Asymptotically local synchronization in interdependent networks with unidirectional interlinks

This paper focuses on event‐triggered sampled‐data synchronization of uncertain complex networks with time‐varying coupled delays. First of all, a discrete event‐triggered sampled‐data control scheme is adopted, which not only makes the state of the system be monitored in discrete time, but also the sampling information is effectively transmitted. The proposed event‐triggered mechanism effectively prevents Zeno behavior. In addition, we also use some novel piecewise time‐dependent Lyapunov–Krasovskii and Wirtinger inequality to handle the time‐varying delays and parameter uncertainties of complex networks. Then, synchronization criteria are given for uncertain complex networks. Finally, the simulation results show that the control scheme can significantly reduce the number of transmitted signals while maintaining the uncertain complex networks synchronization.

Synchronization in complex networks is influenced by higher-order interactions and non-Gaussian perturbations, yet their mechanisms remain unclear. We investigate the synchronization and spike dynamics in a higher-order Kuramoto model subjected to Lévy noise. Using the mean order parameter, mean first-passage time, and basin stability, we identify boundaries distinguishing synchronization and incoherence. The stability index governs the tail heaviness of the probability density function for Lévy noise, while the scale parameter affects the magnitude. Synchronization weakens as the stability index decreases, and even completely disappears when the scale parameter exceeds a critical threshold. By varying coupling, we find bifurcations and hysteresis. Lévy noise smooths the synchronization transitions and requires stronger coupling compared to Gaussian white noise. We then define spikes as extreme excursions of the order parameter and study their statistical and spectral properties. The maximum number of spikes is observed at small-scale parameters. A generalized spectral analysis based on an edit distance algorithm measures the similarity between spike sequences and identifies spike patterns. These findings deepen the understanding of synchronization and extreme events in complex networks driven by non-Gaussian noise. Synchronization in complex networks, crucial for understanding phenomena in systems like neuronal networks and power grids, is affected by higher-order interactions and non-Gaussian perturbations. The authors explore a higher-order Kuramoto model with Lévy noise, revealing how stability and scale parameters influence synchronization, offering insights into network dynamics and extreme events.

Critical infrastructures, including those that concern the nation's economy and security such as electrical power systems, water distribution systems and transportation systems, are becoming more and more interdependent with each other. Although they bring unprecedented improvements on efficiency and flexibility, the interdependent relations enable failures in one network to propagate and impact the performance of other coupled networks. Cascading failures is one such phenomenon that creates dramatic damages to critical infrastructures, where a small initial shock can get escalated due to the intricate dependencies and result in system-wide collapses. This dissertation aims to understand and mitigate the root cause of the seemingly unexpected large-scale cascading failures by characterizing and modeling the inherent dependencies between and within different networks. A main finding is that allocating the available redundancies uniformly across the system maximizes the robustness against random failures. We support this thesis statement with different networks and attack types: flow-carrying networks under random and targeted attack, interdependent flow-carrying networks under random attacks, and interdependent cyber-physical networks under random attacks. In the flow redistribution network, we propose a global and equal flow redistribution model to capture the cascading failure dynamics. In the case of random attacks, we derive the final system size and critical attack size, and prove that the optimal robustness is reached when system redundancy is allocated uniformly. For targeted attacks, we propose the optimization problem of finding the best k lines to attack so as to minimize the number of alive lines at the steady-state, to reveal the worst-case attack vulnerability of the system. In interdependent flow-carrying networks, we study a model where the flow of a failed line is redistributed partially within the network that the failed line belongs to, with the rest being shed to other coupled networks. Analyzing the cascading failures in this model, we show that interdependence has a multi-faceted impact on system robustness in that as the level of coupling increases, the chance for both networks to survive or collapse concurrently increases, whereas it becomes more difficult for each component network to survive on its own. To understand the robustness of integrated cyber-physical systems (CPSs), we develop a novel interdependent system model to capture the inherently different failure cascade characteristics of each component network; i.e., the cyber and the physical networks are governed by different cascade rules to be able to function. We demonstrate the ability of our model to capture the unexpected nature of large-scale cascading failures in CPSs, and provide insights on improving system robustness by proposing optimal redundancy allocation schemes.

The synchronization in complex network model with time-delay nodes is investigated. The model is applicable for two sorts of complex networks. With linear feedback controller and adaptive feedback controller, positive definite functions are designed respectively. The sufficient conditions of synchronization are given by Lyapunov stability theory. Finally, numerical simulations show the effectiveness of the synchronization in complex networks.

Fixed-time synchronization of discontinuous complex networks with time-varying delay and disturbances via aperiodically semi-intermittent control is investigated in this paper. Firstly, a novel lemma is introduced to prove the fixed-time stability. Then, based on the new lemma, we design a controller with less terms than that in existing literature in work time (control time) and rest time, deriving some sufficient conditions to realize synchronization of a discontinuous complex network with disturbances within a settling time. Finally, a numerical example is provided to illustrate the correctness and effectiveness of our main result.

