Introduction to the Modeling and Analysis of Complex Systems

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5.3. COBWEB PLOTS FOR ONE-DIMENSIONAL ITERATIVE MAPS 73x maxx 3x 2x 1x tx minx minx 0x t-1x 1x 2x maxFigure 5.8: Drawing a cobweb plot (5).plot(rng, map(f, rng), ’k’)### drawing trajectoryhorizontal = [result[0]]vertical = [result[0]]for x in result[1:]:horizontal.append(vertical[-1])vertical.append(x)horizontal.append(x)vertical.append(x)plot(horizontal, vertical, ’b’)show()Exercise 5.7 Using Python, draw a cobweb plot of the logistic growth model withr = 2.5, K = 1, N 0 = 0.1.
74 CHAPTER 5. DISCRETE-TIME MODELS II: ANALYSISx maxx 3x 2x 1x tx minx minx 0x t-1x 1x 2x maxFigure 5.9: Drawing a cobweb plot (6).5.4 Graph-Based Phase Space Visualization of Discrete-State Discrete-Time ModelsThe cobweb plot approach discussed above works only for one-dimensional systems,because we can’t embed such plots for any higher dimensional systems in a 3-D physicalspace. However, this dimensional restriction vanishes if the system’s states are discreteand finite. For such a system, you can always enumerate all possible state transitionsand create the entire phase space of the system as a state-transition graph, which can bevisualized reasonably well even within a 2-D visualization space.Here is an example. Let’s consider the following second-order (i.e., two-dimensional)difference equation:x t = x t−1 x t−2 mod 6 (5.18)The “mod 6” at the end of the equation means that its right hand side is always aremainder of the division of x t−1 x t−2 by 6. This means that the possible state of x islimited only to 0, 1, 2, 3, 4, or 5, i.e., the state of the system (x, y) (where y is the previous
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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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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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Chapter 10Interactive Simulation of
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10.2. INTERACTIVE SIMULATION WITH P
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10.4. SIMULATION WITHOUT PYCX 181Co
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10.4. SIMULATION WITHOUT PYCX 183re
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Chapter 11Cellular Automata I: Mode
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11.1. DEFINITION OF CELLULAR AUTOMA
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11.1. DEFINITION OF CELLULAR AUTOMA
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11.2. EXAMPLES OF SIMPLE BINARY CEL
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11.3. SIMULATING CELLULAR AUTOMATA
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11.5. EXAMPLES OF BIOLOGICAL CELLUL
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Chapter 12Cellular Automata II: Ana
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12.2. PHASE SPACE VISUALIZATION 211
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12.2. PHASE SPACE VISUALIZATION 213
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12.3. MEAN-FIELD APPROXIMATION 215E
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12.3. MEAN-FIELD APPROXIMATION 217T
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12.4. RENORMALIZATION GROUP ANALYSI
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228 CHAPTER 13. CONTINUOUS FIELD MO
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OutIn232 CHAPTER 13. CONTINUOUS FIE
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296 CHAPTER 15. BASICS OF NETWORKSg
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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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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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