Introduction to the Modeling and Analysis of Complex Systems

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16.2. SIMULATING DYNAMICS ON NETWORKS 341Figure 16.6: Schematic illustration of coupled oscillators. Light bulbs that changelight intensity periodically at different frequencies (left) may synchronize globally ifconnected (right).
342 CHAPTER 16. DYNAMICAL NETWORKS I: MODELINGRepresenting oscillatory behavior naturally requires two or more dynamical variables,as we learned earlier. But purely harmonic oscillation can be reduced to a simple linearmotion with constant velocity, if the behavior is interpreted as a projection of uniform circularmotion and if the state is described in terms of angle θ. This turns the reaction termin Eq. (16.11) into just a constant angular velocity ω i . In the meantime, this representationalso affects the diffusion term, because the difference between two θ’s is no longercomputable by simple arithmetic subtraction like θ j − θ i , because there are infinitely manyθ’s that are equivalent (e.g., 0, 2π, −2π, 4π, −4π, etc.). So, the difference between the twostates in the diffusion term has to be modified to focus only on the actual “phase difference”between the two nodes. You can imagine marching soldiers on a circular track as avisual example of this model (Fig. 16.7). Two soldiers who are far apart in θ may actuallybe close to each other in physical space (i.e., oscillation phase). The model should be setup in such a way that they try to come closer to each other in the physical space, not inthe angular space.To represent this type of phase-based interaction among coupled oscillators, a Japanesephysicist Yoshiki Kuramoto proposed the following very simple, elegant mathematicalmodel in the 1970s [70]:∑dθ idt = ω j∈Ni + αisin(θ j − θ i )(16.12)|N i |The angular difference part was replaced by a sine function of angular difference, whichbecomes positive if j is physically ahead of i, or negative if j is physically behind i onthe circular track (because adding or subtracting 2π inside sin won’t affect the result).These forces that soldier i receives from his or her neighbors will be averaged and usedto determine the adjustment of his or her movement. The parameter α determines thestrength of the couplings among the soldiers. Note that the original Kuramoto model useda fully connected network topology, but here we are considering the same dynamics on anetwork of a nontrivial topology.Let’s do some simulations to see if our networked soldiers can self-organize to marchin sync. We can take the previous code for network diffusion (Code 16.7) and revise itfor this Kuramoto model. There are several changes we need to implement. First, eachnode needs not only its dynamic state (θ i ) but also its static preferred velocity (ω i ), thelatter of which should be initialized so that there are some variations among the nodes.Second, the visualization function should convert the states into some bounded values,since angular position θ i continuously increases toward infinity. We can use sin(θ i ) forthis visualization purpose. Third, the updating rule needs to be revised, of course, toimplement Eq. (16.12).
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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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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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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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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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