Introduction to the Modeling and Analysis of Complex Systems

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12.4. RENORMALIZATION GROUP ANALYSIS TO PREDICT PERCOLATION... 221A portion ofthe forest?Scale sFigure 12.4: Key property to be analyzed in the renormalization group analysis of theepidemic/forest fire CA model, i.e., the probability for a portion of the forest (of scales) to conduct the fire from one side to another.
222 CHAPTER 12. CELLULAR AUTOMATA II: ANALYSISSingle cell level:p 1 = q2x2 block level:p 2 = F(p 1 )= p 14+ 4p 13 (1 – p 1 )+ 4p 12 (1 – p 1 ) 2Figure 12.5: How to estimate the conductance probability p 2 at scale s = 2 using p 1 .four different ways to transmit the fire through the area. But if there is only one tree, thereis no way to transmit the fire, so there is no need to consider such cases. This is theexhaustive list of possible situations where the fire will be conducted through a scale-2block. We can calculate the probability for each situation and add them up, to obtain thefollowing relationship between p 1 and p 2 :p 2 = Φ(p 1 ) = p 4 1 + 4p 3 1(1 − p 1 ) + 4p 2 1(1 − p 1 ) 2 (12.12)And this is the place where a very strong assumption/approximation comes in. Let’sassume that the relationship above applies to 4 × 4 blocks (Fig. 12.6), 8 × 8 blocks, etc.,all the way up to infinitely large portions of the forest, so thatp s+1 = Φ(p s ) = p 4 s + 4p 3 s(1 − p s ) + 4p 2 s(1 − p s ) 2 for all s. (12.13)Of course this is not correct, but some kind of approximation needs to be made inorder to study complex systems analytically.Exercise 12.8 Why is it not correct to assume that the relationship between the1×1 blocks and the 2×2 blocks can be applied to 2×2 and 4×4 (and larger)? Look atFig. 12.6 carefully and figure out some configurations that violate this assumption.
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Introduction to the Modeling and An
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iiiAbout the TextbookIntroduction t
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New York. He is also a Fellow of th
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PrefaceThis is an introductory text
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a comprehensive view of the related
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Chen, Hal Lewis, Vlad Miskovic, Chu
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xviCONTENTS5 Discrete-Time Models I
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xviiiCONTENTS15.3 Constructing Netw
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Chapter 1Introduction1.1 Complex Sy
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1.1. COMPLEX SYSTEMS IN A NUTSHELL
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1.2. TOPICAL CLUSTERS 7The possibil
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1.2. TOPICAL CLUSTERS 9these topica
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12 CHAPTER 2. FUNDAMENTALS OF MODEL
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Part IISystems with a Small Number
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30 CHAPTER 3. BASICS OF DYNAMICAL S
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36 CHAPTER 4. DISCRETE-TIME MODELS
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Chapter 6Continuous-Time Models I:
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6.2. CLASSIFICATIONS OF MODEL EQUAT
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6.3. CONNECTING CONTINUOUS-TIME MOD
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6.4. SIMULATING CONTINUOUS-TIME MOD
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6.4. SIMULATING CONTINUOUS-TIME MOD
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6.5. BUILDING YOUR OWN MODEL EQUATI
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112 CHAPTER 7. CONTINUOUS-TIME MODE
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Chapter 8Bifurcations8.1 What Are B
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8.2. BIFURCATIONS IN 1-D CONTINUOUS
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8.3. HOPF BIFURCATIONS IN 2-D CONTI
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8.3. HOPF BIFURCATIONS IN 2-D CONTI
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8.4. BIFURCATIONS IN DISCRETE-TIME
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Chapter 9Chaos9.1 Chaos in Discrete
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9.1. CHAOS IN DISCRETE-TIME MODELS
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Next phase space9.3. LYAPUNOV EXPON
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9.3. LYAPUNOV EXPONENT 159easily co
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9.3. LYAPUNOV EXPONENT 16110Lyapuno
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9.4. CHAOS IN CONTINUOUS-TIME MODEL
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Page 192 and 193: Chapter 10Interactive Simulation of
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272 CHAPTER 14. CONTINUOUS FIELD MO
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296 CHAPTER 15. BASICS OF NETWORKSg
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298 CHAPTER 15. BASICS OF NETWORKSi
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300 CHAPTER 15. BASICS OF NETWORKS5
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302 CHAPTER 15. BASICS OF NETWORKSE
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304 CHAPTER 15. BASICS OF NETWORKSC
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306 CHAPTER 15. BASICS OF NETWORKSC
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308 CHAPTER 15. BASICS OF NETWORKSt
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312 CHAPTER 15. BASICS OF NETWORKSs
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314 CHAPTER 15. BASICS OF NETWORKSn
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316 CHAPTER 15. BASICS OF NETWORKSL
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318 CHAPTER 15. BASICS OF NETWORKSJ
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320 CHAPTER 15. BASICS OF NETWORKSF
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322 CHAPTER 15. BASICS OF NETWORKSr
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326 CHAPTER 16. DYNAMICAL NETWORKS
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Chapter 17Dynamical Networks II: An
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17.1. NETWORK SIZE, DENSITY, AND PE
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17.1. NETWORK SIZE, DENSITY, AND PE
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17.2. SHORTEST PATH LENGTH 37717.2
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17.2. SHORTEST PATH LENGTH 379Eccen
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17.3. CENTRALITIES AND CORENESS 381
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17.4. CLUSTERING 387while the numer
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17.5. DEGREE DISTRIBUTION 3891.00.8
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17.5. DEGREE DISTRIBUTION 391er = n
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17.5. DEGREE DISTRIBUTION 393Pk = [
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17.5. DEGREE DISTRIBUTION 395logkda
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17.6. ASSORTATIVITY 397Assortativit
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17.6. ASSORTATIVITY 399>>> nx.degre
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17.7. COMMUNITY STRUCTURE AND MODUL
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406CHAPTER 18. DYNAMICAL NETWORKS I
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428 CHAPTER 19. AGENT-BASED MODELSE
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430 CHAPTER 19. AGENT-BASED MODELS
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432 CHAPTER 19. AGENT-BASED MODELST
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434 CHAPTER 19. AGENT-BASED MODELSI
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436 CHAPTER 19. AGENT-BASED MODELSt
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438 CHAPTER 19. AGENT-BASED MODELSF
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440 CHAPTER 19. AGENT-BASED MODELSY
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442 CHAPTER 19. AGENT-BASED MODELS3
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444 CHAPTER 19. AGENT-BASED MODELSn
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450 CHAPTER 19. AGENT-BASED MODELSF
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452 CHAPTER 19. AGENT-BASED MODELSe
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454 CHAPTER 19. AGENT-BASED MODELSf
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458 CHAPTER 19. AGENT-BASED MODELSi
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460 CHAPTER 19. AGENT-BASED MODELSw
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462 CHAPTER 19. AGENT-BASED MODELSi
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464 CHAPTER 19. AGENT-BASED MODELSB
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466 BIBLIOGRAPHY[12] “The Human C
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468 BIBLIOGRAPHY[39] H. Sayama, “
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470 BIBLIOGRAPHY[65] P. Holme and J
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Indexaction potential, 202actor, 29
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INDEX 475equilibrium point, 61, 111
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INDEX 477partial differential equat
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Introduction to the Modeling and Analysis of Complex Systems

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