DOCSLIB.ORG

Graph Theory in the Information Age Fan Chung

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Graph Theory in the Information Age Fan Chung Graph Theory in the Information Age Fan Chung nthepastdecade,graphtheoryhasgone through a remarkable shift and a profound transformation. The change is in large part due to the humongous amount of informa- tion that we are confronted with. A main way Ito sort through massive data sets is to build and examine the network formed by interrelations. For example, Google’s successful Web search al- gorithms are based on the WWW graph, which contains all Web pages as vertices and hyper- links as edges. There are all sorts of information networks, such as biological networks built from biological databases and social networks formed by email, phone calls, instant messaging, etc., as well as various types of physical networks. Of particular interest to mathematicians is the col- laboration graph, which is based on the data from Mathematical Reviews.Inthecollaborationgraph, every mathematician is a vertex, and two mathe- maticians who wrote a joint paper are connected by an edge. Figure 1 illustrates a portion of the collaboration Figure 1. An induced subgraph of the graph consisting of about 5,000 vertices, repre- collaboration graph. senting mathematicians with Erd˝os number 2 (i.e., mathematicians who wrote a paper with a coauthor of Paul Erd˝os). been used in a wide range of areas. However, Graph theory has two hundred years of history never before have we confronted graphs of not studying the basic mathematical structures called only such tremendous sizes but also extraordinary graphs. A graph G consists of a collection V of richness and complexity, both at a theoretical and vertices and a collection E of edges that connect apracticallevel.Numerouschallengingproblems pairs of vertices. In the past, graph theory has have attracted the attention and imagination of Fan Chung is professor of mathematics at the University researchers from physics, computer science, engi- of California, San Diego. Her email address is fan@ucsd. neering, biology, social science, and mathematics. edu. The new area of “network science” emerged, call- ing for a sound scientific foundation and rigorous ∗This article is based on the Noether Lecture given at the AMS-MAA-SIAM Annual Meeting, January 2009, Wash- analysis for which graph theory is ideally suited. In ington D. C. the other direction, examples of real-world graphs 726 Notices of the AMS Volume 57,Number6 lead to central questions and new directions for graph in G(n, p) has the same expected degree research in graph theory. at every vertex, and therefore G(n, p) does not These real-world networks are massive and capture some of the main behaviors of real-world complex but illustrate amazing coherence. Empir- graphs. Nevertheless, the approaches and methods ically, most real-world graphs have the following in classical random graph theory provide the properties: foundation for the study of random graphs with sparsity—The number of edges is within general degree distributions. • aconstantmultipleofthenumberof Many random graph models have been pro- vertices. posed in the study of information network graphs, small world phenomenon—Any two ver- but there are basically two different approaches. • tices are connected by a short path. Two The “online” model mimics the growth or decay of vertices having a common neighbor are adynamicallychangingnetwork,andthe“offline” more likely to be neighbors. model of random graphs consists of specified fam- power law degree distribution—The degree ilies of graphs as the probability spaces together • of a vertex is the number of its neighbors. with some specified probability distribution. The number of vertices with degree j (or One online model is the so-called preferential β attachment scheme,whichcanbedescribedas having j neighbors) is proportional to j− for some fixed constant β. “the rich get richer”. The preferential attachment scheme has been receiving much attention in the To deal with these information networks, many recent study of complex networks [11, 57], but its basic questions arise: What are basic structures of history can be traced back to Vilfredo Pareto in such large networks? How do they evolve? What 1896, among others. At each tick of the clock (so are the underlying principles that dictate their to speak), a new edge is added, with each of its behavior? How are subgraphs related to the large endpoints chosen with probability proportional to (and often incomplete) host graph? What are the their degrees. It can be proved [15, 31, 57] that the main graph invariants that capture the myriad preferential attachment scheme leads to a power properties of such large graphs? law degree distribution. There are several other To answer these problems, we first delve into the online models, including the duplication model wealth of knowledge from the past, although it is (which seems to be more feasible for biological often not enough. In the past thirty years there has networks, see [35]), as well as many recent exten- been a great deal of progress in combinatorial and sions, such as adding more parameters concerning probabilistic methods, as well as spectral methods. the “talent” or “fitness” of each node [50]. However, traditional probabilistic methods mostly There are two main offline graph models consider the same probability distribution for all for graphs with general degree distribution—the vertices or edges while real graphs are uneven configuration model and random graphs with ex- and clustered. The classical algebraic and ana- pected degree sequences. A random graph in lytic methods are efficient in