Networks Of Dynamical Systems Research Articles

Competitive Epidemic Spreading Over Networks

In this letter, we consider an epidemic model for two competitive viruses spreading over a metapopulation network, termed the ‘bivirus model’ for convenience. The dynamics are described by a networked continuous-time dynamical system, with each node representing a population and edges representing infection pathways for the viruses. We survey existing results on the bivirus model beginning with the nature of the equilibria, including whether they are isolated, and where they exist within the state space with the corresponding interpretation in the context of epidemics. We identify key convergence results, including the conclusion that for generic system parameters, global convergence occurs for almost all initial conditions. Conditions relating to the stability properties of various equilibria are also presented. In presenting these results, we also recall some of the key tools and theories used to secure them. We conclude by discussing the various open problems, ranging from control and network optimization, to further characterization of equilibria, and finally extensions such as modeling three or more viruses.

A Novel Model-Assisted Decentralized Multi-Agent Reinforcement Learning for Joint Optimization of Hybrid Beamforming in Massive MIMO mmWave Systems

To meet the explosive increasing demands for high data rates, hybrid beamforming has become a promising paradigm for massive MIMO millimeter wave (mmWave) systems. In this paper, we study the joint optimization of hybrid beamforming for maximizing the achievable sum rate in networked dynamic massive MIMO mmWave systems. To address this problem, we propose a novel Model-Assisted Decentralized Multi-Agent Reinforcement Learning (MAD-MARL) algorithm, which incorporates the following three designs: 1) a model-based prediction approach, which can improve the policy learning speed greatly and also predict each agent's potential future so as to alleviate the impact of delayed information sharing in a networked system, 2) an attentional prediction sharing approach, which is to effectively reduce the non-stationarity of the environment by combining the proposed model-based prediction and self-attention strategy for reaching global consensus and cooperative inter-agent coordination in a distributed manner, and 3) a decentralized model-free learning approach, which trains the agents with the assistance of model-based predictions. Extensive simulations are conducted and the numerical results demonstrate that MAD-MARL can significantly increase the learning speed and also greatly improve the overall performance compared with existing work.

Pinning Control of Hypergraphs

A standard assumption in control of network dynamical systems is that its nodes interact through pairwise interactions, which can be described by means of a directed graph. However, in several contexts, multibody, directed interactions may occur, thereby requiring the use of directed hypergraphs rather then digraphs. For the first time, we propose a strategy, inspired by the classic pinning control on graphs, that is tailored for controlling network systems coupled through a directed hypergraph. By drawing an analogy with signed graphs, we provide sufficient conditions for controlling the network onto the desired trajectory provided by the pinner, and a dedicated algorithm to design the control hyperedges.

Synchronizing network systems in the presence of limited resources via edge snapping.

In this work, we propose a multilayer control protocol for the synchronization of network dynamical systems under limited resources. In addition to the layer where the interactions of the system take place, i.e., the backbone network, we propose a second, adaptive layer, where the edges are added or removed according to the edge snapping mechanism. Different from classic edge snapping, the inputs to the edge dynamics are modified to cap the number of edges that can be activated. After studying the local stability of the overall network dynamics, we illustrate the effectiveness of the approach on a network of Rössler oscillators and then show its robustness in a more general setting, exemplified with a model of the Italian high-voltage power grid.

Inferring State-feedback Cooperative Control of Networked Dynamical Systems

In this paper, we study the problem of inferring the state-feedback cooperative control of continuous-time NDSs from noisy state observations. To practice, we first propose a causality-based estimator to obtain the discretized system matrix, adaptive to both stable and explosive state evolution cases. Then, we derive the observation period guarantees and leverage the matrix logarithm to accurately reconstruct the continuous closed-loop matrix from the discrete one, circumventing the insufficiency of conventional sampling-recovery methods in this situation (e.g., Shannon sampling theorem). Finally, we exploit the element-wise coupling relation between the local feedback gain and the unknown interaction topology, and construct a duel-level least squares method to obtain the feedback matrix. Simulations are conducted to verify the proposed inference method.

A new method of adaptive mesoscale control in complex multiagent networked dynamical systems

Centralized strategies applied to large-scale systems require a vast amount of computational and communication resources. In contrast to them, distributed strategies offer higher scalability and reliability. However, communication and coordination among agents tremendously impact performance of systems controlled in the distributed manner. The existing methods lead to clustering, where the coordination between agents is limited to groups of entities to be controlled. The size of these groups are usually known in advance. In turn, many systems exhibit self-organization and dynamically form clustering structure. In that sense, control methods should adapt to such dynamic structures offering the same balance between performance and communication/computational demands. In this paper, we propose a new approach to complex system control based on efficient cluster (mesoscopic) control paradigm. We demonstrate its efficacy in scenarios, where a group of agents should reach a certain goal.

