Time-varying Graphs Research Articles

Resilient First-Order Consensus and Weakly Stable, Higher Order Synchronization of Continuous-Time Networked Multiagent Systems

Local interaction rules for consensus and synchronization are vital for many applications in distributed control of cyber-physical systems. However, most research in this area assumes all nodes (or agents) in the networked system cooperate. This paper considers local interaction rules for resilient first-order consensus and weakly stable, higher order synchronization whenever some of the agents in the network are Byzantine-like adversaries defined in a continuous-time setting. The normal agents have identical dynamics modeled by continuous-time, linear, time-invariant, weakly stable systems. Agents in the networked system influence one another by sharing state or output information according to a directed, time-varying graph. We present a resilient consensus protocol as well as dynamic state and output feedback control laws for the normal agents, to achieve the resilient consensus and synchronization objectives, respectively. We characterize the required network topologies using the property of network robustness. We demonstrate the results in simulation examples to illustrate the resilient synchronization output feedback control law.

A Distributed Algorithm for Computing a Common Fixed Point of a Finite Family of Paracontractions

A distributed algorithm is described for finding a common fixed point of a family of m>1 nonlinear maps M_i : R^n -> R^n assuming that each map is a paracontraction and that at least one such common fixed point exists. The common fixed point is simultaneously computed by m agents assuming each agent i knows only M_i, the current estimates of the fixed point generated by its neighbors, and nothing more. Each agent recursively updates its estimate of a fixed point by utilizing the current estimates generated by each of its neighbors. Neighbor relations are characterized by a time-varying directed graph N(t). It is shown under suitably general conditions on N(t), that the algorithm causes all agents estimates to converge to the same common fixed point of the m nonlinear maps.

Distributed Algorithm for Solving Convex Inequalities

In this note, a distributed subgradient-based algorithm is proposed for continuous-time multi-agent systems to search a feasible solution to convex inequalities. The algorithm involves each agent achieving a state constrained by its own inequalities while exchanging local information with other agents under a time-varying directed communication graph. With the validity of a mild connectivity condition associated with the communication graph, it is shown that all agents will reach agreement asymptotically and the consensus state is in the solution set of the inequalities. Furthermore, the method is also extended to solving the distributed optimization problem of minimizing the sum of local objective functions subject to convex inequalities. Simulation examples are presented to demonstrate the effectiveness of the theoretical results.

A lifting method for analyzing distributed synchronization on the unit sphere

Abstract This paper introduces a new lifting method for analyzing convergence of continuous-time distributed synchronization/consensus systems on the unit sphere. Points on the d -dimensional unit sphere are lifted to the ( d + 1 ) -dimensional Euclidean space. The consensus protocol on the unit sphere is the classical one, where agents move toward weighted averages of their neighbors in their respective tangent planes. Only local and relative state information is used. The directed interaction graph topologies are allowed to switch as a function of time. The dynamics of the lifted variables are governed by a nonlinear consensus protocol for which the weights contain ratios of the norms of state variables. We generalize previous convergence results for hemispheres. For a large class of consensus protocols defined for switching uniformly quasi-strongly connected time-varying graphs, we show that the consensus manifold is uniformly asymptotically stable relative to closed balls contained in a hemisphere. Compared to earlier projection based approaches used in this context such as the gnomonic projection, which is defined for hemispheres only, the lifting method applies globally. With that, the hope is that this method can be useful for future investigations on global convergence.

Distributed Subgradient-Based Multiagent Optimization With More General Step Sizes

A wider selection of step sizes is explored for the distributed subgradient algorithm for multi-agent optimization problems, for both time-invariant and time-varying communication topologies. The square summable requirement of the step sizes commonly adopted in the literature is removed. The step sizes are only required to be positive, vanishing and non-summable. It is proved that in both unconstrained and constrained optimization problems, the agents' estimates reach consensus and converge to the optimal solution with the more general choice of step sizes. The idea is to show that a weighted average of the agents' estimates approaches the optimal solution, but with different approaches. In the unconstrained case, the optimal convergence of the weighted average of the agents' estimates is proved by analyzing the distance change from the weighted average to the optimal solution and showing that the weighted average is arbitrarily close to the optimal solution. In the constrained case, this is achieved by analyzing the distance change from the agents' estimates to the optimal solution and utilizing the boundedness of the constraints. Then the optimal convergence of the agents' estimates follows because consensus is reached in both cases. These results are valid for both a strongly connected time-invariant graph and time-varying balanced graphs that are jointly strongly connected.

