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E-raamat: Graph Spectra for Complex Networks

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(Technische Universiteit Delft, The Netherlands)
  • Formaat: PDF+DRM
  • Ilmumisaeg: 02-Dec-2010
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9780511984907
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Graph Spectra for Complex NetworksSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 02-Dec-2010
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9780511984907
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Analyzing the behavior of complex networks is an important element in the design of new man-made structures such as communication systems and biologically engineered molecules. Because any complex network can be represented by a graph, and therefore in turn by a matrix, graph theory has become a powerful tool in the investigation of network performance. This self-contained book provides a concise introduction to the theory of graph spectra and its applications to the study of complex networks. Covering a range of types of graphs and topics important to the analysis of complex systems, this guide provides the mathematical foundation needed to understand and apply spectral insight to real-world systems. In particular, the general properties of both the adjacency and Laplacian spectrum of graphs are derived and applied to complex networks. An ideal resource for researchers and students in communications networking as well as in physics and mathematics.

A concise and self-contained 2010 introduction to the theory of graph spectra and its applications to the study of complex networks. Covering a range of types of graphs, this guide provides the mathematical foundation needed to understand and apply spectral insight to real-world communications systems and networks.

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A concise and self-contained 2010 introduction to the theory of graph spectra and its applications to the study of complex networks.
Preface ix
Acknowledgements xiii
Symbols xv
1 Introduction
1(10)
1.1 Interpretation and contemplation
2(3)
1.2 Outline of the book
5(2)
1.3 Classes of graphs
7(3)
1.4 Outlook
10(1)
Part I Spectra of graphs
11(198)
2 Algebraic graph theory
13(16)
2.1 Graph related matrices
13(12)
2.2 Walks and paths
25(4)
3 Eigenvalues of the adjacency matrix
29(38)
3.1 General properties
29(4)
3.2 The number of walks
33(10)
3.3 Regular graphs
43(3)
3.4 Bounds for the largest, positive eigenvalue λ1
46(9)
3.5 Eigenvalue spacings
55(3)
3.6 Additional properties
58(5)
3.7 The stochastic matrix P = Δ-1 A
63(4)
4 Eigenvalues of the Laplacian Q
67(48)
4.1 General properties
67(13)
4.2 Second smallest eigenvalue of the Laplacian Q
80(9)
4.3 Partitioning of a graph
89(7)
4.4 The modularity and the modularity matrix M
96(12)
4.5 Bounds for the diameter
108(1)
4.6 Eigenvalues of graphs and subgraphs
109(6)
5 Spectra of special types of graphs
115(44)
5.1 The complete graph
115(1)
5.2 A small-world graph
115(8)
5.3 A circuit on N nodes
123(1)
5.4 A path of N - 1 hops
124(5)
5.5 A path of h hops
129(1)
5.6 The wheel WN+1
129(1)
5.7 The complete biPartite graph Km, n
129(2)
5.8 A general biPartite graph
131(4)
5.9 Complete multi-Partite graph
135(3)
5.10 An m-fully meshed star topology
138(9)
5.11 A chain of cliques
147(7)
5.12 The lattice
154(5)
6 Density function of the eigenvalues
159(20)
6.1 Definitions
159(2)
6.2 The density when N → ∞
161(2)
6.3 Examples of spectral density functions
163(3)
6.4 Density of a sparse regular graph
166(3)
6.5 Random matrix theory
169(10)
7 Spectra of complex networks
179(30)
7.1 Simple observations
179(2)
7.2 Distribution of the Laplacian eigenvalues and of the degree
181(3)
7.3 Functional brain network
184(1)
7.4 Rewiring Watts-Strogatz small-world graphs
185(2)
7.5 Assortativity
187(9)
7.6 Reconstructability of complex networks
196(3)
7.7 Reaching consensus
199(1)
7.8 Spectral graph metrics
200(9)
Part II Eigensystem and polynomials
209(130)
8 Eigensystem of a matrix
211(52)
8.1 Eigenvalues and eigenvectors
211(8)
8.2 Functions of a matrix
219(3)
8.3 Hermitian and real symmetric matrices
222(8)
8.4 Vector and matrix norms
230(5)
8.5 Non-negative matrices
235(5)
8.6 Positive (semi) definiteness
240(3)
8.7 Interlacing
243(9)
8.8 Eigenstructure of the product AB
252(3)
8.9 Formulae of determinants
255(8)
9 Polynomials with real coefficients
263(50)
9.1 General properties
263(7)
9.2 Transforming polynomials
270(4)
9.3 Interpolation
274(3)
9.4 The Euclidean algorithm
277(5)
9.5 Descartes' rule of signs
282(10)
9.6 The number of real zeros in an interval
292(3)
9.7 Locations of zeros in the complex plane
295(7)
9.8 Zeros of complex functions
302(3)
9.9 Bounds on values of a polynomial
305(1)
9.10 Bounds for the spacing between zeros
306(2)
9.11 Bounds on the zeros of a polynomial
308(5)
10 Orthogonal polynomials
313(26)
10.1 Definitions
313(2)
10.2 Properties
315(2)
10.3 The three-term recursion
317(6)
10.4 Zeros of orthogonal polynomials
323(3)
10.5 Gaussian quadrature
326(5)
10.6 The Jacobi matrix
331(8)
References 339(6)
Index 345
Piet Van Mieghem is a Professor at the Delft University of Technology with a chair in telecommunication networks, and chairman of the Network Architectures and Services (NAS) section. His main research interests lie in the modeling and analysis of complex networks (such as biological, brain, social, infrastructural, etc. networks) and in new Internet-like architectures and algorithms for future communications networks.
Lisainfo e-raamatute kohta Lisainfo e-raamatute kohta
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