Time-varying Directed Networks Research Articles

Abnormal trial-to-trial variability in P300 time-varying directed eeg network of schizophrenia.

Cognitive disturbance in identifying, processing, and responding to salient or novel stimuli are typical attributes of schizophrenia (SCH), and P300 has been proven to serve as a reliable psychosis endophenotype. The instability of neural processing across trials, i.e., trial-to-trial variability (TTV), is getting increasing attention in uncovering how the SCH "noisy" brain organizes during cognition processes. Nevertheless, the TTV in the brain network remains unrevealed, notably how it varies in different task stages. In this study, resorting to the time-varying directed electroencephalogram (EEG) network, we investigated the time-resolved TTV of the functional organizations subserving the evoking of P300. Results revealed anomalous TTV in time-varying networks across the delta, theta, alpha, beta1, and beta2 bands of SCH. The TTV of cross-band time-varying network properties can efficiently recognize SCH (accuracy: 83.39%, sensitivity: 89.22%, and specificity: 74.55%) and evaluate the psychiatric symptoms (i.e., Hamilton's depression scale-24, r = 0.430, p = 0.022, RMSE = 4.891; Hamilton's anxiety scale-14, r = 0.377, p = 0.048, RMSE = 4.575). Our study brings new insights into probing the time-resolved functional organization of the brain, and TTV in time-varying networks may provide a powerful tool for mining the substrates accounting for SCH and diagnostic evaluation of SCH.

Nonparametric Dynamic Granger Causality based on Multi-Space Spectrum Fusion for Time-varying Directed Brain Network Construction.

Nonparametric estimation of time-varying directed networks can unveil the intricate transient organization of directed brain communication while circumventing constraints imposed by prescribed model-driven methods. A robust time-frequency representation - the foundation of its causality inference - is critical for enhancing its reliability. This study proposed a novel method, i.e., nonparametric dynamic Granger causality based on Multi-space Spectrum Fusion (ndGCMSF), which integrates complementary spectrum information from different spaces to generate reliable spectral representations to estimate dynamic causalities across brain regions. Systematic simulations and validations demonstrate that ndGCMSF exhibits superior noise resistance and a powerful ability to capture subtle dynamic changes in directed brain networks. Particularly, ndGCMSF revealed that during instruction response movements, the laterality in the hemisphere ipsilateral to the hemiplegic limb emerges upon instruction onset and diminishes upon task accomplishment. These intrinsic variations further provide reliable features for distinguishing two types of hemiplegia (left vs. right) and assessing motor functions. The ndGCMSF offers powerful functional patterns to derive effective brain networks in dynamically changing operational settings and contributes to extensive areas involving dynamical and directed communications.

The dynamic causality brain network reflects whether the working memory is solidified.

Working memory, which is foundational to higher cognitive function, is the "sketchpad of volitional control." Successful working memory is the inevitable outcome of the individual's active control and manipulation of thoughts and turning them into internal goals during which the causal brain processes information in real time. However, little is known about the dynamic causality among distributed brain regions behind thought control that underpins successful working memory. In our present study, given that correct responses and incorrect ones did not differ in either contralateral delay activity or alpha suppression, further rooting on the high-temporal-resolution EEG time-varying directed network analysis, we revealed that successful working memory depended on both much stronger top-down connections from the frontal to the temporal lobe and bottom-up linkages from the occipital to the temporal lobe, during the early maintenance period, as well as top-down flows from the frontal lobe to the central areas as the delay behavior approached. Additionally, the correlation between behavioral performance and casual interactions increased over time, especially as memory-guided delayed behavior approached. Notably, when using the network metrics as features, time-resolved multiple linear regression of overall behavioral accuracy was exactly achieved as delayed behavior approached. These results indicate that accurate memory depends on dynamic switching of causal network connections and shifting to more task-related patterns during which the appropriate intervention may help enhance memory.

