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Structural and Dynamical Properties of Complex Networks Structural and dynamical properties of complex networks by Gourab Ghoshal A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Physics) in The University of Michigan 2009 Doctoral Committee: Professor Mark E. Newman, Chair Professor Leonard M. Sander Associate Professor Cagliyan Kurdak Associate Professor Michal R. Zochowski Assistant Professor Lada A. Adamic I guess I should warn you, if I turn out to be particularly clear, you’ve probably misunderstood what I said. —Alan Greenspan, 1988 speech, as quoted in The New York Times, October 2005 To see a world in a grain of sand, And a heaven in a wild flower, Hold infinity in the palm of your hand, And eternity in an hour. —William Blake, Auguries of Innocence c Gourab Ghoshal 2009 All Rights Reserved To my parents ii ACKNOWLEDGEMENTS I would like to thank my advisor Dr. Mark Newman for his diligent guidance over the past five years. Mark’s unique insight into various research problems, coupled with his sense of excellence and sharp wit, has been a source of great inspiration. As a direct academic descendant, I hope to employ the lessons learnt under his tutelage in a constructive fashion. I’d also like to thank Dr. Leonard Sander, Dr. Michal Zochowski, Dr. Lada Adamic and Dr. Cagliyan Kurak for serving on my dissertation committee. Special thanks to Brian Karrer, Petter Holme, Lada Adamic, Bob Ziff and an anonymous referee, whose often harsh but richly deserved comments on earlier ver- sions of this manuscript have greatly improved the final product. Extra special thanks to my once and perhaps future officemates, Juyong Park, Petter Holme, Elizabeth Leicht, Brian Karrer and Beth Percha who have proven themselves to be excellent colleagues, friends and fellow-procrastinators. A collective thanks goes out to the faculty, staff and students of the Department of Physics, for providing me with a stimulating and exciting environment to work in, as well as contributing to a very enjoyable stay at Ann Arbor. Finally, I would like to thank my parents, to whom I owe all. The work presented in this dissertation was funded through a combination of grants DMS-0405348, PHY-0200909 and DMS-0804778 from the National Science Foundation, as well as support from the James McDonnell foundation. iii TABLE OF CONTENTS DEDICATION .......................................... ii ACKNOWLEDGEMENTS .................................. iii LIST OF FIGURES ...................................... vii LIST OF TABLES ....................................... x CHAPTER I. Introduction ....................................... 1 1.1 Historicaloverview .............................. .. 3 1.2 InteresttoPhysicists . .. .. .. .. .. .. .. .. .. .. .. ... 7 1.3 Typesofnetworks................................. 10 1.4 Standard metrics and tools for measuring network structure ......... 12 1.4.1 Adjacencymatrix............................ 12 1.4.2 Degreedistribution .. .. .. .. .. .. .. .. .. .. .. .. 15 1.4.3 Transitivity ............................... 20 1.4.4 Assortativityand degree correlations . ..... 22 1.4.5 Centrality ................................ 24 1.4.6 Components............................... 27 1.4.7 Generatingfunctions . 29 1.5 RandomGraphs.................................. 32 1.5.1 Poissonrandomgraphs. 32 1.5.2 Generalizedrandomgraphs . 36 1.6 Networkgrowthmodels.. .. .. .. .. .. .. .. .. .. .. .. 40 1.6.1 The model of Priceand Barab´asi-Albert . ... 41 1.6.2 Extensions to the Barab´asi-Albert model . ..... 47 1.7 Percolationandnetworkresilience . ....... 49 1.8 Outlineofthedissertation . .... 52 II. Network resilience: The case of bicomponents and k-components ..... 55 2.1 Introduction .................................... 55 2.2 Bicomponents ................................... 57 2.2.1 Basicproperties............................. 57 2.2.2 Sizeofthegiantbicomponent . 59 2.2.3 Percolation ............................... 62 2.3 Comparisonwithrealworlddata . ... 65 2.4 Discussion ..................................... 69 III. Random Hypergraphs and their applications .................. 71 iv 3.1 Introduction .................................... 71 3.2 Tripartitegraphs ................................ 73 3.3 Randomtripartitegraphs. ... 77 3.3.1 Themodel................................ 77 3.3.2 Generatingfunctions . 80 3.3.3 Projections ............................... 80 3.3.4 Formationandsizeofthe giantcomponent . .. 82 3.3.5 Othertypesofcomponents . 87 3.3.6 Percolation ............................... 88 3.3.7 Simulations ............................... 90 3.4 Comparisonwithreal-worlddata . .... 92 3.5 Discussion ..................................... 97 IV. Network growth models I: Equilibrium degree distributions for networks with vertex and edge turnover ............................ 