DOCSLIB.ORG

Percolation and Elasticity of Networks

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Percolation and Elasticity of Networks Percolation and Elasticity of Networks From Cellular Structures to Fibre Bundles Perkolation und Elastizität von Netzwerken Von Zellulären Strukturen zu Faserbündeln Der Naturwissenschaftlichen Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg zur Erlangung des Doktorgrades Dr. rer. nat. vorgelegt von Susan Nachtrab aus Rudolstadt Als Dissertation genehmigt von der Naturwissen- schaftlichen Fakultät der Universität Erlangen-Nürnberg. Tag der mündlichen Prüfung: 9. Dezember 2011 Vorsitzender der Promotionskommission: Prof. Dr. Rainer Fink Erstberichterstatter: Prof. Dr. Klaus Mecke Zweitberichterstatter: Prof. Dr. Ben Fabry Abstract A material’s microstructure is a principal determinant of its effective physical properties. Structure-property relationships that provide a functional form for the dependence of a physical property (e.g. elasticity) on the microstucture’s morphology are essential for the physical understanding and also the practical application in material design. This work focuses on materials with spatial mesoscopic network structure. A new model with adjustable network topology is introduced and its percolation properties and effective elastic properties are examined. In an initially four-coordinated network, ordered or disordered, each vertex is separated with probability p to form two two-coordinated vertices, yielding network geometries that change continuously from network structures to bundles of unbranched, interwoven fibres. The percolation properties of this so-called vertex model are studied for a two- dimensional square lattice and a three-dimensional diamond network, revealing a percola- tion transition at p = 1 in both cases. The analysis of the pair-connectedness function and finite size scaling exhibits critical behaviour with critical exponents β = 0:32 ± 0:02, ν = 1:29 ± 0:04 in two dimensions and β = 0:0021 ± 0:0004, ν = 0:54 ± 0:01 in three dimensions. The values of the exponents differ from those of conventional site and bond percolation, indicating that this vertex model belongs to a different universality class. In addition, in three dimensions the critical exponents do not obey the hyperscaling relations of bond percolation, but a heuristic new hyperscaling relation is found. After inflating the network edges to circular cylinders of finite radius, the resulting structure is interpreted as solid material in network shape, henceforth called network solid. Changes of probability p strongly affect the mechanical properties of such network solids. This is demonstrated by calculating the effective linear-elastic bulk and shear moduli using a finite element method based on voxel representations of the structures. Separating a fraction of the network nodes leads to a strong decay of the effective moduli whose functional dependence can be approximated, for p < 0:5, by an exponential decay for both, fixed and periodic boundary conditions. This is verified for ordered (diamond and nbo) as well as for irregular (foam) initial structures. Compression experiments on laser-sintered models based on diamond network solids confirm these results. In case of periodic boundary conditions, a cross-over from an exponential to a power-law decay in (1 − p) close to the critical point at p = 1 is observed. From this, the elastic critical exponent fc can be estimated as fc = 3:0 ± 0:1, which also differs from the site and bond percolation exponent. The morphological analysis of this work has several applications. For linear-elastic solids, it suggests that the network connectivity can be used as design parameter, for example for open-cell metal foams or bone scaffolds, as the elastic properties can be adjusted to a given value while keeping the pore space geometry and thus transport properties almost constant. The results of the percolation analysis are especially relevant for network models of biological or synthetic polymers with varying degree of cross-linking. iii Zusammenfassung Die Mikrostruktur eines Materials hat großen Einfluß auf seine effektiven physikalischen Eigenschaften. Funktionale Abhängigkeiten physikalischer Größen (z.B. der Elastizität) von der Morphologie der Mikrostruktur sind essentiell für das physikalische Verständnis und die praktische Anwendung im Materialdesign. Der Fokus dieser Arbeit liegt auf Materialien, deren Mikrostruktur durch ein Netzwerk dargestellt werden kann. Ein neues Modell mit einstellbarer Netzwerktopologie wird vorgestellt, dessen Perkolationseigen- schaften und effektive elastische Eigenschaften untersucht werden. Ausgehend von einem ursprünglich vier-verbundenen, geordneten oder ungeordneten Netzwerk wird jeder Vertex mit Wahrscheinlichkeit p in zwei zwei-verbundene Vertizes getrennt. Netzwerkgeometrien