Convergence to Zero of Quadratic Lyapunov Functions for Multiagent Systems in Time-Varying Directed Networks

Convergence to Zero of Quadratic Lyapunov Functions for Multiagent Systems in Time-Varying Directed Networks

AB/Push-Pull method for distributed optimization in time-varying directed networks

In this paper, we study the distributed optimization problem for a system of agents embedded in time-varying directed communication networks. Each agent has its own cost function and agents cooperate to determine the global decision that minimizes the summation of all individual cost functions. We consider the so-called push-pull gradient-based algorithm (termed as AB/Push-Pull) which employs both row- and column-stochastic weights simultaneously to track the optimal decision and the gradient of the global cost while ensuring consensus and optimality. We show that the algorithm converges linearly to the optimal solution over a time-varying directed network for a constant stepsize when the agent's cost function is smooth and strongly convex. The linear convergence of the method has been shown in [F. Saadatniaki, R. Xin, and U.A. Khan, Decentralized optimization over time-varying directed graphs with row and column-stochastic matrices, IEEE Trans. Autom. Control 65(11) (2020), pp. 4769–4780], where the multi-step consensus contraction parameters for row- and column-stochastic mixing matrices are not directly related to the underlying graph structure, and the explicit range for the stepsize value is not provided. With respect to [F. Saadatniaki, R. Xin, and U.A. Khan, Decentralized optimization over time-varying directed graphs with row and column-stochastic matrices, IEEE Trans. Autom. Control 65(11) (2020), pp. 4769–4780], the novelty of this work is twofold: (1) we establish the one-step consensus contraction for both row- and column-stochastic mixing matrices with the contraction parameters given explicitly in terms of the graph diameter and other graph properties; and (2) we provide explicit upper bounds for the stepsize value in terms of the properties of the cost functions, the mixing matrices, and the graph connectivity structure.

Differentially Private Distributed Optimization With an Event-Triggered Mechanism

This study concentrates on the differential private distributed optimization problem with an event-triggered mechanism, whose goals include preserving the privacy of agents’ initial states and local cost functions and improving communication efficiency. A distributed event-triggered mechanism is integrated into the differentially private subgradient-push distributed optimization algorithm and then a new algorithm named as DP-ETSP is designed, where the real-time information propagation among agents is avoided. Additionally, under the proposed event-triggered mechanism, an analysis of mean-square consensus and optimality over time-varying directed networks is made when the added Laplace noises meet some specific decaying conditions. Convergence rate results are further established under a specific stepsize, which are equal to the rate of stochastic gradient-push algorithm without event-triggered communication. Moreover, the differential privacy preservation performance is analyzed and the rule for selecting privacy level is discussed. Finally, the feasibility and effectiveness of DP-ETSP are verified in two simulation cases.

Linear convergence rate analysis of a class of exact first-order distributed methods for weight-balanced time-varying networks and uncoordinated step sizes

We analyze a class of exact distributed first order methods under a general setting on the underlying network and step-sizes. In more detail, we allow simultaneously for time-varying uncoordinated step sizes and time-varying directed weight-balanced networks, jointly connected over bounded intervals. The analyzed class of methods subsumes several existing algorithms like the unified Extra and unified DIGing (Jakovetić in IEEE Trans Signal Inf Process Netw 5(1):31–46, 2019), or the exact spectral gradient method (Jakovetić et al. in Comput Optim Appl 74:703–728, 2019) that have been analyzed before under more restrictive assumptions. Under the assumed setting, we establish R-linear convergence of the methods and present several implications that our results have on the literature. Most notably, we show that the unification strategy in Jakovetić (2019) and the spectral step-size selection strategy in Jakovetić et al. (2019) exhibit a high degree of robustness to uncoordinated time-varying step sizes and to time-varying networks.

