The Immense Impact of Reverse Edges on Large Hierarchical Networks

Hierarchical networks are frequently encountered in animal groups,gene networks,and artificial engineering systems such as multiple robots,unmanned vehicle systems,smart grids,wind farm networks,and so forth.The structure of a large directed hierarchical network is often strongly influenced by reverse edges from lower-to higher-level nodes,such as lagging birds’howl in a flock or the opinions of lowerlevel individuals feeding back to higher-level ones in a social group.This study reveals that,for most large-scale real hierarchical networks,the majority of the reverse edges do not affect the synchronization process of the entire network;the synchronization process is influenced only by a small part of these reverse edges along specific paths.More surprisingly,a single effective reverse edge can slow down the synchronization of a huge hierarchical network by over 60%.The effect of such edges depends not on the network size but only on the average in-degree of the involved subnetwork.The overwhelming majority of active reverse edges turn out to have some kind of“bunching”effect on the information flows of hierarchical networks,which slows down synchronization processes.This finding refines the current understanding of the role of reverse edges in many natural,social,and engineering hierarchical networks,which might be beneficial for precisely tuning the synchronization rhythms of these networks.Our study also proposes an effective way to attack a hierarchical network by adding a malicious reverse edge to it and provides some guidance for protecting a network by screening out the specific small proportion of vulnerable nodes.

An Incentive Mechanism for Federated Learning:A Continuous Zero-Determinant Strategy Approach

As a representative emerging machine learning technique, federated learning(FL) has gained considerable popularity for its special feature of “making data available but not visible”. However, potential problems remain, including privacy breaches, imbalances in payment, and inequitable distribution.These shortcomings let devices reluctantly contribute relevant data to, or even refuse to participate in FL. Therefore, in the application of FL, an important but also challenging issue is to motivate as many participants as possible to provide high-quality data to FL. In this paper, we propose an incentive mechanism for FL based on the continuous zero-determinant(CZD) strategies from the perspective of game theory. We first model the interaction between the server and the devices during the FL process as a continuous iterative game. We then apply the CZD strategies for two players and then multiple players to optimize the social welfare of FL, for which we prove that the server can keep social welfare at a high and stable level. Subsequently, we design an incentive mechanism based on the CZD strategies to attract devices to contribute all of their high-accuracy data to FL.Finally, we perform simulations to demonstrate that our proposed CZD-based incentive mechanism can indeed generate high and stable social welfare in FL.

A survey on delayed feedback control of chaos

This paper introduces the basic idea and provides the mathematical formulation of the delayed feedback control （DFC） methodology, which has been widely used in chaos control. Stability analysis including the well-known odd number linfitation of the DFC is reviewed. Some new developments in characterizing the limitation of the DFC are presented. Various modified DFC methods, which are developed in order to overcome the odd number limitation, are also described. Finally, some open problems in this research field are discussed.

Evolving networks: from topology to dynamics

A multi-local-world model is introduced to describe the evolving networks that have a localization property such as the Internet. Based on this model, we show that the traffic load defined by 'betweenness centrality' on the multi-local-world scale-free networks' model also follows a power law form. In this kind of network, a few vertices have heavier loads and so play more important roles than the others in the network.

Searching for Best Network Topologies with Optimal Synchronizability:A Brief Review

The Laplacian eigenvalue spectrum of a complex network contains a great deal of information about the network topology and dynamics,particularly affecting the network synchronization process and performance.This article briefly reviews the recent progress in the studies of network synchronizability,regarding its spectral criteria and topological optimization,and explores the role of higher-order topologies in measuring the optimal synchronizability of large-scale complex networks.

Controllability robustness of complex networks

This article presents an overview on the state-of-the-art development in complex network controllability and its robustness against malicious attacks and random failures.Specifically,it first reviews the concepts of network pinning control and controllability,and then discusses the network controllability robustness against destructive attacks by means of node-and/or edge-removal.The related issue of network connectivity robustness is also discussed.To that end,it furthermore provides an brief overview on the recent development of a machine-learning approach for predicting optimal network controllability robustness,which may shed some lights on the understanding of optimal network structures for various design considerations.

