An introduction to grids, graphs, and networks / C. Pozrikidis.

Format:

Book

Author/Creator:

Pozrikidis, C. (Constantine), 1958-

Language:

English

Subjects (All):

Graph theory.

Differential equations, Partial--Numerical solutions.

Differential equations, Partial.

Finite differences.

Genre:

Electronic books.

Physical Description:

1 online resource (299 pages) : illustrations

Place of Publication:

Oxford ; New York : Oxford University Press, [2014]

System Details:

text file

Summary:

An Introduction to Grids, Graphs, and Networks provides a concise introduction to graphs and networks at a level that is accessible to scientists, engineers, and students. In a practical approach, the book presents only the necessary theoretical concepts from mathematics, while considering a variety of physical and conceptual configurations as prototypes or examples. The subject is timely, as the performance of networks is recognized as an important topic in the study of complex systems with applications in energy, material, and information grid transport (epitomized by the internet). In the book, Pozrikidis gives an original synthesis of concepts and terms from three distinct fields-mathematics, physics, and engineering- and a formal application of powerful conceptual apparatuses, like lattice Green's function, to areas where they have rarely been used. It is novel in that grids, graphs, and networks are connected using concepts from partial differential equations. This original material has profound implications in the study of networks, and will serve as a resource to readers ranging from undergraduates to experienced scientists. Book jacket.

Contents:

