Synchronization on complex networks of networks

Relationships and representations of brain structures, connectivity, dynamics and functions

The review explores the complex interplay between brain structures and their associated functions, presenting a diversity of hierarchical models that enhances our understanding of these relationships. Central to this approach are structure-function flow diagrams, which offer a visual representation of how specific neuroanatomical structures are linked to their functional roles. These diagrams are instrumental in mapping the intricate connections between different brain regions, providing a clearer understanding of how functions emerge from the underlying neural architecture. The study details innovative attempts to develop new functional hierarchies that integrate structural and functional data. These efforts leverage recent advancements in neuroimaging techniques such as fMRI, EEG, MEG, and PET, as well as computational models that simulate neural dynamics. By combining these approaches, the study seeks to create a more refined and dynamic hierarchy that can accommodate the brain's complexity, including its capacity for plasticity and adaptation. A significant focus is placed on the overlap of structures and functions within the brain. The manuscript acknowledges that many brain regions are multifunctional, contributing to different cognitive and behavioral processes depending on the context. This overlap highlights the need for a flexible, non-linear hierarchy that can capture the brain's intricate functional landscape. Moreover, the study examines the interdependence of these functions, emphasizing how the loss or impairment of one function can impact others. Another crucial aspect discussed is the brain's ability to compensate for functional deficits following neurological diseases or injuries. The investigation explores how the brain reorganizes itself, often through the recruitment of alternative neural pathways or the enhancement of existing ones, to maintain functionality despite structural damage. This compensatory mechanism underscores the brain's remarkable plasticity, demonstrating its ability to adapt and reconfigure itself in response to injury, thereby ensuring the continuation of essential functions. In conclusion, the study presents a system of brain functions that integrates structural, functional, and dynamic perspectives. It offers a robust framework for understanding how the brain's complex network of structures supports a wide range of cognitive and behavioral functions, with significant implications for both basic neuroscience and clinical applications.

Bipartite secure synchronization criteria for coupled quaternion-valued neural networks with signed graph

This study explores the bipartite secure synchronization problem of coupled quaternion-valued neural networks (QVNNs), in which variable sampled communications and random deception attacks are considered. Firstly, by employing the signed graph theory, the mathematical model of coupled QVNNs with structurally-balanced cooperative-competitive interactions is established. Secondly, by adopting non-decomposition method and constructing a suitable unitary Lyapunov functional, the bipartite secure synchronization (BSS) criteria for coupled QVNNs are obtained in the form of quaternion-valued LMIs. It is essential to mention that the structurally-balanced topology is relatively strong, hence, the coupled QVNNs with structurally-unbalanced graph are further studied. The structurally-unbalanced graph is treated as an interruption of the structurally-balanced graph, the bipartite secure quasi-synchronization (BSQS) criteria for coupled QVNNs with structurally-unbalanced graph are derived. Finally, two simulations are given to illustrate the feasibility of the suggested BSS and BSQS approaches.

DoS attacks resilience of heterogeneous complex networks via dynamic event-triggered impulsive scheme for secure quasi-synchronization

This paper addresses the secure quasi-synchronization issue of heterogeneous complex networks (HCNs) under aperiodic denial-of-service (DoS) attacks with dynamic event-triggered impulsive scheme (ETIS). The heterogeneity of networks and the aperiodic DoS attacks, which hinder communication channels and synchronization goals, present challenges to the analysis of secure quasi-synchronization. The ETIS leverages impulsive control and dynamic event-triggered scheme (ETS) to handle the network heterogeneity and the DoS attacks. We give specific bounds on the attack duration and frequency that the network can endure, and obtain synchronization criteria that relate to event parameters, attack duration, attack frequency, and impulsive gain by the variation of parameter formula and recursive methods. Moreover, we prove that the dynamic ETS significantly reduces the controller updates, saves energy without sacrificing the system decay rate, and prevents the Zeno phenomenon. Finally, we validate our control scheme with a numerical example.