Asynchronous Algorithms for Decentralized Resource Allocation Over Directed Networks

In this article, we consider a class of decentralized resource allocation problems over directed networks, where each node only communicates with its in-neighbors and attempts to minimize its own cost when network-wide resource constraints as well as local capacity limits are satisfied. Decentralized optimization to solve this problem has been a significant focus within engineering research due to its advantages in scalability, robustness, and flexibility. Most existing methods are synchronous while few works are devoted to asynchronously solving the problem. The problem becomes even more challenging when the networks are directed. To address the resource allocation problem when the above issues are considered, we propose a novel decentralized asynchronous algorithm based on the gossip-based communication protocol and epigraph strategy. An important feature of the algorithm is that it is implemented in a completely decentralized manner in the case of asynchronous communication and directed networks. We provide theoretical proof to guarantee the convergence of the proposed algorithm, which indicates that it can successfully allocate the optimal resource. When solving the resource allocation problem over time-varying directed networks, we further discuss a related decentralized asynchronous algorithm according to the random sleep protocol. Numerical examples are given to demonstrate the viability and performance of the algorithms.

Distributed nonlinear estimation over time-varying directed networks

In this paper, we investigate distributed estimation with nonlinear sensing functions over time-varying directed networks. Each agent sequentially makes nonlinear and noisy measurements of the objective parameter, and aims to estimate the parameter through local information. To this end, an algorithm is designed by introducing two time-varying matrices corresponding to the network graph. Under the condition of B-bounded strong connectivity, and some mild assumptions on sensing functions, the distributed estimator achieves consistent parameter estimates by selecting an appropriate stepsize. Furthermore, we also analyze the asymptotic property of the weighted estimate error sequence, including the asymptotic mean and asymptotic covariance. It is shown that the performance is consistent with that of a widely used centralized estimator and a distributed estimator over a fixed directed communication topology. Finally, we give simulation results to verify the effectiveness of the devised algorithm.

Privacy-Protected Decentralized Dual Averaging Push With Edge-Based Correlated Perturbations Over Time-Varying Directed Networks

In this article, we consider a decentralized constrained optimization problem over time-varying directed networks, where nodes in the networks collaboratively minimize the sum of local cost functions subject to a common constraint set. Due to the continued broadcasting of information that involves privacy during the optimization process, most existing decentralized algorithms for settling the problem may suffer from the risk of privacy leakage of normal nodes. The problem becomes even more challenging when the networks are time-varying and directed. To overcome these difficulties, we propose an effective privacy-protected decentralized dual averaging push algorithm that possesses two appealing features. On the one hand, our algorithm adds edge-based correlated perturbation signals to the process of information transmission for protecting privacy. On the other hand, our algorithm involves the use of the push-sum averaging protocol which can overcome the impact of the imbalance caused by time-varying directed networks. We provide rigorous analyses for convergence and privacy protection to illustrate that our algorithm not only protects the privacy of normal nodes but also converges exactly to the optimal solution under weaker initial conditions than the existing decentralized works. Numerical examples are given to demonstrate the viability and performance of our algorithm.

Distributed Nonconvex Event-Triggered Optimization Over Time-Varying Directed Networks

Many problems in industrial smart manufacturing, such as process operational optimization and decision-making, can be regarded as distributed nonconvex optimization problems, whose goal is to utilize distributed nodes to cooperatively search for the minimal value of the global objective function. With the consideration of data transmission mode, transmission condition, and communication waste in industrial applications, it is meaningful to study the distributed nonconvex optimization problem with an event-triggered strategy over time-varying directed networks. To solve such a problem, a distributed nonconvex event-triggered algorithm is proposed in this article. Under some assumptions on local objective functions, gradients, and step sizes, the convergence of the proposed event-triggered algorithm to the local minimum is established theoretically. Moreover, it is obtained that the proposed distributed event-triggered algorithm has a convergence rate of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$O(1/\ln (t))$</tex-math></inline-formula> . Finally, two examples of industrial systems are provided to validate the effectiveness of the proposed algorithm.