Temporal Centrality-Balanced Traffic Management for Space Satellite Networks

Structural centrality metrics of the networks are important features to identify central nodes or links, to assess network vulnerability against removal of nodes or links, and to manage or control the network traffic to reduce the network congestion. Classic graph-based centrality metrics are not effective in dynamic space satellite networks due to the changing connectivity and they cannot effectively capture important temporal properties. In this paper, we employ a new technique called time-varying graphs (TVGs) to express temporal concepts and definitions of space satellite networks and propose several temporal centrality metrics in order to account for the temporal features of the networks. To calculate these temporal centrality metrics, we adopt a modified matrix multiplication algorithm to find the shortest journeys between nodes in the TVG. A temporal centrality-balanced traffic management scheme is further developed to enhance the network performance, which is integrated into a temporal centrality-balanced routing algorithm. To show the performance of the proposed centrality metrics and centrality balanced routing algorithm, we conduct some experiments performed on a scenario of the LEO satellite network. The simulation results show that the proposed centrality metrics are more powerful to analyze the evolution of time-varying centrality and vulnerability of space satellite networks than topological adaptation of vanilla centrality methods. Besides, the proposed centrality-balanced routing algorithm reaches the superior quality values of the network performance (in terms of several factors) in comparison with those attained by the traditional shortest journeys routing algorithm and proves to be more powerful to support the high traffic load.

Distributed primal-dual optimisation method with uncoordinated time-varying step-sizes

ABSTRACTThis paper is concerned with the distributed optimisation problem over a multi-agent network, where the objective function is described by a sum of all the local objectives of agents. The target of agents is to collectively reach an optimal solution while minimising the global objective function. Under the assumption that the information exchange among agents is depicted by a sequence of time-varying undirected graphs, a distributed optimisation algorithm with uncoordinated time-varying step-sizes is presented, which signifies that the step-sizes of agents are not always uniform per iteration. In light of some reasonable assumptions, this paper fully conducts an explicit analysis for the convergence rate of the optimisation method. A striking feature is that the algorithm has a geometric convergence rate even if the step-sizes are time-varying and uncoordinated. Simulation results on two numerical experiments in power systems show effectiveness and performance of the proposed algorithm.

Cluster consensus in multi-agent networks with mutual information exchange

The emergence of new technologies such as the Internet of things and the Cloud transforms the way we interact. Whether it be human to human interaction or human to machine interaction, the size of the networks keeps growing. As the networks get more complex nowadays with many interconnected components, it is necessary to develop distributed scalable algorithms so as to minimize the computation required in decision making in such large-scale systems. In this paper, we consider a setup where each agent in the network updates its opinion by relying on its neighbors’ opinions. The information exchange between the agents is assumed to be mutual. The cluster consensus problem is investigated for networks represented by static or time-varying graphs. Joint and integral connectivity conditions are utilized to determine the number of clusters that are formed, as the interactions among the agents evolve over time. Finally, some numerical examples are given to illustrate the theoretical results.

A Time-Vertex Signal Processing Framework: Scalable Processing and Meaningful Representations for Time-Series on Graphs

An emerging way to deal with high-dimensional noneuclidean data is to assume that the underlying structure can be captured by a graph. Recently, ideas have begun to emerge related to the analysis of time-varying graph signals. This paper aims to elevate the notion of joint harmonic analysis to a full-fledged framework denoted as time-vertex signal processing, that links together the time-domain signal processing techniques with the new tools of graph signal processing. This entails three main contributions: a) We provide a formal motivation for harmonic time-vertex analysis as an analysis tool for the state evolution of simple partial differential equations on graphs; b) we improve the accuracy of joint filtering operators by up-to two orders of magnitude; c) using our joint filters, we construct time-vertex dictionaries analyzing the different scales and the local time-frequency content of a signal. The utility of our tools is illustrated in numerous applications and datasets, such as dynamic mesh denoising and classification, still-video inpainting, and source localization in seismic events. Our results suggest that joint analysis of time-vertex signals can bring benefits to regression and learning.

