Introduction to the Modeling and Analysis of Complex Systems

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16.4. SIMULATING ADAPTIVE NETWORKS 365Adaptive diffusion model The final example in this chapter is the adaptive network versionof the continuous state/time diffusion model, where the weight of a social tie can bestrengthened or weakened according to the difference of the node states across the edge.This new assumption represents a simplified form of homophily, an empirically observedsociological fact that people tend to connect those who are similar to themselves. In contrast,the diffusion of node states can be considered a model of social contagion, anotherempirically observed sociological fact that people tend to become more similar to theirsocial neighbors over time. There has been a lot of debate going on about which mechanism,homophily or social contagion, is more dominant in shaping the structure of socialnetworks. This adaptive diffusion model attempts to combine these two mechanisms toinvestigate their subtle balance via computer simulations.This example is actually inspired by an adaptive network model of a corporate mergerthat my collaborator Junichi Yamanoi and I presented recently [73]. We studied the conditionsfor two firms that recently underwent a merger to successfully assimilate and integratetheir corporate cultures into a single, unified identity. The original model was abit complex agent-based model that involved asymmetric edges, multi-dimensional statespace, and stochastic decision making, so here we use a more simplified, deterministic,differential equation-based version. Here are the model assumptions:• The network is initially made of two groups of nodes with two distinct cultural/ideologicalstates.• Each edge is undirected and has a weight, w ∈ [0, 1], which represents the strengthof the connection. Weights are initially set to 0.5 for all the edges.• The diffusion of the node states occurs according to the following equation:dc idt = α ∑ j∈N i(c j − c i )w ij (16.18)Here α is the diffusion constant and w ij is the weight of the edge between node iand node j. The inclusion of w ij signifies that diffusion takes place faster throughedges with greater weights.• In the meantime, each edge also changes its weight dynamically, according to thefollowing equation:dw ijdt= βw ij (1 − w ij )(1 − γ |c i − c j | ) (16.19)
366 CHAPTER 16. DYNAMICAL NETWORKS I: MODELINGHere β is the rate of adaptive edge weight change, and γ is a parameter that determineshow intolerant, or “picky,” each node is regarding cultural difference. Forexample, if γ = 0, w ij always converges and stays at 1. But if γ is large (typicallymuch larger than 1), the two nodes need to have very similar cultural states in orderto maintain an edge between them, or otherwise the edge weight decreases. Theinclusion of w ij (1−w ij ) is to confine the weight values to the range [0, 1] dynamically.So, to simulate this model, we need a network made of two distinct groups. And this isthe perfect moment to disclose a little more secrets about our favorite Karate Club graph(unless you have found it yourself already)! See the node attribute of the Karate Clubgraph, and you will find the following:Code 16.16:>>> nx.karate_club_graph().node{0: {’club’: ’Mr. Hi’}, 1: {’club’: ’Mr. Hi’}, 2: {’club’: ’Mr. Hi’},3: {’club’: ’Mr. Hi’}, 4: {’club’: ’Mr. Hi’}, 5: {’club’: ’Mr. Hi’},6: {’club’: ’Mr. Hi’}, 7: {’club’: ’Mr. Hi’}, 8: {’club’: ’Mr. Hi’},9: {’club’: ’Officer’}, 10: {’club’: ’Mr. Hi’}, 11: {’club’: ’Mr. Hi’},12: {’club’: ’Mr. Hi’}, 13: {’club’: ’Mr. Hi’}, 14: {’club’: ’Officer’},15: {’club’: ’Officer’}, 16: {’club’: ’Mr. Hi’}, 17: {’club’: ’Mr. Hi’},18: {’club’: ’Officer’}, 19: {’club’: ’Mr. Hi’}, 20: {’club’: ’Officer’},21: {’club’: ’Mr. Hi’}, 22: {’club’: ’Officer’}, 23: {’club’: ’Officer’},24: {’club’: ’Officer’}, 25: {’club’: ’Officer’}, 26: {’club’: ’Officer’},27: {’club’: ’Officer’}, 28: {’club’: ’Officer’}, 29: {’club’: ’Officer’},30: {’club’: ’Officer’}, 31: {’club’: ’Officer’}, 32: {’club’: ’Officer’},33: {’club’: ’Officer’}}Each node has an additional property, called ’club’, and its values are either ’Mr. Hi’or ’Officer’! What are these?The truth is, when Wayne Zachary studied this Karate Club, there was an intensepolitical/ideological conflict between two factions. One was Mr. Hi, a part-time karateinstructor hired by the club, and his supporters, while the other was the club president(Officer) John A., other officers, and their followers. They were in sharp conflict over theprice of karate lessons. Mr. Hi wanted to raise the price, while the club president wantedto keep the price as it was. The conflict was so intense that the club eventually fired Mr. Hi,resulting in a fission of the club into two. The Karate Club graph is a snapshot of the clubmembers’ friendship network right before this fission, and therefore, each node comeswith an attribute showing whether he or she was in Mr. Hi’s camp or the Officer’s camp. If
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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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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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298 CHAPTER 15. BASICS OF NETWORKSi
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300 CHAPTER 15. BASICS OF NETWORKS5
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302 CHAPTER 15. BASICS OF NETWORKSE
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304 CHAPTER 15. BASICS OF NETWORKSC
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306 CHAPTER 15. BASICS OF NETWORKSC
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308 CHAPTER 15. BASICS OF NETWORKSt
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310 CHAPTER 15. BASICS OF NETWORKSE
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312 CHAPTER 15. BASICS OF NETWORKSs
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416CHAPTER 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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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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