Introduction to the Modeling and Analysis of Complex Systems

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13.5. SIMULATION OF CONTINUOUS FIELD MODELS 251The numerator of the formula above has an intuitive interpretation: “Just add the valuesof f in all of your nearest neighbors (e.g., top, bottom, left, and right for a 2-D space) andthen subtract your own value f(x, t) as many times as the number of those neighbors.”Or, equivalently: “Measure the difference between your neighbor and yourself, and thensum up all those differences.” It is interesting to see that a higher-order operator likeLaplacians can be approximated by such a simple set of arithmetic operations once thespace is discretized.You can also derive higher-order spatial derivatives in a similar manner, but I don’t thinkwe need to cover those cases here. Anyway, now we have a basic set of mathematicaltools (Eqs. (13.31), (13.33), (13.34), (13.35)) to discretize PDE-based models so we cansimulate their behavior as CA models.Let’s work on an example. Consider discretizing a simple transport equation withoutsource or sink in a 2-D space, where the velocity of particles is given by a constant vector:∂c= −∇ · (cw) (13.36)∂t( )wxw =(13.37)w yWe can discretize this equation in both time and space, as follows:c(x, y, t + ∆t) ≈ c(x, y, t) − ∇ · (c(x, y, t)w) ∆t (13.38)⎛ ⎞∂ ( ( ))⎜= c(x, y, t) − ⎝∂x ⎟wx∂ ⎠Tc(x, y, t) ∆t (13.39)w y∂y()∂c= c(x, y, t) − w x∂x + w ∂cy ∆t (13.40)∂y(c(x + ∆h, y, t) − c(x − ∆h, y, t)≈ c(x, y, t) − w x2∆h)c(x, y + ∆h, t) − c(x, y − ∆h, t)+w y ∆t (13.41)2∆h= c(x, y, t) − ( w x c(x + ∆h, y, t) − w x c(x − ∆h, y, t)+ w y c(x, y + ∆h, t) − w y c(x, y − ∆h, t) ) ∆t2∆h(13.42)This is now fully discretized for both time and space, and thus it can be used as astate-transition function of CA. We can implement this CA model in Python, using Code11.5 as a template. Here is an example of implementation:
252 CHAPTER 13. CONTINUOUS FIELD MODELS I: MODELINGCode 13.5: transport-ca.pyimport matplotlibmatplotlib.use(’TkAgg’)from pylab import *from mpl_toolkits.mplot3d import Axes3Dn = 100 # size of grid: n * nDh = 1. / n # spatial resolution, assuming space is [0,1] * [0,1]Dt = 0.01 # temporal resolutionwx, wy = -0.01, 0.03 # constant velocity of movementxvalues, yvalues = meshgrid(arange(0, 1, Dh), arange(0, 1, Dh))def initialize():global config, nextconfig# initial configurationconfig = exp(-((xvalues - 0.5)**2 + (yvalues - 0.5)**2) / (0.2**2))nextconfig = zeros([n, n])def observe():global config, nextconfigax = gca(projection = ’3d’)ax.cla()ax.plot_surface(xvalues, yvalues, config, rstride = 5, cstride = 5)ax.grid(False)ax.set_zlim(0, 1)show() #
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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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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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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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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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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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438 CHAPTER 19. AGENT-BASED MODELSF
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440 CHAPTER 19. AGENT-BASED MODELSY
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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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