werden dadurch kontinuierlich zu Bündeln verschlungener Fasern. Somit wird über den Parameter p die mittlere Verbundenheit der Vertizes von vier im ursprünglichen Netzwerk auf zwei bei p = 1 herabgesetzt. Die Perkolationseigenschaften dieses sogenannten Vertex-Modells werden auf dem zweidimensionalen Quadratgitter und dem dreidimensionalen Diamantnetz untersucht. Beide Modelle zeigen einen Perkolationsübergang bei p = 1. Die Analyse der Paar- Verbundenheitsfunktion und das Finite-Size-Scaling offenbaren kritisches Verhalten mit kritischen Exponenten β = 0:32 ± 0:02, ν = 1:29 ± 0:04 in zwei Dimensionen und β = 0:0021 ± 0:0004, ν = 0:54 ± 0:01 in drei Dimensionen. Die Werte der Exponenten unterscheiden sich von konventioneller Site- und Bond-Perkolation, was darauf hindeutet, dass das Vertex-Modell zu einer neuen Universalitätsklasse gehört. In drei Dimensionen werden außerdem neue Hyperskalen-Beziehungen für das Vertex-Modell vorgeschlagen. Werden die Kanten der Netzwerkstrukturen durch Kreiszylindern mit endlichem Ra- dius ersetzt, können sie als Festkörper in Form eines Netzwerkes interpretiert werden. Nachfolgend werden diese Strukturen Netzwerkkörper genannt. Mithilfe einer Finite- Elemente-Methode, die auf voxelierten Repräsentationen der Strukturen basiert, werden die effektiven, linear-elastischen Eigenschaften berechnet, wobei eine Veränderung von p großen Einfluß auf die mechanischen Eigenschaften solcher Netzwerkkörper hat. Die Trennung von Netzwerkknoten führt für p < 0:5 zu einem starken Abfall der effektiven linear-elastischen Moduln, deren funktionale Abhängigkeit von p sowohl für periodische als auch nicht-periodische Randbedingungen durch einen exponentiellen Abfall approximiert werden kann. Dies kann für geordnete (Diamant und nbo) und für ungeordnete (Schaum) Ausgangsstrukturen gezeigt werden. Kompressionsexperimente an Laser-gesinterten, auf der Diamantstruktur basierenden Modellen bestätigen dieses Ergebnis. Im Falle periodischer Randbedingungen kann ein Cross-over Verhalten von einem exponentiellen zu einem algebraischen Abfall in (1 − p) nah am kritischen Punkt p = 1 beobachtet werden. Daraus kann der elastische kritische Exponent fc zu fc = 3:0 ± 0:1 abgeschätzt werden, der sich ebenfalls von dem elastischen kritischen Exponenten der Site- und Bondperkolation unterscheidet. Für die morphologische Analyse dieser Arbeit gibt es verschiedene Anwendungen. Für linear-elastische Festkörper legt sie nahe, die Netzwerkverbundenheit als Designparameter, v z.B. für offenzellige Metallschäume oder Knochengerüste, zu verwenden, da die elasti- schen Eigenschaften auf einen bestimmten Wert eingestellt werden können, während die Porenraumgeometrie und somit die Transporteigenschaften kaum verändert werden. Die Ergebnisse der Perkolationsanalyse sind im Speziellen für Netzwerkmodelle biologischer und synthetischer Polymere mit variablem Vernetzungsgrad relevant. vi Parts of the results of the research described in this thesis have been published or accepted for publication in the following journal articles: Susan Nachtrab, Sebastian C. Kapfer, Christoph H. Arns, Mahyar Madadi, Klaus Mecke, Gerd E. Schröder-Turk, Morphology and Linear-Elastic Moduli of Random Network Solids, Advanced Materials, Volume 23, Issue 22-23, pages 2633–2637, 2011 Susan Nachtrab, Sebastian C. Kapfer, Dominik Rietzel, Dietmar Drummer, Mahyar Madadi, Christoph H. Arns, Andrew M. Kraynik, Gerd E. Schröder-Turk and Klaus Mecke, Tuning Elasticity of Open-Cell Solid Foams and Bone Scaffolds via Randomized Vertex Connectivity, Advanced Engineering Materials, 21 September 2011, DOI = 10.1002/adem.201100145 vii Contents 1 Introduction: Networks and Foams1 1.1 Network Structures for Tissue Engineering.................2 1.2 Solid Open-Cell Foams............................3 1.3 Network Percolation and Elastic Properties................5 2 Network models with topology changes: From Open Cells to Fibre Bundles7 2.1 Graph Representation of Networks......................7 2.2 Ordered Four-Coordinated Networks....................8 2.2.1 Diamond Network..........................8 2.2.2 NbO Network.............................9 2.3 Disordered Four-Coordinated Networks................... 10 2.3.1 Voronoi Diagrams.......................... 10 2.3.2 Liquid Foams.............................. 11 2.4 Disconnection Mechanism: Node Separation................. 14 2.4.1 Shifting Nodes When Separating?.................. 15 2.4.2 Periodic Boundary Conditions..................... 21 2.5 Topological Characterisation.......................... 21 3 Percolation Properties 27 3.1 Cluster Characteristics............................ 30 3.2 Scaling Theory................................ 33 3.2.1 Universality Classes.......................... 34 3.2.2 Fractal Dimension Df of a Cluster................. 35 3.2.3 Finite Size Scaling.........................