Interval Bipartite Synchronization of Multiple Neural Networks in Signed Graphs

Interval bipartite consensus of multiagents described by signed graphs has received extensive concern recently, and the rooted cycles play a critical role in stabilization, while the structurally balanced graphs are essential to achieve bipartite consensus. However, the gauge transformation used in the linear system is no longer feasible in the nonlinear case. This article addresses interval bipartite synchronization of multiple neural networks (NNs) in a signed graph via a Lyapunov-based approach, extending the existing work to a more practical but complicated case. A general matrix M in signed graphs is introduced to construct the novel Lyapunov functions, and sufficient conditions are obtained. We find that the rooted cycles and the structurally balanced graphs are essential to stabilize and achieve bipartite synchronization. More importantly, we discover that the nonrooted cycles are crucial in reaching interval bipartite synchronization, not previously mentioned. Several examples are presented to illustrate interval bipartite synchronization of multiple NNs with signed graphs.

Percolation Theories for Quantum Networks

Quantum networks have experienced rapid advancements in both theoretical and experimental domains over the last decade, making it increasingly important to understand their large-scale features from the viewpoint of statistical physics. This review paper discusses a fundamental question: how can entanglement be effectively and indirectly (e.g., through intermediate nodes) distributed between distant nodes in an imperfect quantum network, where the connections are only partially entangled and subject to quantum noise? We survey recent studies addressing this issue by drawing exact or approximate mappings to percolation theory, a branch of statistical physics centered on network connectivity. Notably, we show that the classical percolation frameworks do not uniquely define the network's indirect connectivity. This realization leads to the emergence of an alternative theory called "concurrence percolation", which uncovers a previously unrecognized quantum advantage that emerges at large scales, suggesting that quantum networks are more resilient than initially assumed within classical percolation contexts, offering refreshing insights into future quantum network design.

Notes on resonant and synchronized states in complex networks

Synchronization and resonance on networks are some of the most remarkable collective dynamical phenomena. The network topology, or the nature and distribution of the connections within an ensemble of coupled oscillators, plays a crucial role in shaping the local and global evolution of the two phenomena. This article further explores this relationship within a compact mathematical framework and provides new contributions on certain pivotal issues, including a closed bound for the average synchronization time in arbitrary topologies; new evidences of the effect of the coupling strength on this time; exact closed expressions for the resonance frequencies in terms of the eigenvalues of the Laplacian matrix; a measure of the effectiveness of an influencer node's impact on the network; and, finally, a discussion on the existence of a resonant synchronized state. Some properties of the solution of the linear swing equation are also discussed within the same setting. Numerical experiments conducted on two distinct real networks-a social network and a power grid-illustrate the significance of these results and shed light on intriguing aspects of how these processes can be interpreted within networks of this kind.

Edge-Based Decentralized Adaptive Pinning Synchronization of Complex Networks Under Link Attacks

This article studies the pinning synchronization problem with edge-based decentralized adaptive schemes under link attacks. The link attacks considered here are a class of malicious attacks to break links between neighboring nodes in complex networks. In such an insecure network environment, two kinds of edge-based decentralized adaptive update strategies (synchronous and asynchronous) on coupling strengths and gains are designed to realize the security synchronization of complex networks. Moreover, by virtue of the edge pinning technique, the corresponding secure synchronization problem is considered under the case where only a small fraction of coupling strengths and gains is updated. These designed adaptive strategies do not require any global information, and therefore, the obtained results in this article are developed in a fully decentralized framework. Finally, a numerical example is provided to verify the availability of the achieved theoretical outcomes.

Neural-Network-Based Distributed Asynchronous Event-Triggered Consensus Tracking of a Class of Uncertain Nonlinear Multi-Agent Systems

This article proposes a neural-network-based adaptive asynchronous event-triggered design strategy for the distributed consensus tracking of uncertain lower triangular nonlinear multi-agent systems under a directed network. Compared with the existing event-triggered recursive consensus tracking designs using multiple neural networks for each follower and continuous communications among followers, the primary contribution of this study is the development of an asynchronous event-triggered consensus tracking methodology based on a single-neural network for each follower under event-driven intermittent communications among followers. To this end, a distributed event-triggered estimator using neighbors' triggered output information is developed to estimate a leader signal. Subsequently, the estimated leader signal is used to design local trackers. Only a triggering law and a single-neural network are used to design the local tracking law of each follower, irrespective of unmatched unknown nonlinearities. The information of each follower and its neighbors is asynchronously and intermittently communicated through a directed network. Thus, the proposed asynchronous event-triggered tracking scheme can save communicational and computational resources. From the Lyapunov stability theorem, the stability of the entire closed-loop system is analyzed and the comparative simulation results demonstrate the effectiveness of the proposed control strategy.

