An Introduction to Grids, Graphs, and Networks

299 Pages·2014·4.876 MB·English

by  C. Pozrikidis

AN INTRODUCTION TO GRIDS,
GRAPHS, AND NETWORKS

AN INTRODUCTION
TO GRIDS, GRAPHS,
AND NETWORKS
C. Pozrikidis
3
3
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LibraryofCongressCataloging-in-PublicationData
Pozrikidis,C.(Constantine),1958–author.
Anintroductiontogrids,graphs,andnetworks/C.Pozrikidis.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN978–0–19–999672–8(alk.paper)
1. Graphtheory.2. Differentialequations,Partial—Numericalsolutions.3. Finitedifferences.I.Title.
QA166.P692014
511’.5—dc23
2013048508
1 3 5 7 9 8 6 4 2
PrintedintheUnitedStatesofAmerica
onacid-freepaper
CONTENTS
Preface xi
1. One-DimensionalGrids 1
1.1. PoissonEquationinOneDimension 1
1.2. DirichletBoundaryConditionatBothEnds 3
1.3. Neumann–DirichletBoundaryConditions 6
1.4. Dirichlet–NeumannBoundaryConditions 8
1.5. NeumannBoundaryConditions 10
1.6. PeriodicBoundaryConditions 13
1.7. One-DimensionalGraphs 16
1.7.1. GraphLaplacian 17
1.7.2. AdjacencyMatrix 18
1.7.3. ConnectivityListsandOrientedIncidenceMatrix 19
1.8. PeriodicOne-DimensionalGraphs 20
1.8.1. PeriodicAdjacencyMatrix 21
1.8.2. PeriodicOrientedIncidenceMatrix 22
1.8.3. FourierExpansions 22
1.8.4. CosineFourierExpansion 24
1.8.5. SineFourierExpansion 24
2. GraphsandNetworks 26
2.1. ElementsofGraphTheory 26
2.1.1. AdjacencyMatrix 26
2.1.2. NodeDegrees 28
2.1.3. TheCompleteGraph 29
2.1.4. ComplementofaGraph 29
2.1.5. ConnectivityListsandtheOrientedIncidenceMatrix 30
vi // CONTENTS
2.1.6. ConnectedandUnconnectedGraphs 30
2.1.7. PairwiseDistanceandDiameter 30
2.1.8. Trees 31
2.1.9. RandomandReal-LifeNetworks 31
2.2. LaplacianMatrix 32
2.2.1. PropertiesoftheLaplacianMatrix 33
2.2.2. CompleteGraph 34
2.2.3. EstimatesofEigenvalues 35
2.2.4. SpanningTrees 36
2.2.5. SpectralExpansion 36
2.2.6. SpectralPartitioning 36
2.2.7. ComplementofaGraph 38
2.2.8. NormalizedLaplacian 38
2.2.9. GraphBreakup 39
2.3. CubicNetwork 39
2.4. FabricatedNetworks 41
2.4.1. Finite-ElementNetworkonaDisk 42
2.4.2. Finite-ElementNetworkonaSquare 43
2.4.3. DelaunayTriangulationofanArbitrarySetofNodes 43
2.4.4. DelaunayTriangulationofaPerturbedCartesianGrid 43
2.4.5. FiniteElementNetworkDescendingfromanOctahedron 44
2.4.6. FiniteElementNetworkDescendingfromanIcosahedron 45
2.5. LinkRemovalandAddition 46
2.5.1. SingleandMultipleLink 47
2.5.2. LinkAddition 49
2.6. InfiniteLattices 50
2.6.1. BravaisLattices 50
2.6.2. ArchimedeanLattices 53
2.6.3. LavesLattices 56
2.6.4. OtherTwo-DimensionalLattices 57
2.6.5. CubicLattices 58
2.7. PercolationThresholds 59
2.7.1. Link(Bond)PercolationThreshold 59
2.7.2. NodePercolationThreshold 61
2.7.3. ComputationofPercolationThresholds 62
3. SpectraofLattices 67
3.1. SquareLattice 67
3.1.1. IsolatedNetwork 68
3.1.2. PeriodicStrip 69
CONTENTS // vii
3.1.3. DoublyPeriodicNetwork 73
3.1.4. DoublyPeriodicShearedNetwork 77
3.2. MöbiusStrips 79
3.2.1. HorizontalStrip 80
3.2.2. VerticalStrip 83
3.2.3. KleinBottle 84
3.3. HexagonalLattice 86
3.3.1. IsolatedNetwork 87
3.3.2. DoublyPeriodicNetwork 89
3.3.3. AlternativeNodeIndexing 92
3.4. ModifiedUnionJackLattice 93
3.4.1. IsolatedNetwork 94
3.4.2. DoublyPeriodicNetwork 95
3.5. HoneycombLattice 98
3.5.1. IsolatedNetwork 99
