater than two. We numerically test our analytical predictions on the recently introduced model of network geometry called complex network manifolds, which displays a tunable spectral dimension.

Solitary states in multiplex networks owing to competing interactions

Recent researches in network science demonstrate the coexistence of different types of interactions among the individuals within the same system. A wide range of situations appear in ecological and neuronal systems that incorporate positive and negative interactions. Also, there are numerous examples of systems that are best represented by the multiplex configuration. The present article investigates a possible scenario for the emergence of a newly observed remarkable phenomenon named as solitary state in coupled dynamical units in which one or a few units split off and behave differently from the other units. For this, we consider dynamical systems connected through a multiplex architecture in the presence of both positive and negative couplings. We explore our findings through analysis of the paradigmatic FitzHugh-Nagumo system in both equilibrium and periodic regimes on the top of a multiplex network having positive inter-layer and negative intra-layer interactions. We further substantiate our proposition using a periodic Lorenz system with the same scheme and show that an opposite scheme of competitive interactions may also work for the Lorenz system in the chaotic regime.

Spectral properties of complex networks

This review presents an account of the major works done on spectra of adjacency matrices drawn on networks and the basic understanding attained so far. We have divided the review under three sections: (a) extremal eigenvalues, (b) bulk part of the spectrum, and (c) degenerate eigenvalues, based on the intrinsic properties of eigenvalues and the phenomena they capture. We have reviewed the works done for spectra of various popular model networks, such as the Erdős-Rényi random networks, scale-free networks, 1-d lattice, small-world networks, and various different real-world networks. Additionally, potential applications of spectral properties for natural processes have been reviewed.

Complex Network Geometry and Frustrated Synchronization

The dynamics of networks of neuronal cultures has been recently shown to be strongly dependent on the network geometry and in particular on their dimensionality. However, this phenomenon has been so far mostly unexplored from the theoretical point of view. Here we reveal the rich interplay between network geometry and synchronization of coupled oscillators in the context of a simplicial complex model of manifolds called Complex Network Manifold. The networks generated by this model combine small world properties (infinite Hausdorff dimension) and a high modular structure with finite and tunable spectral dimension. We show that the networks display frustrated synchronization for a wide range of the coupling strength of the oscillators, and that the synchronization properties are directly affected by the spectral dimension of the network.

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.

Competitive influence maximization and enhancement of synchronization in populations of non-identical Kuramoto oscillators

Many networked systems have evolved to optimize performance of function. Much literature has considered optimization of networks by central planning, but investigations of network formation amongst agents connecting to achieve non-aligned goals are comparatively rare. Here we consider the dynamics of synchronization in populations of coupled non-identical oscillators and analyze adaptations in which individual nodes attempt to rewire network topology to optimize node-specific aims. We demonstrate that, even though individual nodes’ goals differ very widely, rewiring rules in which each node attempts to connect to the rest of the network in such a way as to maximize its influence on the system can enhance synchronization of the collective. The observed speed-up of consensus finding in this competitive dynamics might explain enhanced synchronization in real world systems and shed light on mechanisms for improved consensus finding in society.

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.

Asymmetric couplings enhance the transition from chimera state to synchronization

Chimera state has been well studied recently, but little attention has been paid to its transition to synchronization. We study this topic here by considering two groups of adaptively coupled Kuramoto oscillators. By searching the final states of different initial conditions, we find that the system can easily show a chimera state with robustness to initial conditions, in contrast to the sensitive dependence of chimera state on initial conditions in previous studies. Further, we show that, in the case of symmetric couplings, the behaviors of the two groups are always complementary to each other, i.e., robustness of chimera state, except a small basin of synchronization. Interestingly, we reveal that the basin of synchronization will be significantly increased when either the coupling of inner groups or that of intergroups are asymmetric. This transition from the attractor of chimera state to the attractor of synchronization is closely related to both the phase delay and the asymmetric degree of coupling strengths, resulting in a diversity of attractor's patterns. A theory based on the Ott-Antonsen ansatz is given to explain the numerical simulations. This finding may be meaningful for the control of competition between two attractors in biological systems, such as the cardiac rhythm and ventricular fibrillation, etc.

Betweenness Centrality-Based Consensus Protocol for Second-Order Multiagent Systems With Sampled-Data

This paper designs a new leader-following consensus protocol for second-order multiagent systems with time-varying sampling. For the first time in designing a leader-following protocol, the concept of betweenness centrality is adopted to analyze the information flow in the consensus problem for multiagent systems. By construction of a suitable Lyapunov-Krasovskii functional, some criteria for designing consensus protocols of such systems are established in terms of linear matrix inequalities which can be easily solved by various effective optimization algorithms. One numerical example is given to illustrate the validity of the proposed argument.