1 One-Dimensional Grids 1

1.1 Poisson Equation in One Dimension 1

1.2 Dirichlet Boundary Condition at Both Ends 3

1.3 Neumann-Dirichlet Boundary Conditions 6

1.4 Dirichlet-Neumann Boundary Conditions 8

1.5 Neumann Boundary Conditions 10

1.6 Periodic Boundary Conditions 13

1.7 One-Dimensional Graphs 16

1.7.1 Graph Laplacian 17

1.7.2 Adjacency Matrix 18

1.7.3 Connectivity Lists and Oriented Incidence Matrix 19

1.8 Periodic One-Dimensional Graphs 20

1.8.1 Periodic Adjacency Matrix 21

1.8.2 Periodic Oriented Incidence Matrix 22

1.8.3 Fourier Expansions 22

1.8.4 Cosine Fourier Expansion 24

1.8.5 Sine Fourier Expansion 24

2 Graphs and Networks 26

2.1 Elements of Graph Theory 26

2.1.1 Adjacency Matrix 26

2.1.2 Node Degrees 28

2.1.3 The Complete Graph 29

2.1.4 Complement of a Graph 29

2.1.5 Connectivity Lists and the Oriented Incidence Matrix 30

2.1.6 Connected and Unconnected Graphs 30

2.1.7 Pairwise Distance and Diameter 30

2.1.8 Trees 31

2.1.9 Random and Real-Life Networks 31

2.2 Laplacian Matrix 32

2.2.1 Properties of the Laplacian Matrix 33

2.2.2 Complete Graph 34

2.2.3 Estimates of Eigenvalues 35

2.2.4 Spanning Trees 36

2.2.5 Spectral Expansion 36

2.2.6 Spectral Partitioning 36

2.2.7 Complement of a Graph 38

2.2.8 Normalized Laplacian 38

2.2.9 Graph Breakup 39

2.3 Cubic Network 39

2.4 Fabricated Networks 41

2.4.1 Finite-Element Network on a Disk 42

2.4.2 Finite-Element Network on a Square 43

2.4.3 Delaunay Triangulation of an Arbitrary Set of Nodes 43

2.4.4 Delaunay Triangulation of a Perturbed Cartesian Grid 43

2.4.5 Finite Element Network Descending from an Octahedron 44

2.4.6 Finite Element Network Descending from an Icosahedron 45

2.5 Link Removal and Addition 46

2.5.1 Single and Multiple Link 47

2.5.2 Link Addition 49

2.6 Infinite Lattices 50

2.6.1 Bravais Lattices 50

2.6.2 Archimedean Lattices 53

2.6.3 Laves Lattices 56

2.6.4 Other Two-Dimensional Lattices 57

2.6.5 Cubic Lattices 58

2.7 Percolation Thresholds 59

2.7.1 Link (Bond) Percolation Threshold 59

2.7.2 Node Percolation Threshold 61

2.7.3 Computation of Percolation Thresholds 62

3 Spectra of Lattices 67

3.1 Square Lattice 67

3.1.1 Isolated Network 68

3.1.2 Periodic Strip 69

3.1.3 Doubly Periodic Network 73

3.1.4 Doubly Periodic Sheared Network 77

3.2 Möbius Strips 79

3.2.1 Horizontal Strip 80

3.2.2 Vertical Strip 83

3.2.3 Klein Bottle 84

3.3 Hexagonal Lattice 86

3.3.1 Isolated Network 87

3.3.2 Doubly Periodic Network 89

3.3.3 Alternative Node Indexing 92

3.4 Modified Union Jack Lattice 93

3.4.1 Isolated Network 94

3.4.2 Doubly Periodic Network 95

3.5 Honeycomb Lattice 98

3.5.1 Isolated Network 99

3.5.2 Brick Representation 101

3.5.3 Doubly Periodic Network 102

3.5.4 Alternative Node Indexing 110

3.6 Kagomé Lattice 111

3.6.1 Isolated Network 112

3.6.2 Doubly Periodic Network 115

3.7 Simple Cubic Lattice 122

3.8 Body-Centered Cubic (bcc) Lattice 124

3.9 Face-Centered Cubic (fee) Lattice 126

4 Network Transport 130

4.1 Transport Laws and Conventions 130

4.1.1 Isolated and Embedded Networks 130

4.1.2 Nodal Sources 131

4.1.3 Linear Transport 132

4.1.4 Nonlinear Transport 133

4.2 Uniform Conductances 133

4.2.1 Isolated Networks 134

4.2.2 Embedded Networks 134

4.3 Arbitrary Conductances 135

4.3.1 Scaled Conductance Matrix 136

4.3.2 Weighed Adjacency Matrix 136

4.3.3 Weighed Node Degrees 137

4.3.4 Kirchhoff Matrix 138

4.3.5 Weighed Oriented Incidence Matrix 139

4.3.6 Properties of the Kirchhoff Matrix 139

4.3.7 Normalized Kirchhoff Matrix 140

4.3.8 Summary of Notation 141

4.4 Nodal Balances in Arbitrary Networks 142

4.4.1 Isolated Networks 142

4.4.2 Embedded Networks and the Modified Kirchhoff Matrix 142

4.4.3 Properties of the Modified Kirchhoff Matrix 143

4.5 Lattices 145

4.5.1 Square Lattice 145

4.5.2 Mobius Strip 149

4.5.3 Hexagonal Lattice 150

4.5.4 Modified Union Jack Lattice 150

4.5.5 Simple Cubic Lattice 151

4.6 Finite Difference Grids 153

4.7 Finite Element Grids 156

4.7.1 One-Dimensional Grid 156

4.7.2 Two-Dimensional Grid 157

5 Green's Functions 161

5.1 Embedded Networks 161

5.1.1 Green's Function Matrix 162

5.1.2 Normalized Green's Function 163

5.2 Isolated Networks 164

5.2.1 Moore-Penrose Green's Function 164

5.2.2 Spectral Expansion 166

5.2.3 Normalized Moore-Penrose Green's Function 167

5.2.4 One-Dimensional Network 168

5.2.5 Periodic One-Dimensional Network 169

5.2.6 Free-Space Green's Function in One Dimension 171

5.2.7 Complete Network 171

5.2.8 Discontiguous Networks 172

5.3 Lattice Green's Functions 173

5.3.1 Periodic Green's Functions 173

5.3.2 Free-Space Green's Functions 175

5.4 Square Lattice 177

5.4.1 Periodic Green's Function 177

5.4.2 Free-Space Green's Function 179

5.4.3 Helmholtz Equation Green's Function 190

5.4.4 Kirchhoff Green's Function 190

5.5 Hexagonal Lattice 191

5.5.1 Periodic Green's Function 191

5.5.2 Free-Space Green's Function 192

5.6 Modified Union Jack Lattice 196

5.6.1 Periodic Green's Function 197

5.6.2 Free-Space Green's Function 198

5.7 Honeycomb Lattice 200

5.7.1 Periodic Green's Function 201

5.7.2 Free-Space Green's Function 203

5.8 Simple Cubic Lattice 206

5.8.1 Periodic Green's Function 206

5.8.2 Free-Space Green's Function 207

5.9 Body-Centered Cubic (bcc) Lattice 209

5.10 Face-Centered Cubic (fee) Lattice 211

5.11 Free-Space Lattice Green's Functions 212

5.11.1 Probability Lattice Green's Function 213

5.12 Finite Difference Solution in Terms of Green's Functions 216

6 Network Performance 220

6.1 Pairwise Resistance 220

6.1.1 Embedded Networks 221

6.1.2 Isolated Networks 223

6.1.3 One-Dimensional Network 225

6.1.4 One-Dimensional Periodic Network 226

6.1.5 Infinite Lattices 226

6.1.6 Triangle Inequality 227

6.1.7 Random Walks 227

6.2 Mean Pairwise Resistance 228

6.2.1 Spectral Representation 228

6.2.2 Complete Network 229

6.2.3 One-Dimensional Isolated Network 229

6.2.4 One-Dimensional Periodic Network 230

6.2.5 Periodic Lattice Patches 231

6.3 Damaged Networks 234

6.3.1 Damaged Kirchhoff Matrix 235

6.3.2 Embedded Networks 236

6.3.3 One Damaged Link 238

6.3.4 Clipped Links 240

6.3.5 Isolated Networks 240

6.4 Reinforced Networks 240

6.5 Damaged Lattices 242

6.5.1 One Damaged Link 242

6.5.2 Effective-Medium Theory 245

6.5.3 Percolation Threshold 246

6.6 Damaged Square Lattice 247

6.7 Damaged Honeycomb Lattice 251

6.8 Damaged Hexagonal Lattice 255

6.8.1 Longitudinal Transport 255

6.8.2 Lateral Transport 257.

Notes:

Includes bibliographical references and index.

Description based on print version record.

Local Notes:

Electronic reproduction. Palo Alto, Calif. : ebrary, 2014. Available via World Wide Web. Access may be limited to ebrary affiliated libraries.

Other Format:

Print version: Pozrikidis, C. Introduction to grids, graphs, and networks.

ISBN:

9780199996735

OCLC:

870588975

Access Restriction:

Restricted for use by site license.