3.5.2. BrickRepresentation 101
3.5.3. DoublyPeriodicNetwork 102
3.5.4. AlternativeNodeIndexing 110
3.6. KagoméLattice 111
3.6.1. IsolatedNetwork 112
3.6.2. DoublyPeriodicNetwork 115
3.7. SimpleCubicLattice 122
3.8. Body-CenteredCubic(bcc)Lattice 124
3.9. Face-CenteredCubic(fcc)Lattice 126
4. NetworkTransport 130
4.1. TransportLawsandConventions 130
4.1.1. IsolatedandEmbeddedNetworks 130
4.1.2. NodalSources 131
4.1.3. LinearTransport 132
4.1.4. NonlinearTransport 133
4.2. UniformConductances 133
4.2.1. IsolatedNetworks 134
4.2.2. EmbeddedNetworks 134
4.3. ArbitraryConductances 135
4.3.1. ScaledConductanceMatrix 136
4.3.2. WeighedAdjacencyMatrix 136
viii // CONTENTS
4.3.3. WeighedNodeDegrees 137
4.3.4. KirchhoffMatrix 138
4.3.5. WeighedOrientedIncidenceMatrix 139
4.3.6. PropertiesoftheKirchhoffMatrix 139
4.3.7. NormalizedKirchhoffMatrix 140
4.3.8. SummaryofNotation 141
4.4. NodalBalancesinArbitraryNetworks 142
4.4.1. IsolatedNetworks 142
4.4.2. EmbeddedNetworksandtheModifiedKirchhoffMatrix 142
4.4.3. PropertiesoftheModifiedKirchhoffMatrix 143
4.5. Lattices 145
4.5.1. SquareLattice 145
4.5.2. MöbiusStrip 149
4.5.3. HexagonalLattice 150
4.5.4. ModifiedUnionJackLattice 150
4.5.5. SimpleCubicLattice 151
4.6. FiniteDifferenceGrids 153
4.7. FiniteElementGrids 156
4.7.1. One-DimensionalGrid 156
4.7.2. Two-DimensionalGrid 157
5. Green’sFunctions 161
5.1. EmbeddedNetworks 161
5.1.1. Green’sFunctionMatrix 162
5.1.2. NormalizedGreen’sFunction 163
5.2. IsolatedNetworks 164
5.2.1. Moore–PenroseGreen’sFunction 164
5.2.2. SpectralExpansion 166
5.2.3. NormalizedMoore–PenroseGreen’sFunction 167
5.2.4. One-DimensionalNetwork 168
5.2.5. PeriodicOne-DimensionalNetwork 169
5.2.6. Free-SpaceGreen’sFunctioninOneDimension 171
5.2.7. CompleteNetwork 171
5.2.8. DiscontiguousNetworks 172
5.3. LatticeGreen’sFunctions 173
5.3.1. PeriodicGreen’sFunctions 173
5.3.2. Free-SpaceGreen’sFunctions 175
5.4. SquareLattice 177
5.4.1. PeriodicGreen’sFunction 177
5.4.2. Free-SpaceGreen’sFunction 179
CONTENTS // ix
5.4.3. HelmholtzEquationGreen’sFunction 190
5.4.4. KirchhoffGreen’sFunction 190
5.5. HexagonalLattice 191
5.5.1. PeriodicGreen’sFunction 191
5.5.2. Free-SpaceGreen’sFunction 192
5.6. ModifiedUnionJackLattice 196
5.6.1. PeriodicGreen’sFunction 197
5.6.2. Free-SpaceGreen’sFunction 198
5.7. HoneycombLattice 200
5.7.1. PeriodicGreen’sFunction 201
5.7.2. Free-SpaceGreen’sFunction 203
5.8. SimpleCubicLattice 206
5.8.1. PeriodicGreen’sFunction 206
5.8.2. Free-SpaceGreen’sFunction 207
5.9. Body-CenteredCubic(bcc)Lattice 209
5.10. Face-CenteredCubic(fcc)Lattice 211
5.11. Free-SpaceLatticeGreen’sFunctions 212
5.11.1. ProbabilityLatticeGreen’sFunction 213
5.12. FiniteDifferenceSolutioninTermsofGreen’sFunctions 216
6. NetworkPerformance 220
6.1. PairwiseResistance 220
6.1.1. EmbeddedNetworks 221
6.1.2. IsolatedNetworks 223
6.1.3. One-DimensionalNetwork 225
6.1.4. One-DimensionalPeriodicNetwork 226
6.1.5. InfiniteLattices 226
6.1.6. TriangleInequality 227
6.1.7. RandomWalks 227
6.2. MeanPairwiseResistance 228
6.2.1. SpectralRepresentation 228
6.2.2. CompleteNetwork 229
6.2.3. One-DimensionalIsolatedNetwork 229
6.2.4. One-DimensionalPeriodicNetwork 230
6.2.5. PeriodicLatticePatches 231
6.3. DamagedNetworks 234
6.3.1. DamagedKirchhoffMatrix 235
6.3.2. EmbeddedNetworks 236