Finite-Time Passivity and Synchronization of Multi-Weighted Complex Dynamical Networks Under PD Control

This article focuses on finite-time passivity (FTP) and finite-time synchronization (FTS) for complex dynamical networks with multiple state/derivative couplings based on the proportional-derivative (PD) control method. Several criteria of FTP for complex dynamical networks with multiple state couplings (CDNMSCs) are formulated by utilizing the PD controller and constructing an appropriate Lyapunov function. Furthermore, FTP is further used to investigate the FTS in CDNMSCs under the PD controller. In addition, the FTP and FTS for complex dynamical networks with multiple derivative couplings (CDNMDCs) are also studied by exploiting the PD control method and some inequality techniques. Finally, two numerical examples are worked out to demonstrate the validity of the presented PD controllers.

Asymptotically local synchronization in interdependent networks with unidirectional interlinks

Synchronization in complex networks has been investigated for decades. Due to the particularity of the interlinks between networks, the synchronization in interdependent networks has received increasing interest. Since the interlinks are not always symmetric in interdependent networks, we focus on the synchronization in unidirectional interdependent networks to study the control scheme. The mathematical model is put forward and some factors are taken into consideration, such as different coupling functions and strengths. Firstly, the feasibility of the control scheme is proved theoretically by using Lyapunov stability theory and verified by simulations. Then, we find that the synchronization could be maintained in one sub-network by utilizing our control scheme while the nodes in the other sub-network are in chaos. The result indicates that the influence of interlinks can be decreased and the proposed scheme can guarantee the synchronization in one sub-network at least. Moreover, we also discuss the robust of our control scheme against the cascading failure. The scheme is verified by simulations to be effective while the disturbances occur.

Intralayer Synchronization of Multiplex Dynamical Networks via Pinning Impulsive Control

These days, the synchronization of multiplex networks is an emerging and important research topic. Grounded framework and theory about synchronization and control on multiplex networks are yet to come. This article studies the intralayer synchronization on a multiplex network (i.e., a set of networks connected through interlayer edges), via the pinning impulsive control method. The topologies of different layers are independent of each other, and the individual dynamics of nodes in different layers are different as well. Supra-Laplacian matrices are adopted to represent the topological structures of multiplex networks. Two cases are considered according to impulsive sequences of multiplex networks: 1) pinning controllers are applied to all the layers simultaneously at the instants of a common impulse sequence and 2) pinning controllers are applied to each layer at the instants of distinct impulse sequences. Using the Lyapunov stability theory and the impulsive control theory, several intralayer synchronization criteria for multiplex networks are obtained, in terms of the supra-Laplacian matrix of network topology, self-dynamics of nodes, impulsive intervals, and the pinning control effect. Furthermore, the algorithms for implementing pinning schemes at every impulsive instant are proposed to support the obtained criteria. Finally, numerical examples are presented to demonstrate the effectiveness and correctness of the proposed schemes.

Distributed Event-Triggered Control for Cooperative Output Regulation of Multiagent Systems With an Online Estimation Algorithm

In this article, the cooperative output regulation problem of heterogeneous multiagent systems has been investigated. It is assumed that only a few agents could know the system matrix of the exosystem and no agent knows the topological information. Under these conditions, a novel distributed online algorithm is proposed to estimate the information relevant to the topology. Based on this algorithm, a distributed event-triggered adaptive observer is designed such that each agent can observe the exosystem. It is theoretically shown that the proposed distributed controller will make the multiagent system achieve cooperative output regulation asymptotically. Finally, a simulation is presented to show the effectiveness of the result.

The reconstruction on the game networks with binary-state and multi-state dynamics

Reconstruction of network is to infer the relationship among nodes using observation data, which is helpful to reveal properties and functions of complex systems. In view of the low reconstruction accuracy based on small data and the subjectivity of threshold to infer adjacency matrix, the paper proposes two models: the quadratic compressive sensing (QCS) and integer compressive sensing (ICS). Then a combined method (CCS) is given based on QCS and ICS, which can be used on binary-state and multi-state dynamics. It is found that CCS is usually a superior method comparing with compressive sensing, LASSO on several networks with different structures and scales. And it can infer larger node correctly than the other two methods. The paper is conducive to reveal the hidden relationship with small data so that to understand, predicate and control a vast intricate system.

