Introduction to the Modeling and Analysis of Complex Systems

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17.5. DEGREE DISTRIBUTION 391er = nx.erdos_renyi_graph(n, 0.01)ws = nx.watts_strogatz_graph(n, 10, 0.01)ba = nx.barabasi_albert_graph(n, 5)Pk = [float(x) / n for x in nx.degree_histogram(er)]domain = range(len(Pk))loglog(domain, Pk, ’-’, label = ’Erdos-Renyi’)Pk = [float(x) / n for x in nx.degree_histogram(ws)]domain = range(len(Pk))loglog(domain, Pk, ’--’, label = ’Watts-Strogatz’)Pk = [float(x) / n for x in nx.degree_histogram(ba)]domain = range(len(Pk))loglog(domain, Pk, ’:’, label = ’Barabasi-Albert’)xlabel(’k’)ylabel(’P(k)’)legend()show()The result is shown in Fig. 17.9, which clearly illustrates differences between the threenetwork models used in this example. The Erdős-Rényi random network model has abell-curved degree distribution, which appears as a skewed mountain in the log-log scale(blue solid line). The Watts-Strogatz model is nearly regular, and thus it has a very sharppeak at the average degree (green dashed line; k = 10 in this case). The Barabási-Albertmodel has a power-law degree distribution, which looks like a straight line with a negativeslope in the log-log scale (red dotted line).Moreover, it is often visually more meaningful to plot not the degree distribution itselfbut its complementary cumulative distribution function (CCDF), defined as follows:∞∑F (k) = P (k ′ ) (17.29)k ′ =kThis is a probability for a node to have a degree k or higher. By definition, F (0) = 1 andF (k max + 1) = 0, and the function decreases monotonically along k. We can revise Code17.15 to draw CCDFs:Code 17.16: ccdfs-loglog.py
392CHAPTER 17. DYNAMICAL NETWORKS II: ANALYSIS OF NETWORK TOPOLOGIES10 0Erdos-RenyiWatts-StrogatzBarabasi-Albert10 -1P(k)10 -210 -310 0 10 1 10 2 10 3Figure 17.9: Visual output of Code 17.15.kfrom pylab import *import networkx as nxn = 1000er = nx.erdos_renyi_graph(n, 0.01)ws = nx.watts_strogatz_graph(n, 10, 0.01)ba = nx.barabasi_albert_graph(n, 5)Pk = [float(x) / n for x in nx.degree_histogram(er)]domain = range(len(Pk))ccdf = [sum(Pk[k:]) for k in domain]loglog(domain, ccdf, ’-’, label = ’Erdos-Renyi’)Pk = [float(x) / n for x in nx.degree_histogram(ws)]domain = range(len(Pk))ccdf = [sum(Pk[k:]) for k in domain]loglog(domain, ccdf, ’--’, label = ’Watts-Strogatz’)
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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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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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8.4. 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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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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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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326 CHAPTER 16. DYNAMICAL NETWORKS
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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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