dealing with highly the configuration model with degree sequences symmetric structures, whereas real-world graphs d1,d2,...,dn is defined by choosing a random are quite the opposite. Guided by examples of matching on !i di “pseudo nodes”, where the real-world graphs, we are compelled to improvise, pseudo nodes are partitioned into parts of sizes extend and create new theory and methods. Here di ,fori 1,...,n.Eachpartisassociatedwitha we will discuss the new developments in several vertex. By= using results of Molloy and Reed [58, 59], topics in graph theory that are rapidly developing. it can be shown [2] that under some mild condi- The topics include a general random graph theory tions, a random power law graph with exponent for any given degree distribution, percolation in β almost surely has no giant component if β β0 general host graphs, PageRank for representing ≥ where β0 is a solution to the equation involving the quantitative correlations among vertices, and the Riemann zeta function ζ(β 2) 2ζ(β 1) 0. game aspects of graphs. The general random graph− model− G(−w) =with expected degree sequence w (w1,w2,...,wn) Random Graph Theory for General Degree follows the spirit of the Erd˝os-Rényi= model. The Distributions probability of having an edge between the ith and The primary subject in the study of random graph jth vertices is defined to be wi wj /Vol (G),where w theory is the classical random graph G(n, p), Vol (G) denotes !i wi .Furthermore,inG( ) each introduced by Erd˝os and Rényi in 1959 [38, 39] edge is chosen independently of the others, and (also independently by Gilbert [44]). In G(n, p), therefore the analysis can be carried out. It was every pair of a set of n vertices is chosen to proved in [28] that if the expected average degree be an edge with probability p.Inaseriesof is strictly greater than 1 in a random graph papers, Erd˝osand Rényi gave an elegant and in G(w),thenthereisagiantcomponent(i.e., comprehensive analysis describing the evolution aconnectedcomponentofvolumeapositive of G(n, p) as p increases. Note that a random fraction of that of the whole graph). Furthermore, June/July 2010 Notices of the AMS 727 the giant component almost surely has volume the host graph and vice versa. It is of interest to δVol (G) O(√n log3.5 n),whereδ is the unique understand the connections between a graph and nonzero root+ of the following equation [29]: its subgraphs. What invariants of the host graph n n canorcannotbetranslatedtoitssubgraphs?Under wi δ (1) " wi e− (1 δ) " wi . what conditions can we predict the behavior of all = − i 1 i 1 or any subgraphs? Can a sparse subgraph have = = Because of the robustness of the G(w) model, very different behavior from its host graph? Here many properties can be derived. For example, a we discuss some of the work in this direction. random graph in G(w) has average distance almost Many information networks or social networks surely equal to (1 o(1)) log n ,andthediameter have very small diameters (in the range of log n), as + log w˜ dictated by the so-called small world phenomenon. is almost surely ( log n ),wherew˜ w 2/ w log w˜ !i i !i i However, in a recent paper by Liben-Nowell and Θ =w provided some mild conditions on are satisfied Kleinberg [52], it was observed that the tree-like [27]. For the range 2 <β<3, where the power law subgraphs derived from some chain-letter data exponents β for numerous real networks reside, seem to have relatively large diameter. In the the power law graph can be roughly described as study of the Erd˝os-Rényi graph model G(n, p),it an “octopus” with a dense subgraph having small was shown [60] that the diameter of a random diameter O(log log n) as the core, while the overall spanning tree is of order √n,incontrastwith diameter is O(log n) and the average distance is the fact that the diameter of the host graph Kn O(log log n) (see [31]). is 1. Aldous [4] proved that in a regular graph For the spectra of power law graphs, there are G with a certain spectral bound σ ,thediameter basically two competing approaches.
Load more

Recommended publications
  • Processes on Complex Networks. Percolation
    Chapter 5 Processes on complex networks. Percolation 77 Up till now we discussed the structure of the complex networks. The actual reason to study this structure is to understand how this structure influences the behavior of random processes on networks. I will talk about two such processes. The first one is the percolation process. The second one is the spread of epidemics. There are a lot of open problems in this area, the main of which can be innocently formulated as: How the network topology influences the dynamics of random processes on this network. We are still quite far from a definite answer to this question. 5.1 Percolation 5.1.1 Introduction to percolation Percolation is one of the simplest processes that exhibit the critical phenomena or phase transition. This means that there is a parameter in the system, whose small change yields a large change in the system behavior. To define the percolation process, consider a graph, that has a large connected component. In the classical settings, percolation was actually studied on infinite graphs, whose vertices constitute the set Zd, and edges connect each vertex with nearest neighbors, but we consider general random graphs. We have parameter ϕ, which is the probability that any edge present in the underlying graph is open or closed (an event with probability 1 − ϕ) independently of the other edges. Actually, if we talk about edges being open or closed, this means that we discuss bond percolation. It is also possible to talk about the vertices being open or closed, and this is called site percolation.
