## Adaptive Networks: Theory, Models and Applications (Google eBook)Adding one and one makes two, usually. But sometimes things add up to more than the sum of their parts. This observation, now frequently expressed in the maxim “more is different”, is one of the characteristic features of complex systems and, in particular, complex networks. Along with their ubiquity in real world systems, the ability of networks to exhibit emergent dynamics, once they reach a certain size, has rendered them highly attractive targets for research. The resulting network hype has made the word “network” one of the most in uential buzzwords seen in almost every corner of science, from physics and biology to economy and social sciences. The theme of “more is different” appears in a different way in the present v- ume, from the viewpoint of what we call “adaptive networks.” Adaptive networks uniquely combine dynamics on a network with dynamical adaptive changes of the underlying network topology, and thus they link classes of mechanisms that were previously studied in isolation. Here adding one and one certainly does not make two, but gives rise to a number of new phenomena, including highly robust se- organization of topology and dynamics and other remarkably rich dynamical beh- iors. |

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### Contents

1 | |

12 The Interplay Between State and Topology | 2 |

13 Timescale Separation and Critical Phenomena | 3 |

14 SelfOrganization of Nontrivial Network Topologies | 4 |

15 Adaptive Networks with Inseparable Timescales | 5 |

16 In this Book | 6 |

References | 8 |

RealWorld Examples of Adaptive Networks | 9 |

921 Continuous Opinions | 194 |

922 TwoValued Choice and Irreversible Discord | 195 |

923 The Inﬂuence of Bounded Tolerance | 197 |

924 Asymmetric Insertion of Inﬂuence | 199 |

925 Other Approaches | 201 |

931 The Adaptive SIS Model | 202 |

932 Other Approaches | 206 |

94 Summary and Outlook | 207 |

Social Group Dynamics in Networks | 11 |

22 Construction of the Networks | 13 |

23 Finding Communities | 15 |

232 Preferential Attachment at the Level of Communities | 16 |

233 The Static Communities | 20 |

234 Validating the Communities | 22 |

24 Evolving Communities | 24 |

25 Statistical Properties of the Community Dynamics | 27 |

252 Stationarity and Lifetime | 29 |

253 Predicting Community Break Up | 30 |

254 Merging of Communities | 32 |

26 Conclusion | 34 |

References | 35 |

TimeDependent Complex Networks Dynamic Centrality Dynamic Motifs and Cycles of Social Interactions | 39 |

32 Dynamic Centrality in Spatial Proximity Social Networks | 44 |

33 Dynamic Network Motifs and Cycles of Social Interaction | 46 |

34 Summary | 48 |

References | 49 |

Adaptive Biological Networks | 51 |

42 Network Development in Mycelial Fungi | 52 |

43 Predicted Transport Characteristics of the Mycelial Network | 54 |

44 Comparison Between Predicted Transport and Experimental Transport | 58 |

46 Network Robustness | 61 |

48 Universal Features of Biological Networks? | 65 |

References | 67 |

SelfOrganization of Adaptive Networks | 71 |

SelfOrganized Criticality and Adaptation in Discrete Dynamical Networks | 72 |

52 Dynamics of Random Boolean Networks and Random Threshold Networks | 77 |

522 Random Boolean Networks | 79 |

524 Basic Dynamical Properties of RBNs and RTNs | 80 |

53 Network SelfOrganization from Coevolution of Dynamics and Topology | 83 |

532 Adaptive Thresholds Time Scale Separation Leads to Complex Topologies | 90 |

533 Extension to Random Boolean Networks | 93 |

534 CorrelationBased Rewiring in Neural Networks | 96 |

54 Summary and Outlook | 101 |

References | 103 |

SelfOrganization and Complex Networks | 107 |

62 Scale Invariance and SelfOrganization | 109 |

622 SelfOrganized Criticality | 111 |

63 Complex Networks | 115 |

632 Network Models | 116 |

64 A SelfOrganized Network Model | 122 |

641 Motivation | 123 |

642 Deﬁnition | 124 |

643 Analytical Solution | 125 |

644 Particular Cases | 128 |

65 Conclusions | 132 |

References | 133 |

SelfOrganization of Network Structure in CoupledMap Systems | 137 |

72 Adaptive Network of LogisticMap Units | 138 |

721 Model Formulation | 139 |

722 Unit Dynamics | 140 |

723 Connection Dynamics | 143 |

724 Network Structure | 145 |

725 Dynamic Networks in the Desynchronized Phase | 147 |

73 Adaptive Network of Bursting Units | 153 |

732 Unit Dynamics | 154 |

733 Connection Dynamics | 155 |

734 Mechanism of Structure Formation | 158 |

75 Summary and Discussion | 160 |

References | 162 |

Dynamical Optimization and Synchronization in Adaptive Complex Networks | 164 |

82 Phase Synchronization in the Kuramoto Model | 167 |

83 Complete Synchronization and Enhanced Synchronizability in Adaptive Complex Networks | 175 |

832 Enhanced Synchronizability in Adaptive Complex Networks | 180 |

84 Conclusions | 186 |

References | 187 |

Contact Processes and Epidemiology on Adaptive Networks | 189 |

Contact Processes and Moment Closure on Adaptive Networks | 191 |

92 Opinion Formation Theme and Variations | 193 |

References | 208 |

Noise Induced Dynamics in Adaptive Networks with Applications to Epidemiology | 209 |