Filtering for Discrete-Time Takagi-Sugeno Fuzzy Nonhomogeneous Markov Jump Systems With Quantization Effects

This article deals with the problem of H_∞ and l₂-l_∞ filtering for discrete-time Takagi-Sugeno fuzzy nonhomogeneous Markov jump systems with quantization effects, respectively. The time-varying transition probabilities are in a polytope set. To reduce conservativeness, a mode-dependent logarithmic quantizer is considered in this article. Based on the fuzzy-rule-dependent Lyapunov function, sufficient conditions are given such that the filtering error system is stochastically stable and has a prescribed H_∞ or l₂-l_∞ performance index, respectively. Finally, a practical example is provided to illustrate the effectiveness of the proposed fuzzy filter design methods.

Adaptive Tracking Control for Perturbed Strict-Feedback Nonlinear Systems Based on Optimized Backstepping Technique

In this article, an adaptive optimized control scheme based on neural networks (NNs) is developed for a class of perturbed strict-feedback nonlinear systems. An optimized backstepping (OB) technique is employed for breaking through the limitation of the matching condition. The disturbance of existing nonlinear systems may degrade system performance or even lead to instability. In order to improve the system's robustness, a disturbance observer is constructed to compensate for the impact coming from the external disturbance. Because the proposed optimized scheme needs to train the adaptive parameters not only for reinforcement learning (RL) but also for the disturbance observer, it will become more challenging no matter designing the control algorithm or deriving the adaptive updating laws. Finally, by virtue of the Lyapunov stability theory, it is proved that all internal signals of the closed-loop systems are semiglobal uniformly ultimately bounded (SGUUB). Simulation results are provided to illustrate the validity of the devised method.

On Consensus Index of Triplex Star-like Networks: A Graph Spectra Approach

In this article, the consensus-related performances of the triplex multi-agent systems with star-related structures, which can be measured by the algebraic connectivity and network coherence, have been studied by the characterization of Laplacian spectra. Some notions of graph operations are introduced to construct several triplex networks with star substructures. The methods of graph spectra are applied to derive the network coherence, and some asymptotic behaviors of the indices have been derived. It is found that the operations of adhering star topologies will make the first-order coherence increase a constant value under the triplex structures as parameters tend to infinity, and the second-order coherence have some equality relations as the node related parameters tend to infinity. Finally, the consensus related indices of the triplex systems with the same number of nodes but non-isomorphic graph structures have been compared and simulated to verify the results.

Finite-Time Passivity and Synchronization of Complex Dynamical Networks With State and Derivative Coupling

In this article, two kinds of complex dynamical networks (CDNs) with state and derivative coupling are investigated, respectively. First, some important concepts about finite-time passivity (FTP), finite-time output strict passivity, and finite-time input strict passivity are introduced. By making use of state-feedback controllers and adaptive state-feedback controllers, several sufficient conditions are given to guarantee the FTP of these two network models. On the other hand, based on the obtained FTP results, some finite-time synchronization criteria for the CDNs with state and derivative coupling are gained. Finally, two simulation examples are proposed to verify the availability of the derived results.

Target Controllability in Multilayer Networks via Minimum-Cost Maximum-Flow Method

In this article, to maximize the dimension of controllable subspace, we consider target controllability problem with maximum covered nodes set in multiplex networks. We call such an issue as maximum-cost target controllability problem. Likewise, minimum-cost target controllability problem is also introduced which is to find minimum covered node set and driver node set. To address these two issues, we first transform them into a minimum-cost maximum-flow problem based on graph theory. Then an algorithm named target minimum-cost maximum-flow (TMM) is proposed. It is shown that the proposed TMM ensures the target nodes in multiplex networks to be controlled with the minimum number of inputs as well as the maximum (minimum) number of covered nodes. Simulation results on Erdős-Rényi (ER-ER) networks, scale-free (SF-SF) networks, and real-life networks illustrate satisfactory performance of the TMM.