Load more

Recommended publications
  • Emergence of Disconnected Clusters in Heterogeneous Complex Systems István A
    www.nature.com/scientificreports OPEN Emergence of disconnected clusters in heterogeneous complex systems István A. Kovács1,2* & Róbert Juhász2 Percolation theory dictates an intuitive picture depicting correlated regions in complex systems as densely connected clusters. While this picture might be adequate at small scales and apart from criticality, we show that highly correlated sites in complex systems can be inherently disconnected. This fnding indicates a counter-intuitive organization of dynamical correlations, where functional similarity decouples from physical connectivity. We illustrate the phenomenon on the example of the disordered contact process (DCP) of infection spreading in heterogeneous systems. We apply numerical simulations and an asymptotically exact renormalization group technique (SDRG) in 1, 2 and 3 dimensional systems as well as in two-dimensional lattices with long-ranged interactions. We conclude that the critical dynamics is well captured by mostly one, highly correlated, but spatially disconnected cluster. Our fndings indicate that at criticality the relevant, simultaneously infected sites typically do not directly interact with each other. Due to the similarity of the SDRG equations, our results hold also for the critical behavior of the disordered quantum Ising model, leading to quantum correlated, yet spatially disconnected, magnetic domains. Correlated clusters emerge in a broad range of systems, ranging from magnetic models to out-of-equilibrium systems1–3. Apart from the critical point, the correlation length is fnite, limiting the spatial separation between highly correlated sites, leading to spatially localized, fnite clusters. Although at the critical point the correlation length diverges, our traditional intuition is driven by percolation processes, indicating that highly correlated critical clusters remain connected, while broadly varying in their size.
    [Show full text]
  • Arxiv:1504.02898V2 [Cond-Mat.Stat-Mech] 7 Jun 2015 Keywords: Percolation, Explosive Percolation, SLE, Ising Model, Earth Topography
    Recent advances in percolation theory and its applications Abbas Ali Saberi aDepartment of Physics, University of Tehran, P.O. Box 14395-547,Tehran, Iran bSchool of Particles and Accelerators, Institute for Research in Fundamental Sciences (IPM) P.O. Box 19395-5531, Tehran, Iran Abstract Percolation is the simplest fundamental model in statistical mechanics that exhibits phase transitions signaled by the emergence of a giant connected component. Despite its very simple rules, percolation theory has successfully been applied to describe a large variety of natural, technological and social systems. Percolation models serve as important universality classes in critical phenomena characterized by a set of critical exponents which correspond to a rich fractal and scaling structure of their geometric features. We will first outline the basic features of the ordinary model. Over the years a variety of percolation models has been introduced some of which with completely different scaling and universal properties from the original model with either continuous or discontinuous transitions depending on the control parameter, di- mensionality and the type of the underlying rules and networks. We will try to take a glimpse at a number of selective variations including Achlioptas process, half-restricted process and spanning cluster-avoiding process as examples of the so-called explosive per- colation. We will also introduce non-self-averaging percolation and discuss correlated percolation and bootstrap percolation with special emphasis on their recent progress. Directed percolation process will be also discussed as a prototype of systems displaying a nonequilibrium phase transition into an absorbing state. In the past decade, after the invention of stochastic L¨ownerevolution (SLE) by Oded Schramm, two-dimensional (2D) percolation has become a central problem in probability theory leading to the two recent Fields medals.
