Distributed Estimation of Graph Spectrum

In this paper, we develop a two-stage distributed algorithm that enables nodes in a graph to cooperatively estimate the spectrum of a matrix $W$ associated with the graph, which includes the adjacency and Laplacian matrices as special cases. In the first stage, the algorithm uses a discrete-time linear iteration and the Cayley-Hamilton theorem to convert the problem into one of solving a set of linear equations, where each equation is known to a node. In the second stage, if the nodes happen to know that $W$ is cyclic, the algorithm uses a Lyapunov approach to asymptotically solve the equations with an exponential rate of convergence. If they do not know whether $W$ is cyclic, the algorithm uses a random perturbation approach and a structural controllability result to approximately solve the equations with an error that can be made small. Finally, we provide simulation results that illustrate the algorithm.Comment: 15 pages, 2 figure

Connectivity and Consensus in Multi-Agent Systems with Uncertain Links

In the analysis and design of a multi-agent system (MAS), studying the graph representing the system is essential. In particular, when the communication links in a MAS are subject to uncertainty, a random graph is used to model the system. This type of graph is represented by a probability matrix, whose elements reflect the probability of the existence of the corresponding edges in the graph. This probability matrix needs to be adequately estimated. In this thesis, two approaches are proposed to estimate the probability matrix in a random graph. This matrix is time-varying and is used to determine the network configuration at different points in time. For evaluating the probability matrix, the connectivity of the network needs to be assessed first. It is to be noted that connectivity is a requirement for the convergence of any consensus algorithm in a network. The probability matrix is used in this work to study the consensus problem in a leader-follower asymmetric MAS with uncertain communication links. We propose a novel robust control approach to obtain an approximate agreement among agents under some realistic assumptions. The uncertainty is formulated as disturbance, and a controller is developed to debilitate it. Under the proposed controller, it is guaranteed that the consensus error satisfies the global L2-gain performance in the presence of uncertainty. The designed controller consists of two parts: one for time-varying links and one for time-invariant links. Simulations demonstrate the effectiveness of the proposed methods

Connectivity and Consensus in Multi-Agent Systems with Uncertain Links

In the analysis and design of a multi-agent system (MAS), studying the graph representing the system is essential. In particular, when the communication links in a MAS are subject to uncertainty, a random graph is used to model the system. This type of graph is represented by a probability matrix, whose elements reflect the probability of the existence of the corresponding edges in the graph. This probability matrix needs to be adequately estimated. In this thesis, two approaches are proposed to estimate the probability matrix in a random graph. This matrix is time-varying and is used to determine the network configuration at different points in time. For evaluating the probability matrix, the connectivity of the network needs to be assessed first. It is to be noted that connectivity is a requirement for the convergence of any consensus algorithm in a network. The probability matrix is used in this work to study the consensus problem in a leader-follower asymmetric MAS with uncertain communication links. We propose a novel robust control approach to obtain an approximate agreement among agents under some realistic assumptions. The uncertainty is formulated as disturbance, and a controller is developed to debilitate it. Under the proposed controller, it is guaranteed that the consensus error satisfies the global L2-gain performance in the presence of uncertainty. The designed controller consists of two parts: one for time-varying links and one for time-invariant links. Simulations demonstrate the effectiveness of the proposed methods

Distributed estimation of algebraic connectivity of directed networks

In directed network, algebraic connectivity is defined as the second smallest eigenvalue of graph Laplacian, and it captures the most conservative estimate of convergence rate and synchronicity of networked systems. In this paper, distributed estimation of algebraic connectivity of directed and connected graphs is studied using a decentralized power iteration scheme. Specifically, the proposed scheme is introduced in discrete time domain in order to take advantage of the discretized nature of information flow among networked systems and it shows that, with the knowledge of the first left eigenvector associated with trivial eigenvalue of graph Laplacian, distributed estimation of algebraic connectivity becomes possible. Moreover, it is revealed that the proposed estimation scheme still performs in estimating the complex eigenvalues. Simulation results demonstrate the effectiveness of the proposed scheme. (C) 2013 Elsevier B.V. All rights reserved

Distributed Estimation Of Algebraic Connectivity Of Directed Networks

In directed network, algebraic connectivity is defined as the second smallest eigenvalue of graph Laplacian, and it captures the most conservative estimate of convergence rate and synchronicity of networked systems. In this paper, distributed estimation of algebraic connectivity of directed and connected graphs is studied using a decentralized power iteration scheme. Specifically, the proposed scheme is introduced in discrete time domain in order to take advantage of the discretized nature of information flow among networked systems and it shows that, with the knowledge of the first left eigenvector associated with trivial eigenvalue of graph Laplacian, distributed estimation of algebraic connectivity becomes possible. Moreover, it is revealed that the proposed estimation scheme still performs in estimating the complex eigenvalues. Simulation results demonstrate the effectiveness of the proposed scheme. © 2013 Elsevier B.V. All rights reserved