Introduction to the Modeling and Analysis of Complex Systems

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7.4. ASYMPTOTIC BEHAVIOR OF CONTINUOUS-TIME LINEAR DYNAMICAL... 123Or:(a − λ)(d − λ) − bc = λ 2 − (a + d)λ + ad − bc (7.54)= λ 2 − Tr(A)λ + det(A) = 0 (7.55)Here, Tr(A) is the trace of matrix A, i.e., the sum of its diagonal components. The solutionsof the equation above areλ = Tr(A) ± √ Tr(A) 2 − 4 det(A). (7.56)2Between those two eigenvalues, which one is dominant? Since the radical on the numeratorgives either a non-negative real value or an imaginary value, the one with a “plus” signalways has the greater real part. Now we can find the conditions for which this system isstable. The real part of this dominant eigenvalue is given as follows:⎧Tr(A) ⎪⎨if Tr(A) 2 < 4 det(A)Re(λ d ) =⎪⎩2Tr(A) + √ Tr(A) 2 − 4 det(A)2If Tr(A) 2 < 4 det(A), the stability condition is simplyif Tr(A) 2 ≥ 4 det(A)(7.57)Tr(A) < 0. (7.58)If Tr(A) 2 ≥ 4 det(A), the stability condition is derived as follows:Tr(A) + √ Tr(A) 2 − 4 det(A) < 0 (7.59)√Tr(A)2 − 4 det(A) < −Tr(A) (7.60)Since the radical on the left hand side must be non-negative, Tr(A) must be negative, atleast. Also, by squaring both sides, we obtainTr(A) 2 − 4 det(A) < Tr(A) 2 , (7.61)− 4 det(A) < 0, (7.62)det(A) > 0. (7.63)By combining all the results above, we can summarize how the two-dimensional lineardynamical system’s stability depends on Tr(A) and det(A) in a simple diagram as shownin Fig. 7.5. Note that this diagram is applicable only to two-dimensional systems, and it isnot generalizable for systems that involve three or more variables.
124 CHAPTER 7. CONTINUOUS-TIME MODELS II: ANALYSISdet(A)4 det(A) = Tr(A) 2 4 det(A) > Tr(A) 2 4 det(A) < Tr(A) 2StablepointRe(l d ) < 0Im(l d ) = 0Stablespiral focusRe(l d ) < 0Im(l d ) ≠ 00Unstablespiral focusRe(l d ) > 0Im(l d ) ≠ 0Tr(A)UnstablepointRe(l d ) > 0Im(l d ) = 0Figure 7.5: Overview of how the stability of a two-dimensional continuous-time lineardynamical system depends on Tr(A) and det(A).
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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 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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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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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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438 CHAPTER 19. AGENT-BASED MODELSF
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440 CHAPTER 19. AGENT-BASED MODELSY
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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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