Introduction to the Modeling and Analysis of Complex Systems

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14.1. FINDING EQUILIBRIUM STATES 271432c(x)10-1-2-3 -2 -1 0 1 2 3xFigure 14.1: Example of equilibrium states of the PDE model Eq. (14.3) with D =C 1 = C 2 = 1. The black solid curve shows c eq (x), while the cyan (dotted) and pink(dashed) curves represent the increase/decrease caused by the diffusion term andthe source/sink term, respectively. Those two terms are balanced perfectly in thisequilibrium state.
272 CHAPTER 14. CONTINUOUS FIELD MODELS II: ANALYSISwhich will make all the derivatives (both temporal and spatial ones) become zero. For example,consider obtaining homogeneous equilibrium states of the following Turing patternformation model:∂u∂t = a(u − h) + b(v − k) + D u∇ 2 u (14.10)∂v∂t = c(u − h) + d(v − k) + D v∇ 2 v (14.11)The only thing you need to do is to replace the spatio-temporal functions u(x, t) and v(x, t)with the constants u eq and v eq , respectively:∂u eq= a(u eq − h) + b(v eq − k) + D u ∇ 2 u eq (14.12)∂t∂v eq= c(u eq − h) + d(v eq − k) + D v ∇ 2 v eq (14.13)∂tNote that, since u eq and v eq no longer depend on either time or space, the temporal derivativeson the left hand side and the Laplacians on the right hand side both go away. Thenwe obtain the following:0 = a(u eq − h) + b(v eq − k) (14.14)0 = c(u eq − h) + d(v eq − k) (14.15)By solving these equations, we get (u eq , v eq ) = (h, k), as we expected.Note that we can now represent this equilibrium state as a “point” in a two-dimensional(u, v) vector space. This is another reason why homogeneous equilibrium states areworth considering; they provide a simpler, low-dimensional reference point to help usunderstand the dynamics of otherwise complex spatial phenomena. Therefore, we willalso focus on the analysis of homogeneous equilibrium states for the remainder of thischapter.Obtain homogeneous equilibrium states of the following “Orego-Exercise 14.2nator” model:ɛ ∂uu − q= u(1 − u) −∂t u + q fv + D u∇ 2 u (14.16)∂v∂t = u − v + D v∇ 2 v (14.17)
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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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322 CHAPTER 15. BASICS OF NETWORKSr
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324 CHAPTER 15. BASICS OF NETWORKSE
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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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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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