Distributed estimation of algebraic connectivity of directed networks

doi:10.1016/j.sysconle.2013.03.002

Systems & Control Letters

Volume 62, Issue 6, June 2013, Pages 517-524

https://doi.org/10.1016/j.sysconle.2013.03.002 Get rights and content

Abstract

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.

Introduction

Because of its emerging potential in networked control systems, distributed estimation of network connectivity became an attractive topic in the past decade [1]. Common challenge in this venue is that each system estimates the global criteria of the overall network using only the local measurements and information received from its connected peers. Such global criteria could be eigenvalues and its associated eigenvectors, or most recently the first left eigenvector (corresponding to eigenvalue 0 of graph Laplacian). Among them, the most prominent and essential one is the second smallest eigenvalue of graph Laplacian, also known as algebraic connectivity, which captures the synchronicity and convergence rate of networked systems [2], [3]. Hence, it shall be both theoretical interesting and practically useful if algebraic connectivity can be known explicitly and locally in the absence of global information.

In the event that network is either undirected or directed but balanced, algebraic connectivity is defined as the second smallest eigenvalue of graph Laplacian (i.e., Fiedler value [4]). Since analytical solution to Fiedler value is known [3], [5], numerical solution can thus be readily applied to estimate or improve its value. For instance, a decentralized orthogonal iteration approach is proposed in [6] to estimate the leading $k$ eigenvectors, but this approach is not scalable and also requires a centralized initialization, similar work on this topic can be found in [7]; another breakthrough worth noting is using the Fast Fourier Transform (FFT) [8] by constructing distributed oscillators whose states oscillate at frequencies corresponding to the eigenvalues of graph Laplacian, however FFT is not appropriate for real-time implementation nor for handling switching topologies. The arguably most effective schemes in estimating Fiedler value are [9] and [10]. Specifically, a decentralized power iteration approach is introduced in [9] to estimate components of the Fiedler eigenvector and then the Fiedler value in a continuous-time fashion, but the proposed estimation scheme requires global initialization and cannot handle switching topologies. In [10], power of adjacency matrix are used to calculate the upper and lower boundary of algebraic connectivity. In terms of improving Fiedler value or algebraic connectivity, a centralized semi-definite programming (SDP) solver is proposed in [11] to maximize Fiedler value directly; and a similar approach can be found in [12], where relay locations are selected to optimize the connectivity. As an extension, a decentralized supergradient algorithm is proposed in [13], but this requires the a priori knowledge of Fiedler eigenvector and communication overhead during each iteration. Components of distance-dependent graph Laplacian are optimized in [14] by driving mobile robots to appropriate locations, and algebraic connectivity has been maximized as a result.

However, it should be pointed out that all of the aforementioned results on distributed estimation and control are restricted to undirected or directed but balanced network. For directed networks, the most notable work is focused on distributed estimation of the first left eigenvector and its applications in improving network convergence [15], [16], [17]. To be more precise, a supervisory node with global information of network topology is introduced in [15] to improve network convergence by making time derivatives of cooperative control Lyapunov functions more negative; extensions of this approach can be found in [16], [17], where network performance is enhanced distributively with local estimation of the first left eigenvector, consensus vector, and consequently the cooperative control Lyapunov function. To the best of our knowledge, little is available on distributed estimation of algebraic connectivity of directed network.

In this paper, algebraic connectivity of directed graph is estimated using a decentralized power iteration scheme, whose effectiveness has already been verified in calculating eigenstructure [9], [18]. It is shown that, with local knowledge of the first left eigenvector and affine transformation, algebraic connectivity of directed network can be estimated distributively. It is demonstrated that the proposed scheme overcomes the inherent shortcomings associated with power iteration. That is, despite outputs of power iteration always being real, it is applicable in estimating complex eigenvalues. This paper is organized as follows. In Section 2, preliminary results of graph theory and the relevant mathematical results on the first left eigenvector are summarized for directed and switching networks. In Section 3, estimation of algebraic connectivity of a digraph is formulated and the main findings of this paper are presented. Section 4 focuses upon the proof of the main theorem. More specifically, in Section 4.1, properties of the affine transformation are studied and its relation with respect to algebraic connectivity is explicitly found. Then, in Section 4.2, the proof of main theorem is carried out for digraphs with simple and complex eigenvalues, and effectiveness of the proposed scheme is verified by numerical examples. In Section 5, conclusion of the underlying problem is reached.

Section snippets

Preliminaries on algebraic graph theory

In this paper, we consider a digraph $D = (V, E)$ , where $V = {1, 2, \dots, n}$ and $E$ denote the sets of vertices/nodes and directed edges, respectively. Unless otherwise specified, vertex $j$ is said to be adjacent to vertex $i$ if there exists a directed edge $(j, i) \in E$ with $i$ being tail of the edge and $j$ being the head. Analogously, neighborhood set $N_{i} \subseteq V$ of vertex $i$ is ${k \in V ∣ (k, i) \in E}$ , the set of all vertices that are adjacent to vertex $i$ . Hence, cardinality of $N_{i}$ represents the numbers of connected neighbor(s) of

Problem formulation

In this section, algebraic connectivity of digraph under switching topologies is formulated and its distributed estimation is motivated. Moreover, the main findings of this paper is presented and its proof will be carried out rigorously in the next section. In what follows, we define the time sequences ${t_{k} : k \in ℵ}$ for $ℵ = {0, 1, \dots, \infty}$ , and without loss of any generality, digraph $D$ is assumed to be time invariant during each interval $[t_{k}, t_{k + 1})$ , and its corresponding $A$ or $L$ is piecewise-constant. That

Proof of the main theorem

In this section, proof of Theorem 1 is carried out. Special attention will be paid to decentralized implementation of power iteration and its application in estimating complex eigenvalues. As indicated in (5), classical power iteration dictates that a normalization/deflation should be performed at each step with explicit knowledge of the current estimates. However, in networked control systems, such recursive deflations can not be executed locally nor is it possible. On the other hand, for

Conclusion

This paper investigates distributed estimation of algebraic connectivity of a digraph. The proposed scheme is based on a decentralized power iteration and affine transformation of the first left eigenvector. It is shown that, with knowledge of the first left eigenvector and the proposed consensus observer, distributed estimation of algebraic connectivity is possible, even when eigen-structure is complex. Indeed, the proposed scheme, together with numerical calculation, is also applicable when

Acknowledgments

The authors thank all anonymous reviewers for their constructive comments that improved the quality of this paper. This research is supported by the U.S Department of Energy’s SEGIS program and the U.S. National Science Foundation (CCF-0956501 and CMMI-0825502).

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Distributed estimation of algebraic connectivity of directed networks

Abstract

Introduction

Section snippets

Preliminaries on algebraic graph theory

Problem formulation

Proof of the main theorem

Conclusion

Acknowledgments

Linear Algebra and its Applications

Journal of Computer and System Sciences

Automatica

IEEE Transactions on Automatic Control

Systems & Control Letters

A survey on distributed estimation and control applications using linear consensus algorithms

Networked Control Systems

Algebraic connectivity of directed graphs

Linear and Multilinear Algebra

The diameter and Laplacian eigenvalues of directed graphs

Electronic Journal of Combinatorics

Algebraic connectivity of graphs

Czechoslovak Mathematical Journal

Decentralized Laplacian eigenvalues estimation and collaborative network topology identification