Reduced-Order Observer-Based Dynamic Event-Triggered Adaptive NN Control for Stochastic Nonlinear Systems Subject to Unknown Input Saturation

In this article, a dynamic event-triggered control scheme for a class of stochastic nonlinear systems with unknown input saturation and partially unmeasured states is presented. First, a dynamic event-triggered mechanism (DEM) is designed to reduce some unnecessary transmissions from controller to actuator so as to achieve better resource efficiency. Unlike most existing event-triggered mechanisms, in which the threshold parameters are always fixed, the threshold parameter in the developed event-triggered condition is dynamically adjusted according to a dynamic rule. Second, an improved neural network that considers the reconstructed error is introduced to approximate the unknown nonlinear terms existed in the considered systems. Third, an auxiliary system with the same order as the considered system is constructed to deal with the influence of asymmetric input saturation, which is distinct from most existing methods for nonlinear systems with input saturation. Assuming that the partial state is unavailable in the system, a reduced-order observer is presented to estimate them. Furthermore, it is theoretically proven that the obtained control scheme can achieve the desired objects. Finally, a one-link manipulator system and a three-degree-of-freedom ship maneuvering system are presented to illustrate the effectiveness of the proposed control method.

Cooperation in Social Dilemmas: A Group Game Model with Double-Layer Networks

The combination of complex networks and game theory is one of the most suitable ways to describe the evolutionary laws of various complex systems. In order to explore the evolution of group cooperation in multiple social dilemmas, a model of a group game with a double-layer network is proposed here. Firstly, to simulate a multiplayer game under multiple identities, we combine a double-layer network and public goods game. Secondly, in order to make an individual’s strategy selection process more in line with a practical context, a new strategy learning method that incorporates individual attributes is designed here, referred to as a “public goods game with selection preferences” (PGG-SP), which makes strategic choices that are more humane and diversified. Finally, a co-evolution mechanism for strategies and topologies is introduced based on the double-layer network, which effectively explains the dynamic game process in real life. To verify the role of multiple double-layer networks with a PGG-SP, four types of double-layer networks are applied in this paper. In addition, the corresponding game results are compared between single-layer, double-layer, static, and dynamic networks. Accordingly, the results show that double-layer networks can facilitate cooperation in group games.

Adaptive Bipartite Event-Triggered Output Consensus of Heterogeneous Linear Multiagent Systems Under Fixed and Switching Topologies

This article addresses the adaptive bipartite event-triggered output consensus issue for heterogeneous linear multiagent systems. We consider both cooperative interaction and antagonistic interaction between neighbor agents in both fixed and switching topologies. An adaptive bipartite compensator consisting of time-varying coupling weights and dynamic event-triggered mechanism is first proposed to estimate the leader's state in a fully distributed manner. Different from the existing methods, the proposed compensator has three advantages: 1) it does not depend on any global information of the network graph; 2) it avoids the continuous communication between neighbor agents; and 3) it is applicable for the signed communication topology. Assume that the system states are unmeasurable, and we thus design the state observer. Based on the devised compensator and observer, the distributed control law is developed such that the bipartite event-triggered output consensus problem can be achieved. Moreover, we extend the results in fixed topology to switching topology, which is more challenging in that state estimation is updated in two cases: 1) the interaction graph is switched or 2) the event-triggered mechanism is satisfied. It is proven that no agent exhibits Zeno behavior in both fixed and switching interaction topologies. Finally, two examples are provided to illustrate the feasibility of the theoretical results.

Mixed H2/H∞ State Estimation for Discrete-Time Switched Complex Networks With Random Coupling Strengths Through Redundant Channels

This article investigates the mixed H₂/H_∞ state estimation problem for a class of discrete-time switched complex networks with random coupling strengths through redundant communication channels. A sequence of random variables satisfying certain probability distributions is employed to describe the stochasticity of the coupling strengths. A redundant-channel-based data transmission mechanism is adopted to enhance the reliability of the transmission channel from the sensor to the estimator. The purpose of the addressed problem is to design a state estimator for each node, such that the error dynamics achieves both the stochastic stability (with probability 1) and the prespecified mixed H₂/H_∞ performance requirement. By using the switched system theory, an extensive stochastic analysis is carried out to derive the sufficient conditions ensuring the stochastic stability as well as the mixed H₂/H_∞ performance index. The desired state estimator is also parameterized by resorting to the solutions to certain convex optimization problems. A numerical example is provided to illustrate the validity of the proposed estimation scheme.