Periodicity in Delta-modulated feedback control

The Delta-modulated feedback control of a linear system introduces nonlinearity into the system through switchings between two input values. It has been found that Delta-modulation gives rise to periodic orbits. The existence of periodic points of all orders of Sigma-Delta modulation with ＂leaky＂ integration is completely characterized by some interesting groups of polynomials with ＂sign＂ coefficients. The results are naturally generalized to Sigma-Delta modulations with multiple delays, Delta-modulations in the ＂downlink＂, unbalanced Delta-modulations and systems with two-level quantized feedback. Further extensions relate to the existence of periodic points arising from Delta-modulated feedback control of a stable linear system in an arbitrary direction, for which some necessary and sufficient conditions are given.

Discretization behavior analysis of a switching control system from a unified mathematical approach

A useful unified analysis framework is proposed for exploring the intriguing behaviors of a second-order switching control system. Complex discretization behaviors of the switching control system are explored in detail, and some intrinsic relationships between the system periodic behaviors and their associated symbolic sequences are studied. Keywords Switching control - Chaos - Discretisation - Periodicity This work was supported by the Australian Research Council and the Hong Kong Research Grants Council for their financial supports, under the CERG Grants CityU 1018/01E, 1004/02E, and 1115/03E.

Synchronization of general discrete Lur'e systems

This paper studies a master-slave type of synchronization systems for a general form of two discrete Lur＇e systems. Some simple necessary and/or sufficient conditions for synchronization are derived. They are basically algebraic conditions, and are convement to be applied in engineering applications.

Transitivity and Chaoticity in 1-D Cellular Automata

Recent progress in symbolic dynamics of cellular automata (CA) shows that many CA exhibit rich and complicated Bernoulli-shift properties, such as positive topological entropy, topological transitivity and even mixing. Noticeably, some CA are only transitive, but not mixing on their subsystems. Yet, for one-dimensional CA, this paper proves that not only the shift transitivity guarantees the CA transitivity but also the CA with transitive non-trivial Bernoulli subshift of finite type have dense periodic points. It is concluded that, for one-dimensional CA, the transitivity implies chaos in the sense of Devaney on the non-trivial Bernoulli subshift of finite types.

Computing Persistent Homology by Spanning Trees and Critical Simplices

Topological data analysis can extract effective information from higher-dimensional data.Its mathematical basis is persistent homology.The persistent homology can calculate topological features at different spatiotemporal scales of the dataset,that is,establishing the integrated taxonomic relation among points,lines,and simplices.Here,the simplicial network composed of all-order simplices in a simplicial complex is essential.Because the sequence of nested simplicial subnetworks can be regarded as a discrete Morse function from the simplicial network to real values,a method based on the concept of critical simplices can be developed by searching all-order spanning trees.Employing this new method,not only the Morse function values with the theoretical minimum number of critical simplices can be obtained,but also the Betti numbers and composition of all-order cavities in the simplicial network can be calculated quickly.Finally,this method is used to analyze some examples and compared with other methods,showing its effectiveness and feasibility.

A Linear Programming-Based Reinforcement Learning Mechanism for Incomplete-Information Games

Dear Editor,Recently, with the development of artificial intelligence, game intelligence decision-making has attracted more and more attention.In particular, incomplete-information games(IIG) have gradually become a new research focus, where players make decisions without sufficient information, such as the opponent's strategies or preferences.