Optimization of synchronizability in multiplex networks by rewiring one layer

The mathematical framework of multiplex networks has been increasingly realized as a more suitable framework for modeling real-world complex systems. In this work, we investigate the optimization of synchronizability in multiplex networks by evolving only one layer while keeping other layers fixed. Our main finding is to show the conditions under which the efficiency of convergence to the most optimal structure is almost as good as the case where both layers are rewired during an optimization process. In particular, interlayer coupling strength responsible for the integration between the layers turns out to be a crucial factor governing the efficiency of optimization even for the cases when the layer going through the evolution has nodes interacting much more weakly than those in the fixed layer. Additionally, we investigate the dependency of synchronizability on the rewiring probability which governs the network structure from a regular lattice to the random networks. The efficiency of the optimization process preceding evolution driven by the optimization process is maximum when the fixed layer has regular architecture, whereas the optimized network is more synchronizable for the fixed layer having the rewiring probability lying between the small-world transition and the random structure.

Consistency of heterogeneous synchronization patterns in complex weighted networks

Synchronization within the dynamical nodes of a complex network is usually considered homogeneous through all the nodes. Here we show, in contrast, that subsets of interacting oscillators may synchronize in different ways within a single network. This diversity of synchronization patterns is promoted by increasing the heterogeneous distribution of coupling weights and/or asymmetries in small networks. We also analyze consistency, defined as the persistence of coexistent synchronization patterns regardless of the initial conditions. Our results show that complex weighted networks display richer consistency than regular networks, suggesting why certain functional network topologies are often constructed when experimental data are analyzed.

Synchronization in networks with multiple interaction layers

The structure of many real-world systems is best captured by networks consisting of several interaction layers. Understanding how a multilayered structure of connections affects the synchronization properties of dynamical systems evolving on top of it is a highly relevant endeavor in mathematics and physics and has potential applications in several socially relevant topics, such as power grid engineering and neural dynamics. We propose a general framework to assess the stability of the synchronized state in networks with multiple interaction layers, deriving a necessary condition that generalizes the master stability function approach. We validate our method by applying it to a network of Rössler oscillators with a double layer of interactions and show that highly rich phenomenology emerges from this. This includes cases where the stability of synchronization can be induced even if both layers would have individually induced unstable synchrony, an effect genuinely arising from the true multilayer structure of the interactions among the units in the network.

Auditory Power-Law Activation Avalanches Exhibit a Fundamental Computational Ground State

The cochlea provides a biological information-processing paradigm that we are only beginning to understand in its full complexity. Our work reveals an interacting network of strongly nonlinear dynamical nodes, on which even a simple sound input triggers subnetworks of activated elements that follow power-law size statistics ("avalanches"). From dynamical systems theory, power-law size distributions relate to a fundamental ground state of biological information processing. Learning destroys these power laws. These results strongly modify the models of mammalian sound processing and provide a novel methodological perspective for understanding how the brain processes information.

The role of dimensionality in neuronal network dynamics

Recent results from network theory show that complexity affects several dynamical properties of networks that favor synchronization. Here we show that synchronization in 2D and 3D neuronal networks is significantly different. Using dissociated hippocampal neurons we compared properties of cultures grown on a flat 2D substrates with those formed on 3D graphene foam scaffolds. Both 2D and 3D cultures had comparable glia to neuron ratio and the percentage of GABAergic inhibitory neurons. 3D cultures because of their dimension have many connections among distant neurons leading to small-world networks and their characteristic dynamics. After one week, calcium imaging revealed moderately synchronous activity in 2D networks, but the degree of synchrony of 3D networks was higher and had two regimes: a highly synchronized (HS) and a moderately synchronized (MS) regime. The HS regime was never observed in 2D networks. During the MS regime, neuronal assemblies in synchrony changed with time as observed in mammalian brains. After two weeks, the degree of synchrony in 3D networks decreased, as observed in vivo. These results show that dimensionality determines properties of neuronal networks and that several features of brain dynamics are a consequence of its 3D topology.

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.

Synchronization unveils the organization of ecological networks with positive and negative interactions

Network science has helped to understand the organization principles of the interactions among the constituents of large complex systems. However, recently, the high resolution of the data sets collected has allowed to capture the different types of interactions coexisting within the same system. A particularly important example is that of systems with positive and negative interactions, a usual feature appearing in social, neural, and ecological systems. The interplay of links of opposite sign presents natural difficulties for generalizing typical concepts and tools applied to unsigned networks and, moreover, poses some questions intrinsic to the signed nature of the network, such as how are negative interactions balanced by positive ones so to allow the coexistence and survival of competitors/foes within the same system? Here, we show that synchronization phenomenon is an ideal benchmark for uncovering such balance and, as a byproduct, to assess which nodes play a critical role in the overall organization of the system. We illustrate our findings with the analysis of synthetic and real ecological networks in which facilitation and competitive interactions coexist.