Characterizing the Capability of Vehicular Fog Computing in Large-scale Urban Environment

The worldwide increase of vehicles is demanding the deployment of an intelligent transportation system for the urban environment. Recently, cloud computing technology is utilized to make the vehicles on the roads smarter and offer better driving experience. However, the intrinsic client-server communication model in the cloud-assisted service cannot meet the increasing demands for intensive computing in vehicles. To solve this challenging issue, we investigate another form of computing service, vehicular fog computing (VFC), which is a group of nearby smart vehicles connected via peer-to-peer communication model. Though VFC can provide computing service to any task initiator, its computational capability, i.e., the ability to provide computing service to the initiator, might be severely constrained by the realistic environments including limited communication ranges, high speeds and unpredictable mobility patterns of vehicles. In this paper, we characterize the computational capability (indicated by the product of processor speed and the time length from receiving the task) of VFC in a practical scenario through studying real-world vehicular mobility traces of Beijing. Specially, we propose a time-varying graph model to access the capability of VFC in such a large-scale urban environment with different scenarios. Based on this model, we reveal the temporal and spatial characteristics of the computational capability with different number of task initiators and portray its distribution of the number of connected vehicles and the computational capability. The distribution of the computational capability is also portrayed. Based on these observations, we define two modes to depict two different models of task distribution. Furthermore, we reveal the relationship between the computational capability and system parameters of computation delay, communication radius, and the number of initiators.

Different Topological Properties of EEG-Derived Networks Describe Working Memory Phases as Revealed by Graph Theoretical Analysis.

Several non-invasive imaging methods have contributed to shed light on the brain mechanisms underlying working memory (WM). The aim of the present study was to depict the topology of the relevant EEG-derived brain networks associated to distinct operations of WM function elicited by the Sternberg Item Recognition Task (SIRT) such as encoding, storage, and retrieval in healthy, middle age (46 ± 5 years) adults. High density EEG recordings were performed in 17 participants whilst attending a visual SIRT. Neural correlates of WM were assessed by means of a combination of EEG signal processing methods (i.e., time-varying connectivity estimation and graph theory), in order to extract synthetic descriptors of the complex networks underlying the encoding, storage, and retrieval phases of WM construct. The group analysis revealed that the encoding phase exhibited a significantly higher small-world topology of EEG networks with respect to storage and retrieval in all EEG frequency oscillations, thus indicating that during the encoding of items the global network organization could “optimally” promote the information flow between WM sub-networks. We also found that the magnitude of such configuration could predict subject behavioral performance when memory load increases as indicated by the negative correlation between Reaction Time and the local efficiency values estimated during the encoding in the alpha band in both 4 and 6 digits conditions. At the local scale, the values of the degree index which measures the degree of in- and out- information flow between scalp areas were found to specifically distinguish the hubs within the relevant sub-networks associated to each of the three different WM phases, according to the different role of the sub-network of regions in the different WM phases. Our findings indicate that the use of EEG-derived connectivity measures and their related topological indices might offer a reliable and yet affordable approach to monitor WM components and thus theoretically support the clinical assessment of cognitive functions in presence of WM decline/impairment, as it occurs after stroke.

Stability of Phase Difference Trajectories of Networks of Kuramoto Oscillators with Time-Varying Couplings and Intrinsic Frequencies

We study dynamics of phase-differences (PDs) of coupled oscillators where both the intrinsic frequencies and the couplings vary in time. In case the coupling coefficients are all nonnegative, we prove that the PDs are asymptotically stable if there exists T>0 such that the aggregation of the time-varying graphs across any time interval of length $T$ has a spanning tree. We also consider the situation that the coupling coefficients may be negative and provide sufficient conditions for the asymptotic stability of the PD dynamics. Due to time-variations, the PDs are asymptotic to time-varying patterns rather than constant values. Hence, the PD dynamics can be regarded as a generalisation of the well-known phase-locking phenomena. We explicitly investigate several particular cases of time-varying graph structures, including asymptotically periodic PDs due to periodic coupling coefficients and intrinsic frequencies, small perturbations, and fast-switching near constant coupling and frequencies, which lead to PD dynamics close to a phase-locked one. Numerical examples are provided to illustrate the theoretical results.