Structure Identifiability of an NDS With LFT Parametrized Subsystems

Requirements on subsystems have been made clear in this paper for a linear time invariant (LTI) networked dynamic system (NDS), under which subsystem interconnections can be estimated from external output measurements. In this NDS, subsystems may have distinctive dynamics, and subsystem interconnections are arbitrary. It is assumed that system matrices of each subsystem depend on its (pseudo) first principle parameters (FPPs) through a linear fractional transformation (LFT). It has been proven that if in each subsystem, the transfer function matrix (TFM) from its internal inputs to its external outputs is of full normal column rank (FNCR), while the TFM from its external inputs to its internal outputs is of full normal row rank (FNRR), then the structure of the NDS is identifiable. Moreover, under some particular situations like there are no direct information transmission from an internal input to an internal output in each subsystem, a necessary and sufficient condition is established for NDS structure identifiability. A matrix valued polynomial (MVP) rank based equivalent condition is further derived, which depends affinely on subsystem (pseudo) FPPs and can be independently verified for each subsystem. From this condition, some necessary conditions are obtained for both subsystem dynamics and its (pseudo) FPPs, using the Kronecker canonical form (KCF) of a matrix pencil.

Topology Learning of Linear Dynamical Systems With Latent Nodes Using Matrix Decomposition

In this article, we present a novel approach to reconstruct the topology of networked linear dynamical systems with latent nodes. The network is allowed to have directed loops and bi-directed edges. The main approach relies on the unique decomposition of the inverse of power spectral density matrix (IPSDM) obtained from observed nodes as a sum of sparse and low-rank matrices. We provide conditions and methods for decomposing the IPSDM into sparse and low-rank components. The sparse component yields moral graph (MG) associated with the observed nodes, and the low-rank component retrieves parents, children and spouses (the Markov Blanket) of the hidden nodes. The article provides necessary and sufficient conditions for the unique decomposition of a given skew symmetric matrix into sum of a sparse skew symmetric and a low-rank skew symmetric matrices. For a large class of systems, the unique decomposition of imaginary part of the IPSDM of observed nodes, a skew symmetric matrix, into the sparse and the low-rank components is sufficient to identify the MG of the observed nodes as well as the Markov Blanket of latent nodes. For a large class of systems, all spurious links in the MG formed by the observed nodes can be identified. Assuming conditions on hidden nodes required for identifiability, links between hidden and observed nodes can be reconstructed, thus retrieving the exact topology of the network from the IPSDM. Moreover, for finite data, we provide bounds on entry-wise distance between the true and the estimated IPSDMs.

Dirac synchronization is rhythmic and explosive

Topological signals defined on nodes, links and higher dimensional simplices define the dynamical state of a network or of a simplicial complex. As such, topological signals are attracting increasing attention in network theory, dynamical systems, signal processing and machine learning. Topological signals defined on the nodes are typically studied in network dynamics, while topological signals defined on links are much less explored. Here we investigate Dirac synchronization, describing locally coupled topological signals defined on the nodes and on the links of a network, and treated using the topological Dirac operator. The dynamics of signals defined on the nodes is affected by a phase lag depending on the dynamical state of nearby links and vice versa. We show that Dirac synchronization on a fully connected network is explosive with a hysteresis loop characterized by a discontinuous forward transition and a continuous backward transition. The analytical investigation of the phase diagram provides a theoretical understanding of this topological explosive synchronization. The model also displays an exotic coherent synchronized phase, also called rhythmic phase, characterized by non-stationary order parameters which can shed light on topological mechanisms for the emergence of brain rhythms.

A Region-Based Collaborative Management Scheme for Dynamic Clustering in Green VANET

Green Vehicular Ad-hoc Network (VANET) is a newly-emerged research area that focuses on reducing harmful impacts of vehicular communication equipment on the natural environment. Recent studies have shown that grouping vehicles into clusters for green communications in VANETs can significantly improve networking efficiency and reduce infrastructure costs. As a dynamic network system, maintaining the network connectivity and reducing the communication overlap are two critical challenges for green VANET clustering. However, most existing work studies connectivity and overlap separately, lacking a deep understanding of the relationship between them. To address this issue, we present a comprehensive analysis that jointly considers the two critical factors in one model. Specifically, we first design a state resemblance prediction (SRP) model based on the historical trajectory feature relevance between vehicles; Combined with the SRP model, we propose the region-based collaborative management scheme (RCMS) to establish the dynamic clustering; Lastly, we take extensive experiments to verify the region-based collaborative management scheme for dynamic clustering. The results demonstrate that the proposed clustering algorithm can achieve high networking efficiency and better communication stability.