102 Model | 212 |

103 Bifurcation Structure | 214 |

104 Effect of Recovered Class on Fluctuations | 216 |

105 Delayed Outbreaks | 221 |

106 Lifetime of the Endemic Steady State | 222 |

107 Network Geometry | 223 |

108 Conclusions and Discussion | 225 |

References | 226 |

Social Games on Adaptive Networks | 228 |

A Dynamic Model of Social Network Formation | 231 |

A Baseline Model of Uniform Reinforcement | 234 |

Symmetrized Reinforcement | 235 |

113 Making Enemies | 237 |

1131 The Transfer Model | 238 |

1132 The Resistance Model | 239 |

1133 A Better Model? | 240 |

1142 Analysis of Discounting the Past | 241 |

1143 Introduction of Noise | 242 |

1144 Noise and Discounting | 243 |

115 Reinforcement by Games of Nontrivial Strategy | 244 |

1152 Coevolution of Structure and Strategy | 246 |

116 Conclusion | 247 |

References | 251 |

Evolutionary Games in SelfOrganizing Populations | 252 |

122 Active Linking | 254 |

1221 Linking Dynamics | 255 |

1222 Strategy Dynamics | 256 |

1223 Separation of Timescales | 258 |

1224 Effects of Active Linking | 260 |

123 Individual Based Linking Dynamics | 261 |

1232 Numerical Results | 263 |

1233 Graph Structures Under Individual Based Linking Dynamics | 264 |

124 Discussion | 265 |

References | 266 |

The Diplomats Dilemma Maximal Power for Minimal Effort in Social Networks | 269 |

132 Deﬁnition of the Model | 271 |

1322 Moves | 272 |

1324 Strategy Updates and Stochastic Rewiring | 274 |

133 Numerical Results | 275 |

1333 Effects of Strategies on the Network Topology | 277 |

1334 Transition Probabilities | 282 |

1335 Dependence on System Size and Noise | 283 |

134 Discussion | 284 |

References | 287 |

GraphRewritingBased Approaches | 289 |

GraphRewriting Automata as a Natural Extension of Cellular Automata | 290 |

142 Formulation | 292 |

1421 Rules of GraphRewriting Automata | 293 |

1422 Update Procedure | 294 |

1423 Simulation of GraphRewriting Automata | 295 |

1424 Examples | 296 |

143 Rule Design by HandCoding | 297 |

1431 Design of Selfreplicating Turing Machine | 298 |

144 Rule Search by Evolutionary Computation | 300 |

1442 Simulation Results | 302 |

145 Exhaustive Trial | 303 |

1451 Rule Representation | 304 |

146 Conclusions | 307 |

References | 308 |

Generative Network Automata A Generalized Framework for Modeling Adaptive Network Dynamics Using Graph Rewritings | 311 |

152 About Graph Rewriting | 313 |

153 Deﬁnition of GNA | 314 |

154 Generality of GNA | 317 |

1552 Methods | 320 |

1553 Results | 321 |

156 Conclusion | 329 |

References | 330 |

### Common terms and phrases

adaptive networks agents attractor average connectivity Bak-Sneppen model Barab´asi behavior bifurcation boolean networks Bornholdt cellular automata clusters complex networks Complex Systems conﬁguration connection weights correlation corresponding coupling deﬁned deﬁnition degree distribution dynamic region dynamical networks edge evolution evolutionary evolving ﬁnd ﬁnite ﬁrst ﬁtness ﬁxed ﬂuctuations formation function graph graph-rewriting automata Hopf bifurcation individuals infected inﬂuence initial interactions Kuramoto model Lett linking dynamics matrix MAXC mean ﬁeld mechanism NECSI neighbors network models network structure network topology nodes noise number of nodes observed opinion order parameter oscillators pairs payoff phase transition Phys Physics plot power-law preferential attachment probability Random Boolean random graph rewiring rewriting scale Sect self-organized critical self-replication shown in Fig signiﬁcant simulations social network speciﬁc stag steps stochastic strategy strategy dynamics subGNA SW networks synchronizability synchronization threshold tion update values vertex vertices