Pinning Control and Controllability of Complex Dynamical Networks

In this article, the notion of pinning control for directed networks of dynamical systems is introduced, where the nodes could be either single-input single-output （SISO） or multi-input multi-output （MIMO） dynamical systems, and could be non-identical and nonlinear in general but will be specified to be identical linear time-invariant （LTI） systems here in the study of network controllability. Both state and structural controllability problems will be discussed, illustrating how the network topology, node-system dynamics, external control inputs and inner dynamical interactions altogether affect the controllability of a general complex network of LTI systems, with necessary and sufficient conditions presented for both SISO and MIMO settings. To that end, the controllability of a special temporally switching directed network of linear time-varying （LTV） node systems will be addressed, leaving some more general networks and challenging issues to the end for research outlook.

Enhancing Controllability Robustness of q-Snapback Networks through Redirecting Edges

Te well-known small-world network model was established by randomly rewiring edges,aiming to enhance the synchronizability of an undirected nearest-neighbor regular network.Tis paper demonstrates via extensive numerical simulations that randomly redirecting edges could enhance the robustness of the network controllability for directed snapback networks against both random and intentional node-removal and edge-removal attacks.

Pinning synchronization of networked multi-agent systems: spectral analysis

Pinning synchronization of a networked multi-agent system with a directed communication topology is investigated from a spectral analysis approach. Some new types of synchronized regions for networked systems with different nonlinear agent dynamics and inner coupling structures are discovered. The eigenvalue distributions of the coupling and control matrices for different types of directed networks are obtained. The effects of the network topology, pinning density and pinning strength on the network synchronizability are examined through extensive numerical simulations. It is shown that the synchronizability of the pinned network can be effectively improved by increasing pinning density and pinning strength for some types of synchronized regions, whereas too large the pinning density and pinning strength will lead to desynchronization for other types. It is found that directed random networks are not always easier to synchronize than directed small-world networks, and a denser eigenvalue distribution may not always imply better synchronizability.

Distributed filtering and control of complex networks and systems

In the last two decades,research on distributed filtering and control(FAC)has burgeoned into large-scale complex networks and systems.A distributed networked system,typically composed of a number of cost-effective dynamical nodes(agents)with limited sensing,computing,and communication capabilities,is gaining considerable popularity due to its low power consumption,simple installation,high performance,and strong reliability,as compared with the centralized setting.As a result,various networkinduced FAC systems,techniques,and algorithms have emerged,including node registration and control,network consensus and synchronization,multi-sensor data clustering/fusion,network topological design and analysis,and the like.

Indirect effects among biodiversity loss of mutualistic ecosystems

Drastic reduction in biodiversity has been a severe threat to ecosystems,which is exacerbated when losing few species leads to disastrous and even irreparable consequences.Therefore,revealing the mechanism underlying biodiversity loss is of uttermost importance.In this study,we show that abundant indirect interactions among mutualistic ecosystems are critical in determining species’status.Combining topological and ecological characteristics,we propose an indicator derived from a dynamic model to identify keystone species and quantify their influence,which outperforms widely-used indicators like degree in realistic and simulated networks.Furthermore,we demonstrate that networks with high modularity,heterogeneity,biodiversity,and less intimate interactions tend to have larger indirect effects,which are more amenable in predicting decline of biodiversity with the proposed indicator.These findings shed some light onto the influence of apposite biodiversities,paving the way from complex network theory to ecosystem protection and restoration.

Indirect effects among biodiversity loss of mutualistic ecosystems

Drastic reduction in biodiversity has been a severe threat to ecosystems,which is exacerbated when losing few species leads to disastrous and even irreparable consequences.Therefore,revealing the mechanism underlying biodiversity loss is of uttermost importance.In this study,we show that abundant indirect interactions among mutualistic ecosystems are critical in determining species’status.Combining topological and ecological characteristics,we propose an indicator derived from a dynamic model to identify keystone species and quantify their influence,which outperforms widely-used indicators like degree in realistic and simulated networks.Furthermore,we demonstrate that networks with high modularity,heterogeneity,biodiversity,and less intimate interactions tend to have larger indirect effects,which are more amenable in predicting decline of biodiversity with the proposed indicator.These findings shed some light onto the influence of apposite biodiversities,paving the way from complex network theory to ecosystem protection and restoration.