99 4.1 Introduction .................................... 99 4.2 Themodel .....................................101 4.2.1 Rateequation.............................. 103 4.3 Solutionsforspecificcases . 105 4.3.1 Uniform attachment and constant size . 105 4.3.2 Preferential attachment and constant size . .. .. 108 4.3.3 Preferential attachment in a growing network . .. .. 112 4.4 Discussion .....................................116 V. Network growth models II: Design and generation of networks with de- sired properties .....................................119 5.1 Introduction .................................... 119 5.2 Growing networkswith desired properties . ........120 5.2.1 Example: power-law degree distribution . 124 5.3 Apracticalimplementation. 127 5.4 Exampleapplication.. .. .. .. .. .. .. .. .. .. .. .. .. 131 5.4.1 Definitionoftheproblem . 131 5.4.2 Searchtimeandbandwidth . 132 5.4.3 Searchstrategiesandsearchtime . 133 5.4.4 Bandwidth................................136 5.4.5 Choiceofnetwork. .. .. .. .. .. .. .. .. .. .. .. .. 137 5.4.6 Itemfrequencydistribution . 139 5.4.7 Estimatingnetworksize . 141 5.5 Discussion .....................................143 VI. The Diplomats Dilemma: A Game theory model of Social Networks . 145 6.1 Introduction .................................... 145 6.2 DefinitionoftheModel ............................. 149 6.2.1 Preliminaries .............................. 149 6.2.2 MovesandStrategies. 151 6.2.3 Strategy updates and stochastic rewiring . .. .. 153 6.2.4 Thealgorithm.............................. 155 6.3 Numericalresults ................................ 155 6.3.1 Timeevolution ............................. 155 6.3.2 Examplenetworks .. .. .. .. .. .. .. .. .. .. .. .. 157 6.3.3 Effects of strategies on the network topology . .. .. 159 v 6.3.4 Transitionprobabilities. 165 6.3.5 Dependenceonsystemsizeandnoise . 166 6.4 Discussion .....................................168 VII. Conclusion ........................................173 BIBLIOGRAPHY ........................................ 180 vi LIST OF FIGURES Figure 1.1 An epidemic network of HIV transmission, generated from the data of Potterat et al. [125]. Vertices are individuals, and an edge between any two represents transmission of HIV (either through sexual contact or intravenous drug usage). Picture courtesy of Mark Newman. ..................................... 5 1.2 A visualization of the network structure of the Internet. Vertices represent servers, and edges represent fiber-optic links. Image by the OPTE corporation. ....... 8 1.3 Various ways to represent networks: a) a simple unipartite undirected network; b) a directed network with edges pointing in one direction; c) a network with weighted vertices and edges representing some discrete characteristics; d) a bipartite network with two types of vertices with edges running only between unliketypes. 11 1.4 A simple undirected network consisting of five nodes and four regularedges. 13 1.5 Three different ways to plot the degree distribution as described in the text. From left to right, the degree distribution plotted via linear binning, logarithmic binning and finally the plot of the cumulative distribution. The data is taken from a collaboration network of condensed matter physicists [106]. ............. 19 1.6 A network with three connected components. The sets of two and three vertices are referred to as small components, while the larger set is the giant component. 28 1.7 Visual representation of the sum rule for connected components of vertices reached by following a randomly selected edge as per Eqn (1.43). Each component (squares) can be expressed as the sum of the probabilities of a having a single vertex (circles), a single vertex connected to one component, to two componentsandsoon. 38 2.1 A network containing k-component structure. The nested property of k-components can be seen clearly. In the example above, the entire network is a regular 1- component, within which are nested the bicomponents (lightershade). 58 2.2 Size of the giant component and bicomponent as a function of vertex removal for random networks with exponential(e−λk with λ = 0.4, blue circles) and poisson (mean = 1.5, red squares) degree distributions. Solid lines are the analytical solu- tions and the points are numerical results for networks of size 106 vertices averaged overa100instancesofthenetwork.. ...... 65 2.3 Size of giant bicomponent as vertices are randomly removed from three real-world networks, a metabolic network for C. Elegans [55], a collaboration network of sci- entists working in condensed matter phyiscs [106], and the Western States Power GridoftheUnitedStates[149]. ..... 68 vii 3.1 Vertices in our networks come in three types, represented here by the
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