    [Show full text]
  • Directed Percolation and Directed Animals
    1 Directed Percolation and Directed Animals Deepak Dhar Theoretical Physics Group Tata Institute of Fundamental Research Homi Bhabha Road, Mumbai 400 005, (INDIA) Abstract These lectures provide an introduction to the directed percolation and directed animals problems, from a physicist’s point of view. The probabilistic cellular automaton formulation of directed percolation is introduced. The planar duality of the diode-resistor-insulator percolation problem in two dimensions, and relation of the directed percolation to undirected first passage percolation problem are described. Equivalence of the d-dimensional directed animals problem to (d − 1)-dimensional Yang-Lee edge-singularity problem is established. Self-organized critical formulation of the percolation problem, which does not involve any fine-tuning of coupling constants to get critical behavior is briefly discussed. I. DIRECTED PERCOLATION Although directed percolation (DP ) was defined by Broadbent and Hammersley along with undirected percolation in their first paper in 1957 [1], it did not receive much attention in percolation theory until late seventies, when Blease obtained rather precise estimates of some critical exponents [2] using the fact that one can efficiently generate very long series expansions for this problem on the computer. Since then, it has been studied extensively, both as a simple model of stochastic processes, and because of its applications in different physical situations as diverse as star formation in galaxies [3], reaction-diffusion systems [4], conduction in strong electric field in semiconductors [5], and biological evolution [6]. The large number of papers devoted to DP in current physics literature are due to the fact that the critical behavior of a very large number of stochastic evolving systems are in the DP universality class.
    [Show full text]
  • 1 Exploring Percolative Landscapes: Infinite Cascades of Geometric Phase Transitions P. N. Timonin1, Gennady Y. Chitov2 (1) Phys
    Exploring Percolative Landscapes: Infinite Cascades of Geometric Phase Transitions P. N. Timonin1, Gennady Y. Chitov2 (1) Physics Research Institute, Southern Federal University, 344090, Stachki 194, Rostov-on-Don, Russia, [email protected] (2) Department of Physics, Laurentian University Sudbury, Ontario P3E 2C6, Canada, [email protected] The evolution of many kinetic processes in 1+1 (space-time) dimensions results in 2d directed percolative landscapes. The active phases of these models possess numerous hidden geometric orders characterized by various types of large-scale and/or coarse-grained percolative backbones that we define. For the patterns originated in the classical directed percolation (DP) and contact process (CP) we show from the Monte-Carlo simulation data that these percolative backbones emerge at specific critical points as a result of continuous phase transitions. These geometric transitions belong to the DP universality class and their nonlocal order parameters are the capacities of corresponding backbones. The multitude of conceivable percolative backbones implies the existence of infinite cascades of such geometric transitions in the kinetic processes considered. We present simple arguments to support the conjecture that such cascades of transitions is a generic feature of percolation as well as of many other transitions with nonlocal order parameters. Keywords: kinetic processes, phase transitions, percolation. PACS numbers: 64.60.Cn, 05.70.Jk, 64.60.Fr 1. Introduction The cornerstones of modern civilization are various types of networks: telecommunications, electric power supply, Internet, warehouse logistics, to name a few. Their proper functioning is of great importance and many studies have been devoted to their robustness against various kinds of damages [1, 2].