This paper concentrates on the cluster synchronization such that each community achieves linear-generalized synchronization in complex networks. Here two control schemes are introduced to realize cluster synchronization. One is to adjust the coupling strength of edges which connect different groups automatically, and to use the pinning control simultaneously. The other scheme is to adjust the coupling strength of edges in each group and to select only one node in each group to control. Finally, we present some simulations to demonstrate the correction of our conclusion.

In this paper, pinning synchronization of complex networks is investigated from a moment‐based analysis approach. The analytical expressions for the first three expected moments of the coupling and control matrix for an ER random graph, a Chung‐Lu random network, and an NW small‐world network are derived, respectively. The shape of the eigenvalue distribution of the coupling and control matrix can be estimated based on the obtained expected moments and thus used to predict pinning synchronization in the network. It is shown that the expected moments are associated with the local structural properties of the network and the control mechanism. This allows us to bound the smallest and largest eigenvalues of the spectral distribution without computing the Laplacian eigenvalues of a large‐scale network. Numerical simulations of the above three representative networks composing of chaotic systems, respectively, are shown for illustration and verification.

With continuous deepening of researches on complex network problems, the cascading failure problem of interdependent networks becomes one of the hot research topics in this field at present. This paper establishes a cascading failure model of interdependent networks based on a load-capacity model. And then we focus on the robustness problem of interdependent networks under random attacks based on four different network combinations. At last, an interdependent coupling network constituted of two networks with the node quantity of ten is taken as the example to verify conclusions obtained in this paper. Research results show that under random attacks, robustness of the asymmetric interdependent networks is similar with the previous research results, namely, robustness of the networks decreases with the increase of removal proportion and increases with the increase of tolerance coefficient. In addition, BA-BA interdependent networks and disassortative link networks perform better than other network combinations and coupling manners.

This document contains corrections to the paper entitled “Contraction Theory and the Master Stability Function: Linking Two Approaches to Study Synchronization in Complex Networks.” The results reported here should substitute those included in Section V-B therein.

In this paper, it is mainly concerned with the issue of mean square modified function projective synchronization of uncertain complex network with multi-links and stochastic perturbations. The uncertain terms are the unknown topological structure which exists inside the complex network and the external noise is imposed to the network. In order to achieve the mean square modified function projective synchronization of complex dynamical network, a strict feedback controller and novel adaptive control laws are designed based on Lyapunov stability theory, inequality techniques, and the properties of the Weiner process to obtain several novel synchronization criterion. Furthermore, the effectiveness of the proposed control scheme is validated by numerical simulations.

This paper investigates the stabilization and synchronization of positive complex networks with dynamic links. A class of positive complex networks is constructed by introducing dynamic link between nodes. A controller and the corresponding coupling term with respect to links are designed to achieve the positivity and stability of complex networks and link systems, respectively. Then, a synchronization controller and the corresponding coupling term with respect to links and node states are proposed, respectively. The main contributions are as follows: (i) A novel coupling term is constructed to achieve stability and synchronization of positive complex networks, (ii) A stabilization and synchronization framework is constructed by designing two classes of controllers and coupling terms, and (iii) A tractable design, analysis, and computation method is introduced by virtue of linear programming and copositive Lyapunov function. Finally, a simulation example is provided to verify the effectiveness and feasibility of the proposed approaches.

Cascading failures take place between networks after initial failures occur in interdependent networks. Most previous studies are limited to undirected networks, while many real-world networks are directed. For interdependent directed networks, earlier studies focused on random initial failures. How the interdependent directed networks respond to targeted attacks has not been addressed. In this work, we develop a general theoretical framework for analyzing the robustness in interdependent directed networks under targeted attack on nodes with high or low degrees (in-degrees or out-degrees). We find that for degree heterogeneous networks, targeted attacks perform more efficiently in breaking down the system than random failures, with the degree-based attacks being the most efficient way. Interestingly, we find that system robustness first increases then decreases as the degree heterogeneity increases, showing a peak, which helps understand system robustness and enables the better design of robust infrastructure systems.

This article is concerned with the problem of fixed-time (FXT) and preassigned-time (PAT) synchronization for discontinuous dynamic networks by improving FXT stability and developing simple control schemes. First, some more relaxed conditions for FXT stability are established and several more accurate estimates for the settling time (ST) are obtained by means of some special functions. Based on the improved FXT stability, FXT synchronization for discontinuous networks is discussed by designing a simple controller without a linear feedback term. Besides, the PAT synchronization is also explored by developing several nontrivial control protocols with finite control gains, where the synchronized time can be prespecified according to actual needs and is irrelevant with any initial value and any parameter. Finally, the improved FXT stability and the synchronization for complex networks are confirmed by two numerical examples.

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.

Adaptive finite-time projective synchronization of complex networks with nonidentical nodes and quantized time-varying delayed coupling