Small Worldness in Dense and Weighted Connectomes

The human brain is a heterogeneous network of connected functional regions; however, most brain network studies assume that all brain connections can be described in a framework of binary connections. The brain is a complex structure of white matter tracts connected by a wide range of tract sizes, which suggests a broad range of connection strengths. Therefore, the assumption that the connections are binary yields an incomplete picture of the brain. Various thresholding methods have been used to remove spurious connections and reduce the graph density in binary networks. But these thresholds are arbitrary and make problematic the comparison of networks created at different thresholds. The heterogeneity of connection strengths can be represented in graph theory by applying weights to the network edges. Using our recently introduced edge weight parameter, we estimated the topological brain network organization using a complimentary weighted connectivity framework to the traditional framework of a binary network. To examine the reproducibility of brain networks in a controlled condition, we studied the topological network organization of a single healthy individual by acquiring 10 repeated diffusion-weighted magnetic resonance image datasets, over a 1-month period on the same scanner, and analyzing these networks with deterministic tractography. We applied a threshold to both the binary and weighted networks and determined that the extra degree of freedom that comes with the framework of weighting network connectivity provides a robust result as any threshold level. The proposed weighted connectivity framework provides a stable result and is able to demonstrate the small world property of brain networks in situations where the binary framework is inadequate and unable to demonstrate this network property.

Topological constraints on network control profiles

Network models are designed to capture properties of empirical networks and thereby provide insight into the processes that underlie the formation of complex systems. As new information concerning network structure becomes available, it becomes possible to design models that more fully capture the properties of empirical networks. A recent advance in our understanding of network structure is the control profile, which summarizes the structural controllability of a network in terms of source nodes, external dilations, and internal dilations. Here, we consider the topological properties–and their formation mechanisms—that constrain the control profile. We consider five representative empirical categories of internal-dilation dominated networks, and show that the number of source and sink nodes, the form of the in- and out-degree distributions, and local complexity (e.g., cycles) shape the control profile. We evaluate network models that are sufficient to produce realistic control profiles, and conclude that holistic network models should similarly consider these properties.

Enhancing speed of pinning synchronizability: low-degree nodes with high feedback gains

Controlling complex networks is of paramount importance in science and engineering. Despite recent efforts to improve controllability and synchronous strength, little attention has been paid to the speed of pinning synchronizability (rate of convergence in pinning control) and the corresponding pinning node selection. To address this issue, we propose a hypothesis to restrict the control cost, then build a linear matrix inequality related to the speed of pinning controllability. By solving the inequality, we obtain both the speed of pinning controllability and optimal control strength (feedback gains in pinning control) for all nodes. Interestingly, some low-degree nodes are able to achieve large feedback gains, which suggests that they have high influence on controlling system. In addition, when choosing nodes with high feedback gains as pinning nodes, the controlling speed of real systems is remarkably enhanced compared to that of traditional large-degree and large-betweenness selections. Thus, the proposed approach provides a novel way to investigate the speed of pinning controllability and can evoke other effective heuristic pinning node selections for large-scale systems.

Synchronization in dynamical networks with unconstrained structure switching

We provide a rigorous solution to the problem of constructing a structural evolution for a network of coupled identical dynamical units that switches between specified topologies without constraints on their structure. The evolution of the structure is determined indirectly from a carefully built transformation of the eigenvector matrices of the coupling Laplacians, which are guaranteed to change smoothly in time. In turn, this allows one to extend the master stability function formalism, which can be used to assess the stability of a synchronized state. This approach is independent from the particular topologies that the network visits, and is not restricted to commuting structures. Also, it does not depend on the time scale of the evolution, which can be faster than, comparable to, or even secular with respect to the dynamics of the units.

Event-Triggered State Estimation for Complex Networks With Mixed Time Delays via Sampled Data Information: The Continuous-Time Case

In this paper, the event-triggered state estimation problem is investigated for a class of complex networks with mixed time delays using sampled data information. A novel state estimator is presented to estimate the network states. A new event-triggered transmission scheme is proposed to reduce unnecessary network traffic between the sensors and the estimator, where the sampled data is transmitted to the estimator only when the so-called "event-triggered condition" is satisfied. The purpose of the problem addressed is to design an estimator for the complex network such that the estimation error is ultimately bounded in mean square. By utilizing Lyapunov theory combined with the stochastic analysis approach, sufficient conditions are established to guarantee the ultimate boundedness of the estimation error in mean square. Then, the desired estimator gain matrices are obtained via solving a convex problem. Finally, a numerical example is given to illustrate the effectiveness of the results.