    [Show full text]
  • The Universal Critical Dynamics of Noisy Neurons
    University of Calgary PRISM: University of Calgary's Digital Repository Graduate Studies The Vault: Electronic Theses and Dissertations 2019-05-02 The Universal Critical Dynamics of Noisy Neurons Korchinski, Daniel James Korchinski, D. J. (2019). The Universal Critical Dynamics of Noisy Neurons (Unpublished master's thesis). University of Calgary, Calgary, AB. http://hdl.handle.net/1880/110325 master thesis University of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission. Downloaded from PRISM: https://prism.ucalgary.ca UNIVERSITY OF CALGARY The Universal Critical Dynamics of Noisy Neurons by Daniel James Korchinski A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE GRADUATE PROGRAM IN PHYSICS AND ASTRONOMY CALGARY, ALBERTA May, 2019 c Daniel James Korchinski 2019 Abstract The criticality hypothesis posits that the brain operates near a critical point. Typically, critical neurons are assumed to spread activity like a simple branching process and thus fall into the universality class of directed percolation. The branching process describes activity spreading from a single initiation site, an assumption that can be violated in real neurons where external drivers and noise can initiate multiple concurrent and independent cascades. In this thesis, I use the network structure of neurons to disentangle independent cascades of activity. Using a combination of numerical simulations and mathematical modelling, I show that criticality can exist in noisy neurons but that the presence of noise changes the underly- ing universality class from directed to undirected percolation.
    [Show full text]
  • Directed Percolation and Turbulence
    Emergence of collective modes, ecological collapse and directed percolation at the laminar-turbulence transition in pipe flow Hong-Yan Shih, Tsung-Lin Hsieh, Nigel Goldenfeld University of Illinois at Urbana-Champaign Partially supported by NSF-DMR-1044901 H.-Y. Shih, T.-L. Hsieh and N. Goldenfeld, Nature Physics 12, 245 (2016) N. Goldenfeld and H.-Y. Shih, J. Stat. Phys. 167, 575-594 (2017) Deterministic classical mechanics of many particles in a box statistical mechanics Deterministic classical mechanics of infinite number of particles in a box = Navier-Stokes equations for a fluid statistical mechanics Deterministic classical mechanics of infinite number of particles in a box = Navier-Stokes equations for a fluid statistical mechanics Transitional turbulence: puffs • Reynolds’ original pipe turbulence (1883) reports on the transition Univ. of Manchester Univ. of Manchester “Flashes” of turbulence: Precision measurement of turbulent transition Q: will a puff survive to the end of the pipe? Many repetitions survival probability = P(Re, t) Hof et al., PRL 101, 214501 (2008) Pipe flow turbulence Decaying single puff metastable spatiotemporal expanding laminar puffs intermittency slugs Re 1775 2050 2500 푡−푡 − 0 Survival probability 푃 Re, 푡 = 푒 휏(Re) ) Re,t Puff P( lifetime to N-S Avila et al., (2009) Avila et al., Science 333, 192 (2011) Hof et al., PRL 101, 214501 (2008) 6 Pipe flow turbulence Decaying single puff Splitting puffs metastable spatiotemporal expanding laminar puffs intermittency slugs Re 1775 2050 2500 푡−푡 − 0 Splitting
    [Show full text]
  • Critical Behaviour of Directed Percolation Process in the Presence of Compressible Velocity Field
    Critical Behaviour of Directed Percolation Process in the Presence of Compressible Velocity Field Master’s Thesis (60 HE credits) Author: Viktor Skult´etyˇ † Supervisors: Juha Honkonen, Thors Hans Hansson Affiliation: Fysikum, Stockholm University, Nordic Institute for Theoretical Physics (NORDITA) 2017-04-21, Stockholm †[email protected] secret message Abstract Renormalization group analysis is a useful tool for studying critical behaviour of stochastic systems. In this thesis, field-theoretic renormalization group will be applied to the scalar model representing directed percolation, known as Gribov model, in presence of the random velocity field. Turbulent mixing will be modelled by the compressible form of stochastic Navier-Stokes equation where the compressibility is described by an additional field related to the density. The task will be to find corresponding scaling properties. Acknowledgements First, I would like to thank my supervisors Juha Honkonen and Thors Hans Hansson for great supervision. I express my gratitude to people from the Department of Physics at the University of Helsinki where I have spent a couple of months. I would also like to thank Paolo Muratore-Ginanneschi from the Department of Mathematics for his time and willingness to always help me with my questions. Furthermore, I wish to thank people from the Nordic Institute for Theoretical Physics, especially to Erik Aurell, Ralf Eichhorn for their kind hospitality. My special thanks goes to Tom´aˇsLuˇcivjansk´yfrom Department of Theoretical Physics at Pavol Jozef Saf´arikUniversityˇ in Koˇsice,who was always willing to give me advices on how to approach the problems I stumbled upon during my work. Finally, I would like to thank my family and friends which were supporting me all the time and without whom this thesis would not be possible.
