Preface |
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ix | |
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xi | |
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1 | (16) |
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3 | (3) |
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3 | (2) |
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5 | (1) |
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6 | (8) |
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6 | (4) |
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Nearest neighbour networks |
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10 | (1) |
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Poisson random connection networks |
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11 | (1) |
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12 | (1) |
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Interference limited networks |
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13 | (1) |
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Information-theoretic networks |
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14 | (1) |
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Historical notes and further reading |
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15 | (2) |
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Phase transitions in infinite networks |
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17 | (52) |
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The random tree; infinite growth |
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17 | (4) |
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The random grid; discrete percolation |
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21 | (8) |
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29 | (2) |
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Nearest neighbours; continuum percolation |
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31 | (6) |
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37 | (11) |
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48 | (3) |
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Interference limited networks |
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51 | (15) |
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Mapping on a square lattice |
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56 | (2) |
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Percolation on the square lattice |
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58 | (4) |
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Percolation of the interference model |
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62 | (1) |
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Bound on the percolation region |
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63 | (3) |
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Historical notes and further reading |
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66 | (3) |
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Connectivity of finite networks |
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69 | (31) |
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Preliminaries: modes of convergence and Poisson approximation |
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69 | (2) |
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71 | (6) |
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71 | (1) |
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72 | (5) |
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77 | (11) |
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78 | (3) |
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81 | (7) |
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Nearest neighbours; full connectivity |
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88 | (4) |
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92 | (6) |
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98 | (1) |
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Historical notes and further reading |
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98 | (2) |
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More on phase transitions |
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100 | (21) |
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Preliminaries: Harris-FKG Inequality |
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100 | (1) |
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Uniqueness of the infinite cluster |
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101 | (6) |
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Cluster size distribution and crossing paths |
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107 | (7) |
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Threshold behaviour of fixed size networks |
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114 | (5) |
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Historical notes and further reading |
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119 | (2) |
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Information flow in random networks |
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121 | (36) |
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Information-theoretic preliminaries |
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121 | (10) |
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122 | (2) |
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Additive Gaussian channel |
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124 | (3) |
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Communication with continuous time signals |
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127 | (2) |
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Information-theoretic random networks |
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129 | (2) |
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Scaling limits; single source-destination pair |
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131 | (5) |
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Multiple source-destination pairs; lower bound |
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136 | (10) |
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138 | (1) |
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139 | (3) |
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142 | (4) |
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Multiple source-destination pairs; information-theoretic upper bounds |
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146 | (9) |
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Exponential attenuation case |
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148 | (3) |
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Power law attenuation case |
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151 | (4) |
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Historical notes and further reading |
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155 | (2) |
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Navigation in random networks |
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157 | (28) |
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157 | (2) |
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Discrete short-range percolation (large worlds) |
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159 | (2) |
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Discrete long-range percolation (small worlds) |
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161 | (10) |
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Chemical distance, diameter, and navigation length |
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162 | (5) |
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More on navigation length |
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167 | (4) |
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Continuum long-range percolation (small worlds) |
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171 | (10) |
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The role of scale invariance in networks |
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181 | (1) |
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Historical notes and further reading |
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182 | (3) |
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185 | (5) |
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A.1 Landau's order notation |
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185 | (1) |
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185 | (1) |
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A.3 Ergodicity and the ergodic theorem |
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185 | (2) |
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A.4 Deviations from the mean |
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187 | (1) |
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A.5 The Cauchy-Schwartz inequality |
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188 | (1) |
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A.6 The singular value decomposition |
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189 | (1) |
References |
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190 | (4) |
Index |
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194 | |