Enhancing the stability of the synchronization of multivariable coupled oscillators

Synchronization processes in populations of identical networked oscillators are the focus of intense studies in physical, biological, technological, and social systems. Here we analyze the stability of the synchronization of a network of oscillators coupled through different variables. Under the assumption of an equal topology of connections for all variables, the master stability function formalism allows assessing and quantifying the stability properties of the synchronization manifold when the coupling is transferred from one variable to another. We report on the existence of an optimal coupling transference that maximizes the stability of the synchronous state in a network of Rössler-like oscillators. Finally, we design an experimental implementation (using nonlinear electronic circuits) which grounds the robustness of the theoretical predictions against parameter mismatches, as well as against intrinsic noise of the system.

Consensus of Nonlinear Complex Systems with Edge Betweenness Centrality Measure under Time-Varying Sampled-Data Protocol

This paper proposes a new consensus criterion for nonlinear complex systems with edge betweenness centrality measure. By construction of a suitable Lyapunov-Krasovskii functional, the consensus criterion for such systems is established in terms of linear matrix inequalities (LMIs) which can be easily solved by various effective optimization algorithms. One numerical example is given to illustrate the effectiveness of the proposed methods.

Consistency between functional and structural networks of coupled nonlinear oscillators

In data-based reconstruction of complex networks, dynamical information can be measured and exploited to generate a functional network, but is it a true representation of the actual (structural) network? That is, when do the functional and structural networks match and is a perfect matching possible? To address these questions, we use coupled nonlinear oscillator networks and investigate the transition in the synchronization dynamics to identify the conditions under which the functional and structural networks are best matched. We find that, as the coupling strength is increased in the weak-coupling regime, the consistency between the two networks first increases and then decreases, reaching maximum in an optimal coupling regime. Moreover, by changing the network structure, we find that both the optimal regime and the maximum consistency will be affected. In particular, the consistency for heterogeneous networks is generally weaker than that for homogeneous networks. Based on the stability of the functional network, we propose further an efficient method to identify the optimal coupling regime in realistic situations where the detailed information about the network structure, such as the network size and the number of edges, is not available. Two real-world examples are given: corticocortical network of cat brain and the Nepal power grid. Our results provide new insights not only into the fundamental interplay between network structure and dynamics but also into the development of methodologies to reconstruct complex networks from data.

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.

Synchronization of networked chaotic oscillators under external periodic driving

The dynamical responses of a complex system to external perturbations are of both fundamental interest and practical significance. Here, by the model of networked chaotic oscillators, we investigate how the synchronization behavior of a complex network is influenced by an externally added periodic driving. Interestingly, it is found that by a slight change of the properties of the external driving, e.g., the frequency or phase lag between its intrinsic oscillation and external driving, the network synchronizability could be significantly modified. We demonstrate this phenomenon by different network models and, based on the method of master stability function, give an analysis on the underlying mechanisms. Our studies highlight the importance of external perturbations on the collective behaviors of complex networks, and also provide an alternate approach for controlling network synchronization.

Optimal pinning controllability of complex networks: dependence on network structure

Controlling networked structures has many applications in science and engineering. In this paper, we consider the problem of pinning control (pinning the dynamics into the reference state), and optimally placing the driver nodes, i.e., the nodes to which the control signal is fed. Considering the local controllability concept, a metric based on the eigenvalues of the Laplacian matrix is taken into account as a measure of controllability. We show that the proposed optimal placement strategy considerably outperforms heuristic methods including choosing hub nodes with high degree or betweenness centrality as drivers. We also study properties of optimal drivers in terms of various centrality measures including degree, betweenness, closeness, and clustering coefficient. The profile of these centrality values depends on the network structure. For homogeneous networks such as random small-world networks, the optimal driver nodes have almost the mean centrality value of the population (much lower than the centrality value of hub nodes), whereas the centrality value of optimal drivers in heterogeneous networks such as scale-free ones is much higher than the average and close to that of hub nodes. However, as the degree of heterogeneity decreases in such networks, the profile of centrality approaches the population mean.