    [Show full text]
  • Block Scaling in the Directed Percolation Universality Class
    OPEN ACCESS Recent citations Block scaling in the directed percolation - 25 Years of Self-organized Criticality: Numerical Detection Methods universality class R. T. James McAteer et al - The Abelian Manna model on various To cite this article: Gunnar Pruessner 2008 New J. Phys. 10 113003 lattices in one and two dimensions Hoai Nguyen Huynh et al View the article online for updates and enhancements. This content was downloaded from IP address 170.106.40.139 on 26/09/2021 at 04:54 New Journal of Physics The open–access journal for physics Block scaling in the directed percolation universality class Gunnar Pruessner1 Mathematics Institute, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK E-mail: [email protected] New Journal of Physics 10 (2008) 113003 (13pp) Received 23 July 2008 Published 7 November 2008 Online at http://www.njp.org/ doi:10.1088/1367-2630/10/11/113003 Abstract. The universal behaviour of the directed percolation universality class is well understood—both the critical scaling and the finite size scaling. This paper focuses on the block (finite size) scaling of the order parameter and its fluctuations, considering (sub-)blocks of linear size l in systems of linear size L. The scaling depends on the choice of the ensemble, as only the conditional ensemble produces the block-scaling behaviour as established in equilibrium critical phenomena. The dependence on the ensemble can be understood by an additional symmetry present in the unconditional ensemble. The unconventional scaling found in the unconditional ensemble is a reminder of the possibility that scaling functions themselves have a power-law dependence on their arguments.
    [Show full text]
  • Critical Study on the Absorbing Phase Transition in a Four-State Predator-Prey Model in One Dimension2
    Critical Study on the Absorbing Phase Transition in a Four-State Predator-Prey Model in One Dimension Su-Chan Park Department of Physics, The Catholic University of Korea, Bucheon 420-743, Korea E-mail: [email protected] Abstract. In a recent letter [Chatterjee R, Mohanty P K, and Basu A, 2011 J. Stat. Mech. L05001], it was claimed that a four-state predator-prey (4SPP) model exhibits critical behavior distinct from that of directed percolation (DP). In this Letter, we show that the 4SPP in fact belongs to the DP universality class. PACS numbers: 64.60.ah, 64.60.-i, 64.60.De, 89.75.-k arXiv:1108.5127v1 [cond-mat.stat-mech] 25 Aug 2011 Critical Study on the Absorbing Phase Transition in a Four-State Predator-Prey Model in One Dimension2 Absorbing phase transitions have been studied extensively as prototypical emergent phenomena of nonequilibrium systems and several universality classes have been identified (for a review, see, e. g., Refs. [1, 2, 3, 4]). Among them, the directed percolation (DP) universality class has been studied thoroughly both numerically and analytically. As a result, a so-called ‘DP-conjecture’ [5, 6] has emerged since early 1980’s, asserting that a model should belong to the DP class, if the model exhibits a phase transition from an active, fluctuating phase into a unique absorbing state, if the phase transition should be characterized by a one-component order parameter, if no quenched disorder is involved, if dynamic rules of the model are short-ranged processes, and if the model has neither novel symmetry nor conservation.
    [Show full text]
  • The Role of Topology and Topological Changes in the Mechanical Properties of Epithelia
    The Role of Topology and Topological Changes in the Mechanical Properties of Epithelia by Meryl Spencer A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Physics) in The University of Michigan 2019 Doctoral Committee: Associate Professor David K Lubensky, Chair Professor Mark Newman Professor Jennifer Ogilvie Assistant Professor Kevin Wood Meryl Spencer [email protected] ORCID iD: 0000-0003-1994-4100 c Meryl Spencer 2019 All Rights Reserved ACKNOWLEDGEMENTS This work would have not have been possible without the mentorship of my adviser David. Much of my thesis work relies on analysis of thousands of images taken by Jesus Lopez-Gay. I sustained a major spinal cord injury just as I started to write my thesis. I would not have been physically capable of finishing it, if not for the amazing medical care I received from my entire team at the neuro-ICU and 6A. I could not have afforded the care without medical benefits from my grad union GEO. Thank you from the bottom of my heart everyone who cooked me meals, cleaned my house, and gave me moral support throughout my long recovery. Special thanks to Patricia Klein for organizing my care, and my parents for their constant support. This work probably would have been possible without the support of my fellow graduate students, but it wouldn't have been nearly as fun. I'd like to thank my amazing first-year cohort especially, Paige, Rutu, Karishima, Chrisy and Kevin for working through Jackson with me, as well as Jessie, Joe, Glenn, Peter, Ansel, and Anthony for joining in second-year.