A Necessary and Sufficient Condition of an Interfering Reverse Edge for a Directed Acyclic Graph

A directed acyclic graph (DAG) is a common topology in biological, engineering, and social networks. A network topology is critical in determining a collective behavior of a network dynamic system. For example, the convergence rate of a consensus behavior in a multiagent system relies on the eigenvalues of the Laplacian associated with the network topology. This article aims to analyze the influence of adding a reverse edge into a DAG on convergence rate. It reveals the existence of the so-called interfering reverse edges; adding one single edge in this category can reduce the so-called dominant convergence rate even for a large network. More specifically, a necessary and sufficient condition of an interfering reverse edge is explicitly constructed. According to the condition, a computationally efficient method is proposed to assess an interfering reverse edge.

Colloquium: Multiscale modeling of brain network organization

A complete understanding of the brain requires an integrated description of the numerous scales of neural organization. It means studying the interplay of genes, synapses, and even whole brain regions which ultimately leads to different types of behavior, from perception to action, while asleep or awake. Yet, multiscale brain modeling is challenging, in part because of the difficulty to access simultaneously to information from multiple spatiotemporal scales. While some insights have been gained on the role of specific microcircuits (e.g., thalamocortical), a comprehensive characterization of how changes occurring at one scale can have an impact on other ones, remains poorly understood. Recent efforts to address this gap include the development of new analytical tools mostly adapted from network science and dynamical systems theory. These theoretical contributions provide a powerful framework to analyze and model interconnected complex systems exhibiting interactions within and between different scales, or layers. Here, we present recent advances for the characterization of the multiscale brain organization in terms of structure-function, oscillation frequencies and temporal evolution. Efforts are reviewed on the multilayer network properties underlying higher-order organization of neuronal assemblies, as well as on the identification of multimodal network-based biomarkers of brain pathologies, such as Alzheimer's disease. We conclude this Colloquium with a perspective discussion on how recent results from multilayer network theory, involving generative modeling, controllability and machine learning, could be adopted to address new questions in modern neuroscience.

Fluctuation response patterns of network dynamics – An introduction

Networked dynamical systems, i.e., systems of dynamical units coupled via nontrivial interaction topologies, constitute models of broad classes of complex systems, ranging from gene regulatory and metabolic circuits in our cells to pandemics spreading across continents. Most of such systems are driven by irregular and distributed fluctuating input signals from the environment. Yet how networked dynamical systems collectively respond to such fluctuations depends on the location and type of driving signal, the interaction topology and several other factors and remains largely unknown to date. As a key example, modern electric power grids are undergoing a rapid and systematic transformation towards more sustainable systems, signified by high penetrations of renewable energy sources. These in turn introduce significant fluctuations in power input and thereby pose immediate challenges to the stable operation of power grid systems. How power grid systems dynamically respond to fluctuating power feed-in as well as other temporal changes is critical for ensuring a reliable operation of power grids yet not well understood. In this work, we systematically introduce a linear response theory (LRT) for fluctuation-driven networked dynamical systems. The derivations presented not only provide approximate analytical descriptions of the dynamical responses of networks, but more importantly, also allow to extract key qualitative features about spatio-temporally distributed response patterns. Specifically, we provide a general formulation of a LRT for perturbed networked dynamical systems, explicate how dynamic network response patterns arise from the solution of the linearised response dynamics, and emphasise the role of LRT in predicting and comprehending power grid responses on different temporal and spatial scales and to various types of disturbances. Understanding such patterns from a general, mathematical perspective enables to estimate network responses quickly and intuitively, and to develop guiding principles for, e.g., power grid operation, control and design.