Distributed real-time power management of high-penetrated PV sources with voltage regulation over time-varying networks

The growing penetration of photovoltaic (PV) sources accommodated in Active Distribution Network (ADN) brings various severe voltage limits violation problems. This paper provides power management strategy for PV inverters in order to achieve power sharing as well as voltage regulation by dispatching PV active and reactive power. The proposed power management method can operate in a fully distributed manner where each unit measures local voltage information and communicates with neighboring units via time-varying directed communication networks to make its decisions, minimize the PV operation costs, drive system voltage into an acceptable range and always meet the active and reactive capacity constraints. The effectiveness and performance of the proposed method are validated by running different simulation scenarios on a real-time simulator (OPAL-RT).

Thermodynamic motif analysis for directed stock market networks

In this paper, we present a novel thermodynamically based analysis method for directed networks, and in particular for time-evolving networks in the finance domain. Based on an analogy with a dilute gas in statistical mechanics, we develop a partition function for a network composed of directed motifs. The method relies on the decomposition of directed networks into a series of frequently occurring graphlets, or motifs. According to the connection between a directed network and the dilute gas, the network motifs have the same topological structure as the low-order interactions between particles in the gas. This means that we can use the so-called cluster expansion from statistical mechanics to develop a partition function for the motif decomposition. In prior work, we have reported a detailed analysis of the cluster expansion for the case of undirected graphs, and showed how the resulting motif entropy can be used to analyse time evolving networks [1]. In this paper we extend this work to the case of directed graphs to compute thermodynamic quantities including energy, entropy and temperature for the directed network. The three thermodynamic quantities constitute the thermodynamic framework for the analysis of directed network evolution. We apply our thermodynamic framework to the financial and biological domains to represent real world complex systems as time-varying directed networks. Experimental results successfully demonstrate the effectiveness of the thermodynamic framework in representing the evolution of directed network structure and anomalous event detection.

Strong Structural Controllability of Networks under Time-Invariant and Time-Varying Topological Perturbations

This paper investigates the robustness of strong structural controllability for linear time-invariant and linear time-varying directed networks with respect to structural perturbations, including edge deletions and additions. In this direction, we introduce a new construct referred to as a perfect graph associated with a network with a given set of control nodes. The tight upper bounds on the number of edges that can be added to, or removed from a network, while ensuring strong structural controllability, are then derived. Moreover, we obtain a characterization of critical edge-sets, the maximal sets of edges whose any subset can be respectively added to, or removed from a network, while preserving strong structural controllability. In addition, procedures for combining networks to obtain strongly structurally controllable network-of-networks are proposed. Finally, controllability conditions are proposed for networks whose edge weights, as well as their structures, can vary over time.

Distributed output regulation of heterogeneous uncertain linear agents

We study a multi-agent output regulation problem, where only a few agents have access to the exosystem’s dynamics. We propose a fully distributed controller that solves the problem for linear, heterogeneous, non-minimum-phase, and uncertain agent dynamics as well as time-varying directed networks. The distributed controller consists of two parts: (1) an exosystem generator that locally estimates the exosystem dynamics, and (2) a dynamic compensator that, by locally approaching the internal model of the exosystem, achieves perfect output regulation. Compared to the existing results in the literature, our solution is the first to solve the distributed output regulation problem where not all agents have access to information of the exosystem and the agent dynamics are uncertain and non-minimum-phase.

A Privacy Preserving Distributed Optimization Algorithm for Economic Dispatch Over Time-Varying Directed Networks

The economic dispatch problem (EDP) plays a fundamental and significant role in smart grids. Its purpose is to decide the output power of every generator in smart grids for achieving the minimal generation cost. With advantages in flexibility, robustness, and scalability, it is desirable to apply distributed optimization methods to solve EDPs. In most existing distributed optimization approaches, all generators explicitly exchange their states with neighbors to obtain the optimal solution, which may result in disclosing the privacy information of generators. This problem becomes worse if there are some adversaries aimed at inferring privacy information from the communication network for nefarious purposes. For privacy preservation, a privacy preserving distributed optimization algorithm over time-varying directed communication networks is proposed in this article by adding conditional noises to the exchanged states. It is proved that this proposed algorithm is able to solve the EDP. Moreover, the convergence rate and privacy analysis of the proposed algorithm are also shown in this article. An example is provided to confirm the effectiveness of this proposed algorithm.

