Algebraic connectivity of graphs
01 Jan 1973-Czechoslovak Mathematical Journal (Institute of Mathematics, Academy of Sciences of the Czech Republic)-Vol. 23, Iss: 2, pp 298-305
About: This article is published in Czechoslovak Mathematical Journal.The article was published on 1973-01-01 and is currently open access. It has received 3888 citations till now. The article focuses on the topics: Algebraic graph theory & Irreducible component.
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08 Sep 2000TL;DR: This book presents dozens of algorithms and implementation examples, all in pseudo-code and suitable for use in real-world, large-scale data mining projects, and provides a comprehensive, practical look at the concepts and techniques you need to get the most out of real business data.
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23,600 citations
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.
11,658 citations
Cites methods from "Algebraic connectivity of graphs"
...This is different than the approach pursued in the work of Jadbabaie et al. which strongly relies on matrix theoretic properties and infinite right-convergent products (RCP) of stochastic matrices [36]....
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TL;DR: In this article, the modularity of a network is expressed in terms of the eigenvectors of a characteristic matrix for the network, which is then used for community detection.
Abstract: Many networks of interest in the sciences, including social networks, computer networks, and metabolic and regulatory networks, are found to divide naturally into communities or modules. The problem of detecting and characterizing this community structure is one of the outstanding issues in the study of networked systems. One highly effective approach is the optimization of the quality function known as “modularity” over the possible divisions of a network. Here I show that the modularity can be expressed in terms of the eigenvectors of a characteristic matrix for the network, which I call the modularity matrix, and that this expression leads to a spectral algorithm for community detection that returns results of demonstrably higher quality than competing methods in shorter running times. I illustrate the method with applications to several published network data sets.
10,137 citations
05 Mar 2007
TL;DR: A theoretical framework for analysis of consensus algorithms for multi-agent networked systems with an emphasis on the role of directed information flow, robustness to changes in network topology due to link/node failures, time-delays, and performance guarantees is provided.
Abstract: This paper provides a theoretical framework for analysis of consensus algorithms for multi-agent networked systems with an emphasis on the role of directed information flow, robustness to changes in network topology due to link/node failures, time-delays, and performance guarantees. An overview of basic concepts of information consensus in networks and methods of convergence and performance analysis for the algorithms are provided. Our analysis framework is based on tools from matrix theory, algebraic graph theory, and control theory. We discuss the connections between consensus problems in networked dynamic systems and diverse applications including synchronization of coupled oscillators, flocking, formation control, fast consensus in small-world networks, Markov processes and gossip-based algorithms, load balancing in networks, rendezvous in space, distributed sensor fusion in sensor networks, and belief propagation. We establish direct connections between spectral and structural properties of complex networks and the speed of information diffusion of consensus algorithms. A brief introduction is provided on networked systems with nonlocal information flow that are considerably faster than distributed systems with lattice-type nearest neighbor interactions. Simulation results are presented that demonstrate the role of small-world effects on the speed of consensus algorithms and cooperative control of multivehicle formations
9,715 citations
Cites background from "Algebraic connectivity of graphs"
...Graph Laplacians and their spectral properties [ 20 ]–[23] are important graph-related matrices that play a crucial role in convergence analysis of consensus and alignment algorithms....
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...The second smallest eigenvalue of Laplacian � 2 is called algebraic connectivity of a graph [ 20 ]....
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TL;DR: The major concepts and results recently achieved in the study of the structure and dynamics of complex networks are reviewed, and the relevant applications of these ideas in many different disciplines are summarized, ranging from nonlinear science to biology, from statistical mechanics to medicine and engineering.
9,441 citations
References
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TL;DR: In this paper, the authors give conditions that two graphs be congruent and some theorems on the connectivity of graphs, and conclude with some applications to dual graphs, which can also be proved by topological methods.
Abstract: We give here conditions that two graphs be congruent and some theorems on the connectivity of graphs, and we conclude with some applications to dual graphs. These last theorems might also be proved by topological methods. The definitions and results of a paper by the author on “Non-separable and planar graphs,” † will be made use of constantly. We shall refer to this paper as N. For convenience, we shall say two arcs touch if they have a common vertex.
1,114 citations
Additional excerpts
...Since Йе = 0, it follows that уТ^Му = clx'^Mx = cl{x'^Mx ~ Д2) è 0 by (3)....
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CSAV1
51 citations
"Algebraic connectivity of graphs" refers background in this paper
...Thus the minimum diagonal entry of Й is nonnegative: min тц — ^2(1 — и~^) ^ О i and (2) is proved....
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...Then the second smallest eigenvalue À2 of M satisfies (2) ^2 S ["/('î "- 1)] niin тц ....
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