Introduction to the Modeling and Analysis of Complex Systems

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18.3. SYNCHRONIZABILITY 415(a)1.0(b)1.00.50.50.00.00.50.51.01.0 0.5 0.0 0.5 1.01.01.0 0.5 0.0 0.5 1.01.01.00.50.50.00.00.50.51.01.0 0.5 0.0 0.5 1.01.01.0 0.5 0.0 0.5 1.01.01.00.50.50.00.00.50.51.01.0 0.5 0.0 0.5 1.01.01.0 0.5 0.0 0.5 1.01.01.00.50.50.00.00.50.51.01.0 0.5 0.0 0.5 1.01.01.0 0.5 0.0 0.5 1.01.01.00.50.50.00.00.50.51.01.0 0.5 0.0 0.5 1.01.01.0 0.5 0.0 0.5 1.0Figure 18.2: Visual outputs of Code 18.2. Time flows from top to bottom. (a) Resultwith α = 2, β = 1 (λ 2 < β/α.). (b) Result with α = 2, β = 0.9 (λ 2 > β/α.).
416CHAPTER 18. DYNAMICAL NETWORKS III: ANALYSIS OF NETWORK DYNAMICSis represented by the Laplacian operator ∇ 2 , while for the latter, it is represented by theLaplacian matrix L (note again that their signs are opposite for historical misfortune!). Networkmodels allows us to study more complicated, nontrivial spatial structures, but therearen’t any fundamentally different aspects between these two modeling frameworks. Thisis why the same analytical approach works for both.Note that the synchronizability analysis we covered in this section is still quite limitedin its applicability to more complex dynamical network models. It relies on the assumptionthat dynamical nodes are homogeneous and that they are linearly coupled, so the analysiscan’t generalize to the behaviors of heterogeneous dynamical networks with nonlinearcouplings, such as the Kuramoto model discussed in Section 16.2 in which nodes oscillatein different frequencies and their couplings are nonlinear. Analysis of such networks willneed more advanced nonlinear analytical techniques, which is beyond the scope of thistextbook.18.4 Mean-Field Approximation of Discrete-State NetworksAnalyzing the dynamics of discrete-state network models requires a different approach,because the assumption of smooth, continuous state space, on which the linear stabilityanalysis is based on, no longer applies. This difference is similar to the difference betweencontinuous field models and cellular automata (CA). In Section 12.3, we analyzed CAmodels using mean-field approximation. Since CA are just a special case of discretestatedynamical networks, we should be able to apply the same analysis to dynamicalnetworks as well.In fact, mean-field approximation works on dynamical networks almost in the sameway as on CA. But one important issue we need to consider is how to deal with heterogeneoussizes of the neighborhoods. In CA, every cell has the same number of neighbors,so the mean-field approximation is very easy. But this is no longer the case on networksin which nodes can have any number of neighbors. There are a few different ways to dealwith this issue.In what follows, we will work on a simple binary-state example, the Susceptible-Infected-Susceptible (SIS) model, which we discussed in Section 16.2. As you may remember, thestate transition rules of this model are fairly simple: A susceptible node can get infectedby an infected neighbor node with infection probability p i (per infected neighbor), whilean infected node can recover to a susceptible node with recovery probability p r . In theprevious chapter, we used asynchronous updating in simulations of the SIS model, but
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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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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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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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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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