Distributed Sliding-Mode Tracking Control of Second-Order Nonlinear Multiagent Systems: An Event-Triggered Approach

The event-triggered tracking control problem of second-order multiagent systems in consideration of system nonlinearities is investigated by utilizing the distributed sliding-mode control (SMC) approach. An event-triggered strategy is proposed to decrease the controller sampling frequency and save the network communication resources; the triggering condition is then established for leader-following multiagent systems. In this article, by utilizing the distributed event-based sliding-mode controller, the system state of second-order multiagent systems with system nonlinearities is capable of approaching the integral sliding-mode surface in finite time. A novel integral sliding-mode surface is constructed in this article to guarantee the consensus tracking performance in the existence of system nonlinearities as the state trajectories of second-order integrator systems move on the constructed sliding manifold. By employing the Lyapunov approach, sufficient conditions are deduced to ensure that the consensus tracking performance is obtained for the closed-loop system. Furthermore, it is presented that the triggering scheme can effectively reduce state updates and eliminate the Zeno behavior. A simulation example is provided to testify the validity of our proposed methodology.

Trajectory Tracking on Uncertain Complex Networks via NN-Based Inverse Optimal Pinning Control

A new approach for trajectory tracking on uncertain complex networks is proposed. To achieve this goal, a neural controller is applied to a small fraction of nodes (pinned ones). Such controller is composed of an on-line identifier based on a recurrent high-order neural network, and an inverse optimal controller to track the desired trajectory; a complete stability analysis is also included. In order to verify the applicability and good performance of the proposed control scheme, a representative example is simulated, which consists of a complex network with each node described by a chaotic Lorenz oscillator.

Finite-Time Synchronization and H∞ Synchronization of Multiweighted Complex Networks With Adaptive State Couplings

In this paper, two kinds of multiweighted and adaptive state coupled complex networks (CNs) with or without coupling delays are presented. First, we develop the appropriate state feedback controller and adaptive law for the sake of guaranteeing that the proposed network models without coupling delays can be finite-timely synchronized and H_∞ synchronized. Furthermore, for the multiweighted CNs with coupling delays and adaptive state couplings, some finite-time synchronization and H_∞ synchronization criteria are presented by choosing the appropriate adaptive law and controllers. Eventually, we give two numerical simulations to verify the validity of the theoretical results.

Cluster Synchronization for Neutral Stochastic Delay Networks via Intermittent Adaptive Control

This paper studies the problem of cluster synchronization at exponential rates in both the mean square and almost sure senses for neutral stochastic coupled neural networks with time-varying delay via a periodically intermittent pinning adaptive control strategy. The network topology can be symmetric or asymmetric, with each network node being described by neutral stochastic delayed neural networks. When considering the exponential stabilization in the mean square sense for neutral stochastic delay system, the delay integral inequality approach is used to circumvent the obstacle arising from the coexistence of random disturbance, neutral item, and time-varying delay. The almost surely exponential stabilization is also analyzed with the nonnegative semimartingale convergence theorem. Sufficient criteria on cluster synchronization at exponential rates in both the mean square and almost sure senses of the underlying networks under the designed control scheme are derived. The effectiveness of the obtained theoretical results is illustrated by two examples.

Finite-Time Passivity and Synchronization of Coupled Reaction-Diffusion Neural Networks With Multiple Weights

In this paper, two multiple weighted coupled reaction-diffusion neural networks (CRDNNs) with and without coupling delays are introduced. On the one hand, some finite-time passivity (FTP) concepts are proposed for the spatially and temporally system with different dimensions of output and input. By choosing appropriate Lyapunov functionals and controllers, several sufficient conditions are presented to ensure the FTP of these CRDNNs. On the other hand, the finite-time synchronization (FTS) problem is also discussed for the multiple weighted CRDNNs with and without coupling delays, respectively. Finally, two numeral examples with simulation results are provided to verify the effectiveness of the obtained FTP and FTS criteria.

Adaptive Pinning Synchronization of Fractional Complex Networks with Impulses and Reaction–Diffusion Terms

In this paper, a class of fractional complex networks with impulses and reaction–diffusion terms is introduced and studied. Meanwhile, a class of more general network structures is considered, which consists of an instant communication topology and a delayed communication topology. Based on the Lyapunov method and linear matrix inequality techniques, some sufficient criteria are obtained, ensuring adaptive pinning synchronization of the network under a designed adaptive control strategy. In addition, a pinning scheme is proposed, which shows that the nodes with delayed communication are good candidates for applying controllers. Finally, a numerical example is given to verify the validity of the main results.

