Numeric Study of Magnetic Systems and Networks Público

Abstract

Recent years have seen considerable time and effort devoted to both Magnetic Systems and Complex Networks. In both cases, their properties depend strongly on the way the different elements are connected and on disorder. These similarities make their combined study an interesting and useful exercise with insights gained in one area proving valuable in the other. We started by studying a fully conencted SK-like spin glass model with distance depen- dent bonds. By varying the distance dependence exponent, we easily interpolate between a fully connected and a nearest neigbor model. The way that the varying conenctivity affects the dynamics of the system is analysed. I then moved on to a purely 1D system of Heisenberg spins where I studied the consequences that the Topological Non-connectivity Threshold (TNT) has on systems with long range interactions. We find that this system can display ferromagnetism and hysteretic behavior for experimental times. A system of Potts spins interacting along the edges of k-regular graphs was also studied by comparing the an- alytical expressions for the entropy and free energy obtained through the Large Deviation Cavity Method, with the numerical results of Metropolis and Creutz simulations. In the field of Networks, two classes of regular hierarchical graphs that have proven amenable to several nice analytical approaches are introduced. In particular, the Renor- malization Group analysis of the RW leads to the use of a boundary layer concept, and results in a fractional diffusion coefficient involving the golden ratio. In the structure of Erdos-Renyi graphs I concluded that an optimal bisection requires not only the pruning of all the trees and dangling sub loops that surround the central core of the GCC. Still in this area, I analyzed how weight-weight correlations in networks impacted the transportation properties of networks. We showed that weight-weight correlations have the effect of facili- tating transport. In an interdisciplinary project in the field of human dynamics, I analyzed the access logs to Emory University's website, and showed that one can use these indirect methods to characterize a given population and how it interacts with a website.

Table of Contents

1 Introduction 1 I Magnetic Systems 4 2 Spin Glasses 5 2.1 Disordered Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Frustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Sherrington-Kirkpatrick model . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 1D Ising chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.5 Hysteretic Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.6 Magnetic recording . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.7 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3 Monte-Carlo Methods 21 3.1 Master Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 Importance Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.3 Spin Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.4 Density Of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.5 Microcanonical Monte-Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4 Ensemble Inequivalence 29 4.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2 Large Deviation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 iii CONTENTS iv 4.2.1 Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2.2 Link addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.2.3 Adding a site . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.2.4 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.3 Comparison with numerical simulations . . . . . . . . . . . . . . . . . . . . 36 4.4 Conclusion and perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.5 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5 Topological Non-Connectivity Threshold 38 5.1 Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.2 Probability of Zero Magnetization . . . . . . . . . . . . . . . . . . . . . . . 45 5.3 Magnetization Reversal Times . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.4 Power-law corrections to Arrhenius's law . . . . . . . . . . . . . . . . . . . . 55 5.5 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6 Hanoi Networks 58 6.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.2 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.3 Ising Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 6.3.1 Renormalization Group . . . . . . . . . . . . . . . . . . . . . . . . . 72 6.3.1.1 Leading order evaluation . . . . . . . . . . . . . . . . . . . 76 6.3.2 Monte-Carlo Simulations . . . . . . . . . . . . . . . . . . . . . . . . 78 6.4 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 II Complex Networks 80 7 Graph Bisection 81 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 7.2 Core Peeling Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 7.3 Core Cutting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 7.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 CONTENTS v 8 Weight Correlations 94 8.1 Weight Correlation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 8.2 Correlation Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 8.3 Transport In Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 8.4 Real World Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 8.5 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 9 Web Surfing 104 9.1 The World Wide Web (WWW) . . . . . . . . . . . . . . . . . . . . . . . . . 104 9.2 Network Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 9.3 Preferential Linking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 9.4 Non-stationarities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 9.5 Activity Patterns of the Population . . . . . . . . . . . . . . . . . . . . . . . 116 9.6 Individual Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 9.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 9.8 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 10 Information propagation in networks 123 10.1 Epidemics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 10.2 Broadcasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 10.3 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 11 Final Remarks 131 III Appendices 133 A Logarithmic Binning 134 B Generating Function Formalism 136 C Renormalization Group For Random Walks 138 CONTENTS vi D Condor and DRBL 145 D.1 DRBL installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 D.2 Condor installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 D.3 Using the cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Bibliography 154

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School	Laney Graduate School
Department	Physics
Degree	PhD
Submission	Dissertation
Language	English
Research Field	Physics, General Physics, Electricity and Magnetism Physics, Theory
Palabra Clave	Monte-Carlo Complex Networks Optimization Spin Glasses
Committee Chair / Thesis Advisor	Boettcher, Stefan, Emory University
Committee Members	Haber, Eldad, Emory University Family, Fereydoon, Emory University Borgonovi, Fausto, Univercita Cattolica Percus, Allon, UCLA Segre, Phil, Emory University

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