    [Show full text]
  • Correlation in Complex Networks
    Correlation in Complex Networks by George Tsering Cantwell A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Physics) in the University of Michigan 2020 Doctoral Committee: Professor Mark Newman, Chair Professor Charles Doering Assistant Professor Jordan Horowitz Assistant Professor Abigail Jacobs Associate Professor Xiaoming Mao George Tsering Cantwell [email protected] ORCID iD: 0000-0002-4205-3691 © George Tsering Cantwell 2020 ACKNOWLEDGMENTS First, I must thank Mark Newman for his support and mentor- ship throughout my time at the University of Michigan. Further thanks are due to all of the people who have worked with me on projects related to this thesis. In alphabetical order they are Eliz- abeth Bruch, Alec Kirkley, Yanchen Liu, Benjamin Maier, Gesine Reinert, Maria Riolo, Alice Schwarze, Carlos Serván, Jordan Sny- der, Guillaume St-Onge, and Jean-Gabriel Young. ii TABLE OF CONTENTS Acknowledgments .................................. ii List of Figures ..................................... v List of Tables ..................................... vi List of Appendices .................................. vii Abstract ........................................ viii Chapter 1 Introduction .................................... 1 1.1 Why study networks?...........................2 1.1.1 Example: Modeling the spread of disease...........3 1.2 Measures and metrics...........................8 1.3 Models of networks............................ 11 1.4 Inference.................................
    [Show full text]
  • Network Science
    NETWORK SCIENCE Random Networks Prof. Marcello Pelillo Ca’ Foscari University of Venice a.y. 2016/17 Section 3.2 The random network model RANDOM NETWORK MODEL Pál Erdös Alfréd Rényi (1913-1996) (1921-1970) Erdös-Rényi model (1960) Connect with probability p p=1/6 N=10 <k> ~ 1.5 SECTION 3.2 THE RANDOM NETWORK MODEL BOX 3.1 DEFINING RANDOM NETWORKS RANDOM NETWORK MODEL Network science aims to build models that reproduce the properties of There are two equivalent defini- real networks. Most networks we encounter do not have the comforting tions of a random network: Definition: regularity of a crystal lattice or the predictable radial architecture of a spi- der web. Rather, at first inspection they look as if they were spun randomly G(N, L) Model A random graph is a graph of N nodes where each pair (Figure 2.4). Random network theoryof nodes embraces is connected this byapparent probability randomness p. N labeled nodes are connect- by constructing networks that are truly random. ed with L randomly placed links. Erds and Rényi used From a modeling perspectiveTo a constructnetwork is a arandom relatively network simple G(N, object, p): this definition in their string consisting of only nodes and links. The real challenge, however, is to decide of papers on random net- where to place the links between1) the Start nodes with so thatN isolated we reproduce nodes the com- works [2-9]. plexity of a real system. In this 2)respect Select the a philosophynode pair, behindand generate a random a network is simple: We assume thatrandom this goal number is best betweenachieved by0 and placing 1.
    [Show full text]
  • Fractal Network in the Protein Interaction Network Model
    Journal of the Korean Physical Society, Vol. 56, No. 3, March 2010, pp. 1020∼1024 Fractal Network in the Protein Interaction Network Model Pureun Kim and Byungnam Kahng∗ Department of Physics and Astronomy, Seoul National University, Seoul 151-747 (Received 22 September 2009) Fractal complex networks (FCNs) have been observed in a diverse range of networks from the World Wide Web to biological networks. However, few stochastic models to generate FCNs have been introduced so far. Here, we simulate a protein-protein interaction network model, finding that FCNs can be generated near the percolation threshold. The number of boxes needed to cover the network exhibits a heavy-tailed distribution. Its skeleton, a spanning tree based on the edge betweenness centrality, is a scaffold of the original network and turns out to be a critical branching tree. Thus, the model network is a fractal at the percolation threshold. PACS numbers: 68.37.Ef, 82.20.-w, 68.43.-h Keywords: Fractal complex network, Percolation, Protein interaction network DOI: 10.3938/jkps.56.1020 I. INTRODUCTION consistent with that of the hub-repulsion model [2]. The fractal scaling of a FCN originates from the fractality of its skeleton underneath it [8]. The skeleton is regarded Fractal complex networks (FCNs) have been discov- as a critical branching tree: It exhibits a plateau in the ered in diverse real-world systems [1, 2]. Examples in- mean branching number functionn ¯(d), defined as the clude the co-authorship network [3], metabolic networks average number of offsprings created by nodes at a dis- [4], the protein interaction networks [5], the World-Wide tance d from the root.
