Complex and Adaptive Dynamical Systems: A Primer

356 Pages·2013·6.63 MB·English

by  Claudius Gros (auth.)

3
COMPLEXITY
Claudius Gros
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A Primer
Third Edition
123
Springer Complexity
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Complex Systems are systems that comprise many interacting parts with the ability to
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Markus Kirkilionis, Mathematics Institute and Centre for Complex Systems, University of Warwick,
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FrankSchweitzer,SystemDesign,ETHZurich,Zurich,Switzerland
DidierSornette,EntrepreneurialRisk,ETHZurich,Zurich,Switzerland
StefanThurner,SectionforScienceofComplexSystems,MedicalUniversityofVienna,Vienna,Austria

Claudius Gros
Complex and Adaptive
Dynamical Systems
A Primer
Third Edition
123
Prof.Dr.ClaudiusGros
Universita¨tFrankfurt
Institutfu¨rTheoretischePhysik
Frankfurt
Germany
ISBN978-3-642-36585-0 ISBN978-3-642-36586-7(ebook)
DOI10.1007/978-3-642-36586-7
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013936648
(cid:2)c Springer-VerlagBerlinHeidelberg2013, 2011, 2008
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Contents
1 GraphTheoryandSmall-WorldNetworks............................... 1
1.1 GraphTheoryandReal-WorldNetworks.............................. 1
1.1.1 TheSmall-WorldEffect........................................ 1
1.1.2 BasicGraph-TheoreticalConcepts............................. 3
1.1.3 GraphSpectraandDegreeDistributions ...................... 8
1.2 PercolationinGeneralizedRandomGraphs........................... 14
1.2.1 GraphswithArbitraryDegreeDistributions .................. 15
1.2.2 ProbabilityGeneratingFunctionFormalism .................. 20
1.2.3 DistributionofComponentSizes .............................. 22
1.3 RobustnessofRandomNetworks...................................... 26
1.4 Small-WorldModels.................................................... 30
1.5 Scale-FreeGraphs....................................................... 32
Exercises....................................................................... 37
FurtherReading ............................................................... 39
References..................................................................... 39
2 Chaos,BifurcationsandDiffusion ......................................... 41
2.1 BasicConceptsofDynamicalSystemsTheory........................ 41
2.2 BifurcationsandStability............................................... 47
2.3 TheLogisticMapandDeterministicChaos ........................... 51
2.4 DissipationandAdaption............................................... 56
2.4.1 DissipativeSystemsandStrangeAttractors................... 57
2.4.2 AdaptiveSystems............................................... 61
2.5 DiffusionandTransport................................................. 66
2.5.1 RandomWalks,DiffusionandLe´vyFlights .................. 66
2.5.2 TheLangevinEquationandDiffusion......................... 69
2.6 Noise-ControlledDynamics............................................ 71
2.6.1 StochasticEscape............................................... 72
2.6.2 StochasticResonance........................................... 75
2.7 DynamicalSystemswithTimeDelays................................. 78
vii
viii Contents
Exercises....................................................................... 81
FurtherReading ............................................................... 82
References..................................................................... 83
3 ComplexityandInformationTheory ..................................... 85
3.1 ProbabilityDistributionFunctions..................................... 85
3.1.1 BayesianStatistics.............................................. 88
3.1.2 TheLawofLargeNumbers.................................... 91
3.1.3 TimeSeriesCharacterization................................... 92
3.2 EntropyandInformation................................................ 96
3.2.1 InformationContentofaReal-WorldTimeSeries............ 102
3.2.2 MutualInformation............................................. 103
3.2.3 Kullback-LeiblerDivergence .................................. 108
3.3 ComplexityMeasures................................................... 110
3.3.1 ComplexityandPredictability ................................. 112
3.3.2 AlgorithmicandGenerativeComplexity...................... 114
Exercises....................................................................... 116
FurtherReading ............................................................... 118
References..................................................................... 118
4 RandomBooleanNetworks ................................................ 121
