scispace - formally typeset
Topic

Algebraic connectivity

About: Algebraic connectivity is a(n) research topic. Over the lifetime, 1392 publication(s) have been published within this topic receiving 60848 citation(s).
Papers
More filters

Journal ArticleDOI
TL;DR: A distinctive feature of this work is to address consensus problems for networks with directed information flow by establishing a direct connection between the algebraic connectivity of the network and the performance of a linear consensus protocol.
Abstract: In this paper, we discuss consensus problems for networks of dynamic agents with fixed and switching topologies. We analyze three cases: 1) directed networks with fixed topology; 2) directed networks with switching topology; and 3) undirected networks with communication time-delays and fixed topology. We introduce two consensus protocols for networks with and without time-delays and provide a convergence analysis in all three cases. We establish a direct connection between the algebraic connectivity (or Fiedler eigenvalue) of the network and the performance (or negotiation speed) of a linear consensus protocol. This required the generalization of the notion of algebraic connectivity of undirected graphs to digraphs. It turns out that balanced digraphs play a key role in addressing average-consensus problems. We introduce disagreement functions for convergence analysis of consensus protocols. A disagreement function is a Lyapunov function for the disagreement network dynamics. We proposed a simple disagreement function that is a common Lyapunov function for the disagreement dynamics of a directed network with switching topology. A distinctive feature of this work is to address consensus problems for networks with directed information flow. We provide analytical tools that rely on algebraic graph theory, matrix theory, and control theory. Simulations are provided that demonstrate the effectiveness of our theoretical results.

10,379 citations


Book
03 Dec 1996-
TL;DR: Eigenvalues and the Laplacian of a graph Isoperimetric problems Diameters and eigenvalues Paths, flows, and routing Eigen values and quasi-randomness
Abstract: Eigenvalues and the Laplacian of a graph Isoperimetric problems Diameters and eigenvalues Paths, flows, and routing Eigenvalues and quasi-randomness Expanders and explicit constructions Eigenvalues of symmetrical graphs Eigenvalues of subgraphs with boundary conditions Harnack inequalities Heat kernels Sobolev inequalities Advanced techniques for random walks on graphs Bibliography Index.

6,908 citations


Journal ArticleDOI

3,549 citations


Journal ArticleDOI
TL;DR: Applied to networks of low redundancy, the small-world route produces synchronizability more efficiently than standard deterministic graphs, purely random graphs, and ideal constructive schemes.
Abstract: We quantify the dynamical implications of the small-world phenomenon by considering the generic synchronization of oscillator networks of arbitrary topology. The linear stability of the synchronous state is linked to an algebraic condition of the Laplacian matrix of the network. Through numerics and analysis, we show how the addition of random shortcuts translates into improved network synchronizability. Applied to networks of low redundancy, the small-world route produces synchronizability more efficiently than standard deterministic graphs, purely random graphs, and ideal constructive schemes. However, the small-world property does not guarantee synchronizability: the synchronization threshold lies within the boundaries, but linked to the end of the small-world region.

1,401 citations


1


01 Jan 1991-
Abstract: The paper is essentially a survey of known results about the spectrum of the Laplacian matrix of graphs with special emphasis on the second smallest Lapla- cian eigenvalue 2 and its relation to numerous graph invariants, including connectivity, expanding properties, isoperimetric number, maximum cut, independence number, genus, diameter, mean distance, and bandwidth-type parameters of a graph. Some new results and generalizations are added.

1,374 citations


Network Information
Related Topics (5)
Adjacency matrix

6.7K papers, 147.4K citations

89% related
Laplacian matrix

5.7K papers, 167.3K citations

87% related
Directed graph

12.2K papers, 302.4K citations

87% related
Strongly connected component

2.3K papers, 51.6K citations

87% related
Graph theory

20.8K papers, 691.4K citations

86% related
Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20222
202143
202037
201960
201851
201795

Top Attributes

Show by:

Topic's top 5 most impactful authors

Wai Chee Shiu

12 papers, 179 citations

Russell Merris

11 papers, 3K citations

Jason J. Molitierno

10 papers, 88 citations

Ali Kaveh

9 papers, 102 citations

Steve Kirkland

9 papers, 197 citations