Stochastic Dual Averaging for Decentralized Online Optimization on Time-Varying Communication Graphs

We consider a decentralized online convex optimization problem in a network of agents, where each agent controls only a coordinate (or a part) of the global decision vector. For such a problem, we propose two decentralized stochastic variants ( $\mathsf{SODA}\hbox{-}\mathsf{C}$ and $\mathsf{SODA}\hbox{-}\mathsf{PS}$ ) of Nesterov's dual averaging method $(\mathsf{DA})$ , where each agent only uses a coordinate of the noise-corrupted gradient in the dual-averaging step. We show that the expected regret bounds for both algorithms have sublinear growth of $O(\sqrt{T})$ , with the time horizon $T$ , in scenarios when the underlying communication topology is time-varying. The sublinear regret can be obtained when the stepsize is of the form $1/\sqrt{t}$ and the objective functions are Lipschitz-continuous convex functions with Lipschitz gradients, and the variance of the noisy gradients is bounded. We also provide simulation results of the proposed algorithms on sensor networks to complement our theoretical analysis.

Dynamical SimRank search on time-varying networks

SimRank is an appealing pair-wise similarity measure based on graph structure. It iteratively follows the intuition that two nodes are assessed as similar if they are pointed to by similar nodes. Many real graphs are large, and links are constantly subject to minor changes. In this article, we study the efficient dynamical computation of all-pairs SimRanks on time-varying graphs. Existing methods for the dynamical SimRank computation [e.g., LTSF (Shao et al. in PVLDB 8(8):838–849, 2015) and READS (Zhang et al. in PVLDB 10(5):601–612, 2017)] mainly focus on top-k search with respect to a given query. For all-pairs dynamical SimRank search, Li et al.’s approach (Li et al. in EDBT, 2010) was proposed for this problem. It first factorizes the graph via a singular value decomposition (SVD) and then incrementally maintains such a factorization in response to link updates at the expense of exactness. As a result, all pairs of SimRanks are updated approximately, yielding O({r}^{4}n^2) time and O({r}^{2}n^2) memory in a graph with n nodes, where r is the target rank of the low-rank SVD. Our solution to the dynamical computation of SimRank comprises of five ingredients: (1) We first consider edge update that does not accompany new node insertions. We show that the SimRank update {varvec{Delta }}{} mathbf{S} in response to every link update is expressible as a rank-one Sylvester matrix equation. This provides an incremental method requiring O(Kn^2) time and O(n^2) memory in the worst case to update n^2 pairs of similarities for K iterations. (2) To speed up the computation further, we propose a lossless pruning strategy that captures the “affected areas” of {varvec{Delta }}{} mathbf{S} to eliminate unnecessary retrieval. This reduces the time of the incremental SimRank to O(K(m+|{textsf {AFF}}|)), where m is the number of edges in the old graph, and |{textsf {AFF}}| (le n^2) is the size of “affected areas” in {varvec{Delta }}{} mathbf{S}, and in practice, |{textsf {AFF}}| ll n^2. (3) We also consider edge updates that accompany node insertions, and categorize them into three cases, according to which end of the inserted edge is a new node. For each case, we devise an efficient incremental algorithm that can support new node insertions and accurately update the affected SimRanks. (4) We next study batch updates for dynamical SimRank computation, and design an efficient batch incremental method that handles “similar sink edges” simultaneously and eliminates redundant edge updates. (5) To achieve linear memory, we devise a memory-efficient strategy that dynamically updates all pairs of SimRanks column by column in just O(Kn+m) memory, without the need to store all (n^2) pairs of old SimRank scores. Experimental studies on various datasets demonstrate that our solution substantially outperforms the existing incremental SimRank methods and is faster and more memory-efficient than its competitors on million-scale graphs.

Exploring density regions for analyzing dynamic graph data

Static or dynamic graphs are typically visualized by either node-link diagrams, adjacency matrices, adjacency lists, or hybrids thereof. In particular, for the case of a changing graph structure a viewer wishes to be able to visually compare the graphs in a sequence. Doing such a comparison task rapidly and reliably demands for visually analyzing the dynamic graph for certain dynamic patterns. In this paper we describe a novel dynamic graph visualization that is based on the concept of smooth density fields generated by first splatting the link information of a given graph in a certain layout or visual metaphor. To further visually enhance the time-varying graph structures we add user-adaptable isolines to the resulting dynamic graph representation. The computed visual encoding of the dynamic graph is aesthetically appealing due to its smooth curves and can additionally be used to do comparisons in a long graph sequence, i.e., from an information visualization perspective it serves as an overview representation supporting to start more detailed analysis processes. To demonstrate the usefulness of the technique we explore real-world dynamic graph data by taking into account visual parameters like visual metaphors, node-link layouts, smoothing iterations, number of isolines, and different color codings. In this extended work we additionally incorporate matrix and list splatting while also supporting the selection of density regions with overlaid link information. Moreover, from the selected graph the user can automatically apply region comparisons with other graphs based on global and local density properties. Such a feature is in particular useful for finding commonalities, hence serving as a special filtering function.