Momentum methods for stochastic optimization over time-varying directed networks

Recently, decentralized optimization (DO) has received widespread attentions. Since the scale of DO is usually very large, research on acceleration has become a hotspot. However, the error caused by the global information deficiency for each agent becomes the key problem cumbering the existing elaborately designed accelerated methods from being applied to DO, directly. On the other hand, recent studies illustrate that a group of accelerated methods can be covered from a viewpoint of momentum. In this paper, we propose to follow this methodology to design accelerated algorithms and adapt them to DO over time-varying directed networks, of which the main benefit is that, because the proposed algorithms are derived from momentum, they not only avoid the design of elaborate iterative structures but also inherit the physical interpretability of momentum. Furthermore, we show that our proposed algorithms can achieve sharper convergence rates than their competitors under the same condition. In the end, experiments on a number of benchmark datasets validate well the competitiveness of our algorithms.

Online distributed stochastic learning algorithm for convex optimization in time-varying directed networks

This paper investigates a online learning optimization problem in a distributed manner, where a set of agents aim to cooperatively minimize the sum of local time-varying cost functions while communications among agents depend on a sequence of time-varying directed graphs. For such a problem, we first propose a online distributed stochastic (sub)gradient push-sum algorithm by utilizing distributed optimization methods and push-sum protocols. Then we analyze the regret bounds for the proposed algorithm on the cases when the cost functions are convex and strongly convex, respectively. The bound on the expected regret for convex functions grows sub-linearly with order of O(T),where T is the time horizon. When the cost functions are strongly convex with Lipschitz gradients, the regret bound has an improved rate with order of O(lnT). Numerical simulations on the localization in wireless sensor networks are used to show the effectiveness of the proposed algorithm.

Stochastic gradient-push for economic dispatch on time-varying directed networks with delays

Economic dispatch problem (EDP) is an elementary optimization problem of power systems whose purpose is to minimize the total generation cost while meeting total demands and complying with individual generator output constraints. This paper proposes a distributed optimization algorithm with noisy gradient based on stochastic gradient-push approach to solve the EDP on time-varying directed communication networks potentially with time delays. It shows that the proposed algorithm can be ensured to solve the EDP when the time-varying directed communication networks are uniformly jointly strongly connected. The algorithm with asynchronous step-sizes can deal with the finite time-varying delays on communication links as well. The generation cost functions are considered as strictly convex and strongly convex functions respectively with convergence rate of Olnk/k and Olnk/k. Simulation results are provided to validate the availability of the algorithm.

Distributed Energy Resource Coordination Over Time-Varying Directed Communication Networks

In this paper, we consider the optimal coordination problem for distributed energy resources (DERs), including distributed generators and energy storages. We first propose an algorithm based on the push-sum and gradient method to solve the optimal DER coordination problem in a distributed manner. In the proposed algorithm, each DER only maintains a set of variables and updates them through information exchange with a few neighboring DERs over a time-varying directed communication network. We show that the proposed distributed algorithm with appropriately chosen diminishing step sizes solves the optimal DER coordination problem if the time-varying directed communication network is uniformly jointly strongly connected. Moreover, in order to improve the convergence speed and to reduce the communication burden, we propose an accelerated distributed algorithm with a fixed step size. We show that the new proposed algorithm exponentially solves the optimal DER coordination problem if the cost functions satisfy an additional assumption and the selected step size is less than a certain critical value. Both proposed distributed algorithms are validated and evaluated using the IEEE 39-bus system.