Zero-lag synchronization despite inhomogeneities in a relay system

A novel proposal for the zero-lag synchronization of the delayed coupled neurons, is to connect them indirectly via a third relay neuron. In this study, we develop a Poincaré map to investigate the robustness of the synchrony in such a relay system against inhomogeneity in the neurons and synaptic parameters. We show that when the inhomogeneity does not violate the symmetry of the system, synchrony is maintained and in some cases inhomogeneity enhances synchrony. On the other hand if the inhomogeneity breaks the symmetry of the system, zero lag synchrony can not be preserved. In this case we give analytical results for the phase lag of the spiking of the neurons in the stable state.

The structure and dynamics of multilayer networks

In the past years, network theory has successfully characterized the interaction among the constituents of a variety of complex systems, ranging from biological to technological, and social systems. However, up until recently, attention was almost exclusively given to networks in which all components were treated on equivalent footing, while neglecting all the extra information about the temporal- or context-related properties of the interactions under study. Only in the last years, taking advantage of the enhanced resolution in real data sets, network scientists have directed their interest to the multiplex character of real-world systems, and explicitly considered the time-varying and multilayer nature of networks. We offer here a comprehensive review on both structural and dynamical organization of graphs made of diverse relationships (layers) between its constituents, and cover several relevant issues, from a full redefinition of the basic structural measures, to understanding how the multilayer nature of the network affects processes and dynamics.

Controlling cluster synchronization by adapting the topology

We suggest an adaptive control scheme for the control of in-phase and cluster synchronization in delay-coupled networks. Based on the speed-gradient method, our scheme adapts the topology of a network such that the target state is realized. It is robust towards different initial conditions as well as changes in the coupling parameters. The emerging topology is characterized by a delicate interplay of excitatory and inhibitory links leading to the stabilization of the desired cluster state. As a crucial parameter determining this interplay we identify the delay time. Furthermore, we show how to construct networks such that they exhibit not only a given cluster state but also with a given oscillation frequency. We apply our method to coupled Stuart-Landau oscillators, a paradigmatic normal form that naturally arises in an expansion of systems close to a Hopf bifurcation. The successful and robust control of this generic model opens up possible applications in a wide range of systems in physics, chemistry, technology, and life science.

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.

Matching rules for collective behaviors on complex networks: optimal configurations for vibration frequencies of networked harmonic oscillators

The structure-dynamics-function has become one of central problems in modern sciences, and it is a great challenge to unveil the organization rules for different dynamical processes on networks. In this work, we study the vibration spectra of the classical mass spring model with different masses on complex networks, and pay our attention to how the mass spatial configuration influences the second-smallest vibrational frequency (ω2) and the largest one (ωN). For random networks, we find that ω2 becomes maximal and ωN becomes minimal if the node degrees are point-to-point-positively correlated with the masses. In these cases, we call it point-to-point matching. Moreover, ω2 becomes minimal under the condition that the heaviest mass is placed on the lowest-degree vertex, and ωN is maximal as long as the lightest mass is placed on the highest-degree vertex, and in both cases all other masses can be arbitrarily settled. Correspondingly, we call it single-point matching. These findings indicate that the matchings between the node dynamics (parameter) and the node position rule the global systems dynamics, and sometimes only one node is enough to control the collective behaviors of the whole system. Therefore, the matching rules might be the common organization rules for collective behaviors on networks.

Analysis of remote synchronization in complex networks

A novel regime of synchronization, called remote synchronization, where the peripheral nodes form a phase synchronized cluster not including the hub, was recently observed in star motifs [Bergner et al., Phys. Rev. E 85, 026208 (2012)]. We show the existence of a more general dynamical state of remote synchronization in arbitrary networks of coupled oscillators. This state is characterized by the synchronization of pairs of nodes that are not directly connected via a physical link or any sequence of synchronized nodes. This phenomenon is almost negligible in networks of phase oscillators as its underlying mechanism is the modulation of the amplitude of those intermediary nodes between the remotely synchronized units. Our findings thus show the ubiquity and robustness of these states and bridge the gap from their recent observation in simple toy graphs to complex networks.

Explosive synchronization in weighted complex networks

The emergence of dynamical abrupt transitions in the macroscopic state of a system is currently a subject of the utmost interest. Given a set of phase oscillators networking with a generic wiring of connections and displaying a generic frequency distribution, we show how combining dynamical local information on frequency mismatches and global information on the graph topology suggests a judicious and yet practical weighting procedure which is able to induce and enhance explosive, irreversible, transitions to synchronization. We report extensive numerical and analytical evidence of the validity and scalability of such a procedure for different initial frequency distributions, for both homogeneous and heterogeneous networks, as well as for both linear and nonlinear weighting functions. We furthermore report on the possibility of parametrically controlling the width and extent of the hysteretic region of coexistence of the unsynchronized and synchronized states.