Synchronization of interconnected heterogeneous networks: The role of network sizes

Increasing evidence shows that real networks interact with each other, forming a network of networks (NONs). Synchronization, a ubiquitous process in natural and engineering systems, has fascinatingly gained rising attentions in the context of NONs. Despite efforts to study the synchronization of NONs, it is still a challenge to understand how do the network sizes affect the synchronization and its phase diagram of NONs coupled with nonlinear dynamics. Here, we model such NONs as star-like motifs to analytically derive the critical values of both the internal and the external coupling strengths, at which a phase transition from synchronization to incoherence occurs. Our results show that the critical values strongly depend on the network sizes. Reducing the difference between network sizes will enhance the synchronization of the whole system, which indicates the irrationality of previous studies that assume the network sizes to be the same. The optimal connection strategy also changes as the network sizes change, a discovery contradicting to the previous conclusion that connecting the high-degree nodes of each network is always the most effective strategy to achieve synchronization unchangeably. This finding emphasizes the crucial role of network sizes which has been neglected in the previous studies and could contribute to the design of a global synchronized system.

Generalized State Estimation for Markovian Coupled Networks Under Round-Robin Protocol and Redundant Channels

In this paper, the problem of generalized state estimation for an array of Markovian coupled networks under the round-Robin protocol (RRP) and redundant channels is investigated by using an extended dissipative property. The randomly varying coupling of the networks under consideration is governed by a Markov chain. With the aid of using the RRP, the transmission order of nodes is availably orchestrated. In this case, the probability of occurrence data collisions through a shared constrained network may be reduced. The redundant channels are also used in the signal transmission to deal with the frangibility of networks caused by a single channel in the networks. The network induced phenomena, that is, randomly occurring packet dropouts and randomly occurring quantization are fully considered. The main purpose of the research is to find a desired estimator design approach such that the extended (Ω₁,Ω₂,Ω₃) - γ -stochastic dissipativity property of the estimation error system is guaranteed. In terms of the Lyapunov-Krasovskii methodology, the Kronecker product and an improved matrix decoupling approach, sufficient conditions for such an addressed problem are established by means of handling some convex optimization problems. Finally, the serviceability of the proposed method is explained by providing an illustrated example.

Distributed Event-Triggered Adaptive Control for Cooperative Output Regulation of Heterogeneous Multiagent Systems Under Switching Topology

This paper investigates the cooperative output regulation problem for heterogeneous multiagent systems (MASs) under switching topology. Two novel distributed event-triggered adaptive control strategies based on state feedback and output feedback are developed, which can avoid using the minimal nonzero eigenvalue of Laplacian matrix associated with global system topologies. It is shown that under the proposed control protocols, MASs could achieve asymptotic tracking and disturbance rejection, and meanwhile, the amount of transmission data and communication cost among agents can be reduced. Then, the leader-following consensus problem of MASs is given as an application of our main results. Finally, an example is presented to verify the effectiveness of the proposed control schemes.

Improved Stability and Stabilization Results for Stochastic Synchronization of Continuous-Time Semi-Markovian Jump Neural Networks With Time-Varying Delay

Continuous-time semi-Markovian jump neural networks (semi-MJNNs) are those MJNNs whose transition rates are not constant but depend on the random sojourn time. Addressing stochastic synchronization of semi-MJNNs with time-varying delay, an improved stochastic stability criterion is derived in this paper to guarantee stochastic synchronization of the response systems with the drive systems. This is achieved through constructing a semi-Markovian Lyapunov-Krasovskii functional together as well as making use of a novel integral inequality and the characteristics of cumulative distribution functions. Then, with a linearization procedure, controller synthesis is carried out for stochastic synchronization of the drive-response systems. The desired state-feedback controller gains can be determined by solving a linear matrix inequality-based optimization problem. Simulation studies are carried out to demonstrate the effectiveness and less conservatism of the presented approach.