Consensus With Output Saturations

This paper considers a standard consensus algorithm under output saturations. In this case, the global consensus cannot be realized due to the existence of stable equilibrium points that do not belong to the consensus manifold. Therefore, this paper investigates necessary and sufficient initial conditions for the achievement of consensus that characterize an exact domain of attraction. Specifically, this paper considers single-integrator agents with both fixed and time-varying undirected graphs. Then, we show that the consensus will be achieved if and only if the average of the initial states is within the minimum saturation level.

Impact of Selfishness in Device-to-Device Communication Underlying Cellular Networks

In a device-to-device (D2D) communication underlying cellular network, a user equipment is desired to operate cooperatively and unselfishly to transmit data as relays. However, most users more or less behave in a selfish way, which makes user selfishness a key factor that affects the performance of the whole communication system. We focus on the impact of user selfishness on D2D communications. First, we distinguish user selfishness into two types in accordance with D2D transmission modes, namely the connected transmission and opportunistic transmission modes. The user selfishness in these two types of transmissions are modeled separately. Then, we propose a time-varying graph model that characterizes the impacts of both individual and social selfishness on the D2D communications. Simulation results obtained under the realistic networking settings indicate that the interaction between connected and opportunistic selfishness worsens the impairment caused by individual selfishness. Additionally, when concerning social selfishness, inside-community selfishness can be ignored in some occasions, while otherwise its role is heavily influenced by community numbers.

Survivability in Time-Varying Networks

Time-varying graphs are a useful model for networks with dynamic connectivity such as vehicular networks, yet, despite their great modeling power, many important features of time-varying graphs are still poorly understood. In this paper, we study the survivability properties of time-varying networks against unpredictable interruptions. We first show that the traditional definition of survivability is not effective in time-varying networks, and propose a new survivability framework. To evaluate the survivability of time-varying networks under the new framework, we propose two metrics that are analogous to MaxFlow and MinCut in static networks. We show that some fundamental survivability-related results such as Menger's Theorem only conditionally hold in time-varying networks. Then, we analyze the complexity of computing the proposed metrics and develop approximation algorithms. Finally, we conduct trace-driven simulations to demonstrate the application of our survivability framework in the robust design of a real-world bus communication network.

Time-Varying Graph Signal Reconstruction

Signal processing on graphs is an emerging research field dealing with signals living on an irregular domain that is captured by a graph, and has been applied to sensor networks, machine learning, climate analysis, etc. Existing works on sampling and reconstruction of graph signals mainly studied static bandlimited signals. However, many real-world graph signals are time-varying, and they evolve smoothly, so instead of the signals themselves being bandlimited or smooth on graph, it is more reasonable that their temporal differences are smooth on graph. In this paper, a new batch reconstruction method of time-varying graph signals is proposed by exploiting the smoothness of the temporal difference signals, and the uniqueness of the solution to the corresponding optimization problem is theoretically analyzed. Furthermore, driven by practical applications faced with real-time requirements, huge size of data, lack of computing center, or communication difficulties between two nonneighboring vertices, an online distributed method is proposed by applying local properties of the temporal difference operator and the graph Laplacian matrix. Experiments on a variety of synthetic and real-world datasets demonstrate the excellent performance of the proposed methods.

Controllability of Formations Over Directed Time-Varying Graphs

In this paper, we investigate the controllability of a class of formation control systems. Given a directed graph, we assign an agent to each of its vertices and let the edges of the graph describe the information flow in the system. We relate the strongly connected components of this graph to the reachable set of the formation control system. Moreover, we show that the formation control model is approximately path-controllable over a path-connected, open dense subset as long as the graph is weakly connected and satisfies some mild assumption on the numbers of vertices of the strongly connected components.