    [Show full text]
  • Neutral Evolution of Proteins: the Superfunnel in Sequence Space and Its Relation to Mutational Robustness
    Neutral evolution of proteins: The superfunnel in sequence space and its relation to mutational robustness. Josselin Noirel, Thomas Simonson To cite this version: Josselin Noirel, Thomas Simonson. Neutral evolution of proteins: The superfunnel in sequence space and its relation to mutational robustness.. Journal of Chemical Physics, American Institute of Physics, 2008, 129 (18), pp.185104. 10.1063/1.2992853. hal-00488189 HAL Id: hal-00488189 https://hal-polytechnique.archives-ouvertes.fr/hal-00488189 Submitted on 22 May 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THE JOURNAL OF CHEMICAL PHYSICS 129, 185104 ͑2008͒ Neutral evolution of proteins: The superfunnel in sequence space and its relation to mutational robustness Josselin Noirela͒ and Thomas Simonsonb͒ Laboratoire de Biochimie, École Polytechnique, Route de Saclay, Palaiseau 91128 Cedex, France ͑Received 31 July 2008; accepted 11 September 2008; published online 11 November 2008͒ Following Kimura’s neutral theory of molecular evolution ͓M. Kimura, The Neutral Theory of Molecular Evolution ͑Cambridge University Press, Cambridge, 1983͒͑reprinted in 1986͔͒,ithas become common to assume that the vast majority of viable mutations of a gene confer little or no functional advantage. Yet, in silico models of protein evolution have shown that mutational robustness of sequences could be selected for, even in the context of neutral evolution.
    [Show full text]
  • Fractal Boundaries of Complex Networks
    OFFPRINT Fractal boundaries of complex networks Jia Shao, Sergey V. Buldyrev, Reuven Cohen, Maksim Kitsak, Shlomo Havlin and H. Eugene Stanley EPL, 84 (2008) 48004 Please visit the new website www.epljournal.org TAKE A LOOK AT THE NEW EPL Europhysics Letters (EPL) has a new online home at www.epljournal.org Take a look for the latest journal news and information on: • reading the latest articles, free! • receiving free e-mail alerts • submitting your work to EPL www.epljournal.org November 2008 EPL, 84 (2008) 48004 www.epljournal.org doi: 10.1209/0295-5075/84/48004 Fractal boundaries of complex networks Jia Shao1(a),SergeyV.Buldyrev1,2, Reuven Cohen3, Maksim Kitsak1, Shlomo Havlin4 and H. Eugene Stanley1 1 Center for Polymer Studies and Department of Physics, Boston University - Boston, MA 02215, USA 2 Department of Physics, Yeshiva University - 500 West 185th Street, New York, NY 10033, USA 3 Department of Mathematics, Bar-Ilan University - 52900 Ramat-Gan, Israel 4 Minerva Center and Department of Physics, Bar-Ilan University - 52900 Ramat-Gan, Israel received 12 May 2008; accepted in final form 16 October 2008 published online 21 November 2008 PACS 89.75.Hc – Networks and genealogical trees PACS 89.75.-k – Complex systems PACS 64.60.aq –Networks Abstract – We introduce the concept of the boundary of a complex network as the set of nodes at distance larger than the mean distance from a given node in the network. We study the statistical properties of the boundary nodes seen from a given node of complex networks. We find that for both Erd˝os-R´enyi and scale-free model networks, as well as for several real networks, the boundaries have fractal properties.