4.1 Introduction ............................................................. 121
4.2 RandomVariablesandNetworks....................................... 123
4.2.1 BooleanVariablesandGraphTopologies ..................... 123
4.2.2 CouplingFunctions............................................. 125
4.2.3 Dynamics........................................................ 127
4.3 TheDynamicsofBooleanNetworks................................... 128
4.3.1 TheFlowofInformationThroughtheNetwork............... 129
4.3.2 TheMean-FieldPhaseDiagram ............................... 131
4.3.3 TheBifurcationPhaseDiagram................................ 133
4.3.4 Scale-FreeBooleanNetworks.................................. 137
4.4 CyclesandAttractors................................................... 139
4.4.1 QuenchedBooleanDynamics.................................. 139
4.4.2 TheKD1KauffmanNetwork................................. 142
4.4.3 TheKD2KauffmanNetwork................................. 144
4.4.4 TheKDNKauffmanNetwork ................................ 145
4.5 Applications............................................................. 147
4.5.1 LivingattheEdgeofChaos.................................... 147
4.5.2 TheYeastCellCycle ........................................... 149
4.5.3 ApplicationtoNeuralNetworks ............................... 151
Exercises....................................................................... 153
FurtherReading ............................................................... 154
References..................................................................... 154
Contents ix
5 CellularAutomataandSelf-OrganizedCriticality....................... 157
5.1 TheLandauTheoryofPhaseTransitions.............................. 157
5.2 CriticalityinDynamicalSystems ...................................... 162
5.2.1 1/fNoise......................................................... 166
5.3 CellularAutomata....................................................... 167
5.3.1 Conway’sGameofLife........................................ 168
5.3.2 TheForestFireModel.......................................... 169
5.4 TheSandpileModelandSelf-OrganizedCriticality .................. 171
5.4.1 AbsorbingPhaseTransitions................................... 173
5.5 RandomBranchingTheory............................................. 175
5.5.1 BranchingTheoryofSelf-OrganizedCriticality.............. 175
5.5.2 Galton-WatsonProcesses....................................... 180
5.6 ApplicationtoLong-TermEvolution .................................. 182
Exercises....................................................................... 188
FurtherReading ............................................................... 189
References..................................................................... 190
6 DarwinianEvolution,HypercyclesandGameTheory .................. 191
6.1 Introduction ............................................................. 191
6.2 MutationsandFitnessinaStaticEnvironment........................ 193
6.3 DeterministicEvolution ................................................ 197
6.3.1 EvolutionEquations............................................ 197
6.3.2 BeanbagGenetics:EvolutionsWithoutEpistasis............. 200
6.3.3 EpistaticInteractionsandtheErrorCatastrophe.............. 202
6.4 FinitePopulationsandStochasticEscape.............................. 206
6.4.1 StrongSelectivePressureandAdaptiveClimbing............ 207
6.4.2 AdaptiveClimbingVersusStochasticEscape................. 210
6.5 PrebioticEvolution...................................................... 212
6.5.1 QuasispeciesTheory............................................ 212
6.5.2 HypercyclesandAutocatalyticNetworks ..................... 213
6.6 MacroecologyandSpeciesCompetition............................... 217
6.7 CoevolutionandGameTheory......................................... 220
Exercises....................................................................... 225
FurtherReading ............................................................... 226
References..................................................................... 227
7 SynchronizationPhenomena ............................................... 229
7.1 FrequencyLocking...................................................... 229
7.2 SynchronizationofCoupledOscillators ............................... 230
7.3 SynchronizationwithTimeDelays..................................... 237
7.4 SynchronizationviaAggregateAveraging............................. 239
7.5 SynchronizationviaCausalSignaling ................................. 243
7.6 SynchronizationandObjectRecognitioninNeuralNetworks ....... 246
7.7 SynchronizationPhenomenainEpidemics............................ 250