Global Pinning Synchronization of Complex Networks With Sampled-Data Communications

This paper investigates the global pinning synchronization problem for a class of complex networks with aperiodic samplings. Combined with the Writinger-based integral inequality, a new less conservative criterion is presented to guarantee the global pinning synchronization of the complex network. Furthermore, a novel condition is proposed under which the complex network is globally pinning synchronized with a given performance index. It is shown that the performance index has a positive correlation with the upper bound of the sampling intervals. Finally, the validity and the advantage of the theoretic results obtained are verified by means of the applications in Chua's circuit and pendulum.

Asynchronous State Estimation for Discrete-Time Switched Complex Networks With Communication Constraints

This paper is concerned with the asynchronous state estimation for a class of discrete-time switched complex networks with communication constraints. An asynchronous estimator is designed to overcome the difficulty that each node cannot access to the topology/coupling information. Also, the event-based communication, signal quantization, and the random packet dropout problems are studied due to the limited communication resource. With the help of switched system theory and by resorting to some stochastic system analysis method, a sufficient condition is proposed to guarantee the exponential stability of estimation error system in the mean-square sense and a prescribed performance level is also ensured. The characterization of the desired estimator gains is derived in terms of the solution to a convex optimization problem. Finally, the effectiveness of the proposed design approach is demonstrated by a simulation example.

Variance-Constrained State Estimation for Nonlinearly Coupled Complex Networks

This paper studies the state estimation problem for nonlinearly coupled complex networks. A variance-constrained state estimator is developed by using the structure of the extended Kalman filter, where the gain matrix is determined by optimizing an upper bound matrix for the estimation error covariance despite the linearization errors and coupling terms. Compared with the existing estimators for linearly coupled complex networks, a distinct feature of the proposed estimator is that the gain matrix can be derived separately for each node by solving two Riccati-like difference equations. By using the stochastic analysis techniques, sufficient conditions are established which guarantees the state estimation error is bounded in mean square. A numerical example is provided to show the effectiveness and applicability of the proposed estimator.

Containment Control for Second-Order Multiagent Systems Communicating Over Heterogeneous Networks

The containment control is studied for the second-order multiagent systems over a heterogeneous network where the position and velocity interactions are different. We consider three cases that multiple leaders are stationary, moving at the same constant speed, and moving at the same time-varying speed, and develop different containment control algorithms for each case. In particular, for the former two cases, we first propose the containment algorithms based on the well-established ones for the homogeneous network, for which the position interaction topology is required to be undirected. Then, we extend the results to the general setting with the directed position and velocity interaction topologies by developing a novel algorithm. For the last case with time-varying velocities, we introduce two algorithms to address the containment control problem under, respectively, the directed and undirected interaction topologies. For most cases, sufficient conditions with regard to the interaction topologies are derived for guaranteeing the containment behavior and, thus, are easy to verify. Finally, six simulation examples are presented to illustrate the validity of the theoretical findings.

Minimal-Approximation-Based Distributed Consensus Tracking of a Class of Uncertain Nonlinear Multiagent Systems With Unknown Control Directions

A minimal-approximation-based distributed adaptive consensus tracking approach is presented for strict-feedback multiagent systems with unknown heterogeneous nonlinearities and control directions under a directed network. Existing approximation-based consensus results for uncertain nonlinear multiagent systems in lower-triangular form have used multiple function approximators in each local controller to approximate unmatched nonlinearities of each follower. Thus, as the follower's order increases, the number of the approximators used in its local controller increases. However, the proposed approach employs only one function approximator to construct the local controller of each follower regardless of the order of the follower. The recursive design methodology using a new error transformation is derived for the proposed minimal-approximation-based design. Furthermore, a bounding lemma on parameters of Nussbaum functions is presented to handle the unknown control direction problem in the minimal-approximation-based distributed consensus tracking framework and the stability of the overall closed-loop system is rigorously analyzed in the Lyapunov sense.

A Layered Event-Triggered Consensus Scheme

This paper studies the dynamics of multiagent systems with a multilayer structure, proposing a novel layered event-triggered scheme (LETS) for consensus. This LETS emphasizes on synchronous information transmission in the same layer but asynchronous message update between different layers, which differs from the existing centralized or distributed event-triggered schemes. Moreover, under the LETS, agents in different layers achieve asymptotical consensus eventually and the Zeno behavior is successfully eliminated. Furthermore, an algorithm is provided to avoid continuous event detection, verified by a numerical example.