    [Show full text]
  • Giant Component in Random Multipartite Graphs with Given
    Giant Component in Random Multipartite Graphs with Given Degree Sequences David Gamarnik ∗ Sidhant Misra † January 23, 2014 Abstract We study the problem of the existence of a giant component in a random multipartite graph. We consider a random multipartite graph with p parts generated according to a d given degree sequence ni (n) which denotes the number of vertices in part i of the multi- partite graph with degree given by the vector d. We assume that the empirical distribution of the degree sequence converges to a limiting probability distribution. Under certain mild regularity assumptions, we characterize the conditions under which, with high probability, there exists a component of linear size. The characterization involves checking whether the Perron-Frobenius norm of the matrix of means of a certain associated edge-biased distribu- tion is greater than unity. We also specify the size of the giant component when it exists. We use the exploration process of Molloy and Reed to analyze the size of components in the random graph. The main challenges arise due to the multidimensionality of the ran- dom processes involved which prevents us from directly applying the techniques from the standard unipartite case. In this paper we use techniques from the theory of multidimen- sional Galton-Watson processes along with Lyapunov function technique to overcome the challenges. 1 Introduction The problem of the existence of a giant component in random graphs was first studied by arXiv:1306.0597v3 [math.PR] 22 Jan 2014 Erd¨os and R´enyi. In their classical paper [ER60], they considered a random graph model on n and m edges where each such possible graph is equally likely.
    [Show full text]
  • 0848736-Bachelorproject Peter Verleijsdonk
    Eindhoven University of Technology BACHELOR Site percolation on the hierarchical configuration model Verleijsdonk, Peter Award date: 2017 Link to publication Disclaimer This document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Student theses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the document as presented in the repository. The required complexity or quality of research of student theses may vary by program, and the required minimum study period may vary in duration. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain Department of Mathematics and Computer Science Site Percolation on the Hierarchical Configuration Model Bachelor Thesis P. Verleijsdonk Supervisors: prof. dr. R.W. van der Hofstad C. Stegehuis (MSc) Eindhoven, March 2017 Abstract This paper extends the research on percolation on the hierarchical configuration model. The hierarchical configuration model is a configuration model where single vertices are replaced by small community structures. We study site percolation on the hierarchical configuration model, as well as the critical percolation value, size of the giant component and distance distribution after percolation.
    [Show full text]
  • Cavity and Replica Methods for the Spectral Density of Sparse Symmetric Random Matrices
    SciPost Phys. Lect. Notes 33 (2021) Cavity and replica methods for the spectral density of sparse symmetric random matrices Vito A R Susca, Pierpaolo Vivo? and Reimer Kühn King’s College London, Department of Mathematics, Strand, London WC2R 2LS, United Kingdom ? [email protected] Abstract We review the problem of how to compute the spectral density of sparse symmetric ran- dom matrices, i.e. weighted adjacency matrices of undirected graphs. Starting from the Edwards-Jones formula, we illustrate the milestones of this line of research, including the pioneering work of Bray and Rodgers using replicas. We focus first on the cavity method, showing that it quickly provides the correct recursion equations both for single instances and at the ensemble level. We also describe an alternative replica solution that proves to be equivalent to the cavity method. Both the cavity and the replica derivations allow us to obtain the spectral density via the solution of an integral equation for an auxiliary probability density function. We show that this equation can be solved using a stochastic population dynamics algorithm, and we provide its implementation. In this formalism, the spectral density is naturally written in terms of a superposition of local contributions from nodes of given degree, whose role is thoroughly elucidated. This paper does not contain original material, but rather gives a pedagogical overview of the topic. It is indeed addressed to students and researchers who consider entering the field. Both the theoretical tools and the numerical algorithms are discussed in detail, highlighting conceptual subtleties and practical aspects. Copyright V.A.
    [Show full text]
  • Arxiv:1907.09957V1 [Physics.Soc-Ph] 23 Jul 2019 2
    Explosive phenomena in complex networks Raissa M. D'Souza,1, 2 Jesus G´omez-Garde~nes,3, 4 Jan Nagler,5 and Alex Arenas6 1Department of Computer Science, Department of Mechanical and Aerospace Engineering, Complexity Sciences Center, University of California, Davis 95616, USA 2Santa Fe Institute, 1399 Hyde Park Rd Santa Fe New Mexico 87501, USA 3Department of Condensed Physics, University of Zaragoza, Zaragoza 50009, Spain 4GOTHAM Lab, Institute for Biocomputation and Physics of Complex Systems (BIFI), University of Zaragoza, Zaragoza 50018, Spain 5Deep Dynamics Group & Centre for Human and Machine Intelligence, Frankfurt School of Finance & Management, Frankfurt, Germany 6Departament d'Enginyeria Inform`atica i Matem`atiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain (Dated: July 24, 2019) The emergence of large-scale connectivity and synchronization are crucial to the structure, func- tion and failure of many complex socio-technical networks. Thus, there is great interest in analyzing phase transitions to large-scale connectivity and to global synchronization, including how to enhance or delay the onset. These phenomena are traditionally studied as second-order phase transitions where, at the critical threshold, the order parameter increases rapidly but continuously. In 2009, an extremely abrupt transition was found for a network growth process where links compete for addition in attempt to delay percolation. This observation of \explosive percolation" was ulti- mately revealed to be a continuous transition in the thermodynamic limit, yet with very atypical finite-size scaling, and it started a surge of work on explosive phenomena and their consequences. Many related models are now shown to yield discontinuous percolation transitions and even hybrid transitions.
