Introduction to the Modeling and Analysis of Complex Systems

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15.6. GENERATING RANDOM GRAPHS 323Random graphs can also be generated by randomizing existing graph topologies.Such network randomization is particularly useful when you want to compare a certainproperty of an observed network with that of a randomized “null” model. For example,imagine you found that information spreading was very slow in the network you werestudying. Then you could test whether or not the network’s specific topology caused thisslowness by simulating information spreading on both your network and on its randomizedcounterparts, and then comparing the speed of information spreading between them.There are several different ways to randomize network topologies. They differ regardingwhich network properties are to be conserved during randomization. Some examples areshown below (here we assume that g is the original network to be randomized):• gnm_random_graph(g.number_of_nodes(), g.number_of_edges()) conserves thenumbers of nodes and edges, nothing else.• random_degree_sequence_graph(g.degree().values()) conserves the sequenceof node degrees, in addition to the numbers of nodes and edges. As noted above,this function may not generate a graph within a given number of trials.• expected_degree_graph(g.degree().values()) conserves the number of nodesand the sequence of “expected” node degrees. This function can always generate agraph very quickly using the efficient Miller-Hagberg algorithm [64]. While the generatednetwork’s degree sequence doesn’t exactly match that of the input (especiallyif the network is dense), this method is often a practical, reasonable approach forthe randomization of large real-world networks.• double_edge_swap(g) conserves the numbers of nodes and edges and the sequenceof node degrees. Unlike the previous three functions, this is not a graphgeneratingfunction, but it directly modifies the topology of graph g. It conducts whatis called a double edge swap, i.e., randomly selecting two edges a − b and c − d,removing those edges, and then creating new edges a − c and b − d (if either pairis already connected, another swap will be attempted). This operation realizes aslight randomization of the topology of g, and by repeating it many times, you cangradually randomize the topology from the original g to a completely randomizednetwork. double_edge_swap(g, k) conducts k double edge swaps. There is alsoconnected_double_edge_swap available, which guarantees that the resulting graphremains connected.
324 CHAPTER 15. BASICS OF NETWORKSExercise 15.15 Randomize the topology of Zachary’s Karate Club graph by usingeach of the following functions and visualize the results:• gnm_random_graph• random_degree_sequence_graph• expected_degree_graph• double_edge_swap (5 times)• double_edge_swap (50 times)
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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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17.1. 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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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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