Disconnection-Aware Attack Detection and Isolation With Separation-Based Detector Reconfiguration

This study addresses incident handling during an adverse event for dynamical networked control systems. Incident handling can be divided into five steps: detection, analysis, containment, eradication, and recovery. For networked control systems, the containment step can be conducted through physical disconnection of an attacked subsystem. In accordance with the disconnection, the equipped attack detection unit should be reconfigured to maintain its detection capability. In particular, separating the detection subunit associated with the disconnected subsystem is considered as a specific reconfiguration scheme in this study. This article poses the problem of disconnection-aware attack detection and isolation with the separation-based detector reconfiguration. The objective is to design an attack detection unit that preserves its detection and isolation capability even under any possible disconnection and separation. The difficulty arises from network topology variation caused by disconnection that can possibly lead to stability loss of the distributed observer inside the attack detection unit. A solution is proposed based on an existing controller design technique referred to as retrofit control. Furthermore, an application to low-voltage power distribution networks with distributed generation is exhibited. Numerical examples evidence the practical use of the proposed method through a benchmark distribution network.

Finite-Time Synchronization Control of Networked Chaotic Complex-Valued Systems With Adaptive Coupling

In the present paper, the finite-time synchronization control problem is addressed for the networked complex-valued dynamical systems with directed and undirected topologies. By combining the two-phase method with delayed impulsive control and designing adaptive coupling weights, several finite-time synchronization results and the corresponding settling times are obtained by using the complex-valued states directly. In particular, the delayed impulsive control is introduced in the first phase to guarantee that the controlled system can globally exponentially converge to the inner of the unit ball, while the finite-time control term is used only in the second phase to ensure the finite-time stability. Finally, two numerical examples are used to show the validity of our control strategy.

Evolutionary dynamics of cooperation in dynamic networked systems with active striving mechanism

How the cooperation between individuals emerges in complex networked systems is a hot issue of common concern in many different fields. Evolutionary game theory has provided a mathematical framework for studying this problem. In social networks, the interaction between individuals is not fixed, but constantly changing. Therefore, this paper considers the coevolution of strategy and network structure when studying the evolutionary dynamics of cooperation. For the linking dynamics, different from the traditional random selection of connection objects, we propose a mechanism that individuals can actively recommend themselves before being selected. Through such active efforts, individuals expect to get more opportunities to be connected. In particular, the attributes of individuals considered in this paper have two characteristics without correlation. One is strategy, that is, cooperation or defection, and they have their own intensities of self recommendation respectively. The other is the high or low response to others’ recommendation, which reflects whether individuals are easy to be persuaded. We make a theoretical analysis of the system under the limit condition that the linking dynamics are much faster than strategy evolution. It is found that when the self recommendation intensity of the defector is greater than that of the cooperator, it is not conducive to the emergence of cooperation. On the contrary, when cooperators make self recommendation more actively than defectors, the response levels of different types of individuals and their proportion in the system jointly determine whether cooperation emerges. We theoretically analyze the conditions that can promote cooperation when the recommendation intensity of cooperators is high. However, the arms race in which both sides actively strive to raise the intensity of self recommendation will lead to an increasing adjustment cycle and form a Red Queen like evolution.

Global structure identifiability and reconstructibility of an NDS with descriptor subsystems

This paper investigates requirements on a networked dynamic system (NDS) such that its subsystem interactions can be solely determined from experiment data or reconstructed from its overall model. The NDS is constituted from several subsystems whose dynamics are described through a descriptor form. Except regularity on each subsystem and the whole NDS, and the well-posedness of the whole NDS, no other restrictions are put on either subsystem dynamics or subsystem interactions. A matrix rank based necessary and sufficient condition is derived for the global identifiability of subsystem interactions, which leads to several conclusions about NDS structure identifiability when there is some a priori information. This matrix also gives an explicit description for the set of subsystem interactions that cannot be differentiated from experiment data only. In addition, a necessary and sufficient condition is obtained for the reconstructibility of subsystem interactions from an NDS descriptor form model. This condition can be verified with each subsystem separately and is therefore attractive in the analysis and synthesis of a large-scale NDS. Simulation results show that rather than increases monotonically with the distance of subsystem interactions to the undifferentiable set, the magnitude of the external output differences between two NDSs with distinct subsystem interactions increases much more rapidly when one of them is close to be unstable. In addition, directions of probing signals are also very important in differentiating external outputs of distinct NDSs. These findings are expected to be helpful in identification experiment designs, etc.