    [Show full text]
  • Network Robustness
    8 ALBERT-LÁSZLÓ BARABÁSI NETWORK SCIENCE NETWORK ROBUSTNESS ACKNOWLEDGEMENTS MÁRTON PÓSFAI SARAH MORRISON GABRIELE MUSELLA AMAL HUSSEINI NICOLE SAMAY PHILIPP HOEVEL ROBERTA SINATRA INDEX Introduction Introduction 1 Percolation Theory 2 Robustness of Scale-free Networks 3 Attack Tolerance 4 Cascading Failures 5 Modeling Cascading Failures 6 Building Robustness 7 Summary: Achilles' Heel 8 Homework 9 ADVANCED TOPICS 8.A Percolation in Scale-free Network 10 ADVANCED TOPICS 8.B Molloy-Reed Criteria 11 Figure 8.0 (cover image) Networks & Art: Facebook Users ADVANCED TOPICS 8.C 12 Created by Paul Butler, a Toronto-based data Critical Threshold Under Random Failures scientist during a Facebook internship in 2010, the image depicts the network connect- ADVANCED TOPICS 8.D ing the users of the social network company. It highlights the links within and across con- Breakdown of a Finite Scale-free Network 13 tinents. The presence of dense local links in the U.S., Europe and India is just as revealing ADVANCED TOPICS 8.E 14 as the lack of links in some areas, like China, Attack and Error Tolerance of Real Networks where the site is banned, and Africa, reflect- ing a lack of Internet access. ADVANCED TOPICS 8.F 15 Attack Threshold ADVANCED TOPICS 8.G 16 The Optimal Degree Distribution This book is licensed under a Homework Creative Commons: CC BY-NC-SA 2.0. PDF V26, 05.09.2014 SECTION 8.1 INTRODUCTION Errors and failures can corrupt all human designs: The failure of a com- ponent in your car’s engine may force you to call for a tow truck or a wiring error in your computer chip can make your computer useless.
    [Show full text]
  • ANATOMY of the GIANT COMPONENT: the STRICTLY SUPERCRITICAL REGIME 1. Introduction the Famous Phase Transition of the Erd˝Os
    ANATOMY OF THE GIANT COMPONENT: THE STRICTLY SUPERCRITICAL REGIME JIAN DING, EYAL LUBETZKY AND YUVAL PERES Abstract. In a recent work of the authors and Kim, we derived a com- plete description of the largest component of the Erd}os-R´enyi random graph G(n; p) as it emerges from the critical window, i.e. for p = (1+")=n where "3n ! 1 and " = o(1), in terms of a tractable contiguous model. Here we provide the analogous description for the supercritical giant component, i.e. the largest component of G(n; p) for p = λ/n where λ > 1 is fixed. The contiguous model is roughly as follows: Take a random degree sequence and sample a random multigraph with these degrees to arrive at the kernel; Replace the edges by paths whose lengths are i.i.d. geometric variables to arrive at the 2-core; Attach i.i.d. Poisson Galton-Watson trees to the vertices for the final giant component. As in the case of the emerging giant, we obtain this result via a sequence of contiguity arguments at the heart of which are Kim's Poisson-cloning method and the Pittel-Wormald local limit theorems. 1. Introduction The famous phase transition of the Erd}osand R´enyi random graph, in- troduced in 1959 [14], addresses the double jump in the size of the largest component C1 in G(n; p) for p = λ/n with λ > 0 fixed. When λ < 1 it is log- arithmic in size with high probability (w.h.p.), when λ = 1 its size has order n2=3 and when λ > 1 it is linear w.h.p.
    [Show full text]