    [Show full text]
  • ESD.342 Random Networks Lecture 13
    Random Networks and Percolation • Percolation, cascades, pandemics • Properties, Metrics of Random Networks • Basic Theory of Random Networks and Cascades • Watts Cascades • Analytic Model of Watts Cascades 2/16/2011 Random networks © Daniel E Whitney 1997-2010 1/46 and cascades Types of Percolation Models • “Short loop” models (Stauffer, Grimmett, Morris) usually assume regular network structure or same nodal degree for all nodes • “Long loop/no loop” models (Newman, Calloway, Watts) usually assume random tree-like structure • “Collective action” models (Schelling, Granovetter) assume k = n (all nodes see all other nodes) • “Local action” models assume z << n (Watts, etc.) • “Threshold models” (Morris, Schelling, Granovetter, Watts) assume that a node changes state when more than a threshold fraction of neighboring nodes have changed state – Threshold models are equivalent to deterministic two-person games (Lopez-Pintado) (Morris) – Disease spreading (SIR) assumes a threshold number of neighbors • Mixed collective/local models have also been proposed (Valente) for diffusion of innovations 2/16/2011 Random networks © Daniel E Whitney 1997-2010 2/46 and cascades Percolation Contexts • Spread of diseases (Watts and others)(local) • Propagation of rumors (Newman, Watts, Calloway)(local)-scary talk about surprises • Success of “blockbusting” (Schelling)(collective) • Decision to join a riot (Granovetter)(collective) • Adoption of innovations (Rogers, Valente)(both) • In each case, nodes are assumed to be different in their susceptibility
    [Show full text]
  • Study of Percolation Threshold on Urban Road Models
    1 HALAMAN JUDUL THESIS – SS142501 FIFTH GENERATION (5G) MOBILE NETWORKS: STUDY OF PERCOLATION THRESHOLD ON URBAN ROAD MODELS NILA NOVITA GAFUR 06211450010019 SUPERVISORS Dr. Elie Cali Dr. Taoufik En-Najjary Dr. Arnaud Guyader MASTER PROGRAM DEPARTMENT OF STATISTICS FACULTY OF MATHEMATICS, COMPUTING AND DATA SCIENCE INSTITUT TEKNOLOGI SEPULUH NOPEMBER SURABAYA 2018 Réseau Mobile 5G: Étude de Seuils de Percolation sur des Modèles de Voirie Urbaine Confidentiel Orange Réalisée par : Nila Novita Gafur Responsable Entreprise : Dr. Elie Cali Dr. Taoufik En-Najjary Responsable Académique : Dr. Arnaud Guyader Rapport du Stage Master de Statistique M2 Université Pierre et Marie Curie 2016/2017 DOCUMENT FOR INSTITUT TEKNOLOGI SEPULUH NOPEMBER (ITS), SURABAYA, INDONESIA This document is research report held in Université Pierre et Marie Curie (UPMC), Paris, France. This report collection aims to meet the master graduation in ITS, Surabaya, Indonesia. I would like to dedicate this thesis to my beloved parents FIFTH GENERATION (5G) MOBILE NETWORKS: STUDY OF PERCOLATION THRESHOLD ON URBAN ROAD MODELS Student’s name : Nila Novita Gafur Student’s ID : 06211450010019 First supervisor : Dr. Elie Cali Second supervisor : Dr. Taoufik En-Najjary Third supervisor : Dr. Arnaud Guyader ABSTRACT Stochastic geometry is a mathematical discipline that combines geometry and probability. In particular, it model complex systems with a large number of elements distributed over a geographical area and has numerous applications in telecommunications. Based on stochastic geometry, mathematical models are designed to represent aspects of wireless networks. Talking about stochastic geometry models of wireless networks will not be detached from the important role of the continuum percolation. That is an extension of the percolation theory at 푅2.
    [Show full text]