Dynamic network biomarker analysis and system pharmacology methods to explore the therapeutic effects and targets of Xiaoyaosan against liver cirrhosis

Ethnopharmacological relevanceXiaoyaosan is a traditional Chinese herbal formula that has long been used to treat liver cirrhosis, liver failure, and hepatocarcinoma (HCC). However, little is known about its mechanism of action and targets in treating chronic liver disease.Aim of the study: This study aimed to detect the critical transition of HCC progression and to explore the regulatory mechanism and targets of Xiaoyaosan treating liver cirrhosis (cirrhosis) using integrative medicinal research involving system biology and pharmacology. Materials and methodsWe recruited chronic liver disease participants to obtain gene expression data and applied the dynamic network biomarker (DNB) method to identify molecular markers and the critical transition. We combined network pharmacology and DNB analysis to locate the potential DNBs (targets). Then we validated the DNBs in the liver cirrhosis rat models using Xiaoyaosan treatment. The expression of genes encoding the four DNBs, including Cebpa, Csf1, Egfr, and Il7r, were further validated in rat liver tissue using Western blot analysis. ResultsWe found EGFR, CEBPA, Csf1, Ccnb1, Rrmm2, C3, Il7r, Ccna2, and Peg10 overlap in the DNB list and Xiaoyaosan-Target-Disease (XTD) network constructed using network pharmacology databases. We investigated the diagnostic ability of each member in the DNB cluster and found EGFR, CEBPA, CSF1, and IL7R had high diagnostic abilities with AUC >0.7 and P-value < 0.05. We validated these findings in rats and found that liver function improved significantly and fibrotic changes were relieved in the Xiaoyaosan treatment group. The expression levels of CSF1 and IL7R in the Xiaoyaosan group were significantly lower than those in the cirrhosis model group. In contrast, CEBPA expression in the Xiaoyaosan group was significantly higher than that in the cirrhosis model group. The expression of EGFR in the Xiaoyaosan group was slightly decreased than in the model group but not significantly. ConclusionUsing the DNB method and network pharmacology approach, this study revealed that CEBPA, IL7R, EGFR, and CSF1 expression was remarkably altered in chronic liver disease and thus, may play an important role in driving the progression of cirrhosis. Therefore, CEBPA, IL7R, EGFR, and CSF1 may be important targets of Xiaoyaosan in treating cirrhosis and can be considered for developing novel therapeutics.

Homeostasis in Networks with Multiple Input Nodes and Robustness in Bacterial Chemotaxis

A biological system achieves homeostasis when there is a regulated quantity that is maintained within a narrow range of values. Here, we consider homeostasis as a phenomenon of network dynamics. In this context, we improve a general theory for the analysis of homeostasis in network dynamical systems with distinguished input and output nodes, called ‘input–output networks.’ The theory allows one to define ‘homeostasis types’ of a given network in a ‘model independent’ fashion, in the sense that the classification depends on the network topology rather than on the specific model equations. Each ‘homeostasis type’ represents a possible mechanism for generating homeostasis and is associated with a suitable ‘subnetwork motif’ of the original network . Our contribution is an extension of the theory to the case of networks with multiple input nodes. To showcase our theory, we apply it to bacterial chemotaxis, a paradigm for homeostasis in biochemical systems. By considering a representative model of Escherichia coli chemotaxis, we verify that the corresponding abstract network has multiple input nodes. Thus showing that our extension of the theory allows for the inclusion of an important class of models that were previously out of reach. Moreover, from our abstract point of view, the occurrence of homeostasis in the studied model is caused by a new mechanism, called input counterweight homeostasis. This new homeostasis mechanism was discovered in the course of our investigation and is generated by a balancing between the several input nodes of the network—therefore, it requires the existence of at least two input nodes to occur. Finally, the framework developed here allows one to formalize a notion of ‘robustness’ of homeostasis based on the concept of ‘genericity’ from the theory dynamical systems. We discuss how this kind of robustness of homeostasis appears in the chemotaxis model.

Research on Regional Collaborative Network Innovation and Enterprises’ Innovation Performance in the Sustainable Development of Yangtze River Delta, China

As one of the most dynamic and sustainable economic regions in China, Yangtze River Delta has been forming an innovation cluster of high-quality development. To explore the driving force during the integration of Yangtze River Delta, this article constructs a driving force model and makes an empirical study by the structural equation modeling method. The model proves that heterogeneity knowledge will positively affect innovation, that innovation performance has a strong positive impact on regional collaborative network innovation, that regional collaborative network innovation exerts a positive effect on knowledge spillover, and that organizational learning can affect innovation performance positively. Based on the above research results, some recommendations are proposed. Those recommendations consist of: (a) to forge a dynamic sustainable and innovative regional collaborative network system lay a foundation of high-quality development; (b) while the two driving forces of heterogeneity knowledge and organizational learning activated, the indices of enterprise innovation performance should be optimized.