DOCSLIB.ORG

On the Size of the Giant Component in Inhomogeneous Random K-Out

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On the Size of the Giant Component in Inhomogeneous Random K-Out 1 Existence and Size of the Giant Component in Inhomogeneous Random K-out Graphs Mansi Sood and Osman Yagan˘ Abstract—Random K-out graphs are receiving attention as nodes, denoted by H(n; K), is constructed as follows. Each a model to construct sparse yet well-connected topologies in of the n nodes draws K edges towards K distinct nodes distributed systems including sensor networks, federated learn- chosen uniformly at random from among all other nodes. ing, and cryptocurrency networks. In response to the growing heterogeneity in emerging real-world networks, where nodes The orientation of the edges is then ignored, yielding an differ in resources and requirements, inhomogeneous random undirected graph. Random K-out graphs achieve connectivity K-out graphs were proposed recently. In this model, first, each with probability approaching one in the limit of large n for all of the n nodes is classified as type-1 (respectively, type-2) with K ≥ 2 [5], [7], with the probability of connectivity growing 2 probability µ (respectively, 1−µ) independently from the others, as 1 − Θ(1=nK −1) for all K ≥ 2 [8]. Consequently, random where 0 < µ < 1. Next, each type-1 (respectively, type-2) node K-out graphs get connected very easily, i.e., with far fewer draws 1 arc towards a node (respectively, Kn arcs towards Kn distinct nodes) selected uniformly at random. The orientation of edges (O(n)) as compared to classical random graph models the arcs is ignored yielding the inhomogeneous random K-out including Erdos-R˝ enyi´ (ER) random graphs (O(n log n)). Ow- graph, denoted by H(n; µ, Kn). It was recently established that ing to the simplicity of graph construction and their unique H(n; µ, Kn) is connected with high probability (whp) if and only connectivity properties, random K-out graphs are receiving if K = !(1). Motivated by practical settings where establishing n increasing attention as a model to construct sparse, yet well- links is costly and only a bounded choice of Kn is feasible (Kn = O(1)), we study the size of the largest connected sub- connected topologies in a fully distributed fashion in ad- network of H(n; µ, Kn). We first show that the trivial condition hoc networks. Random K-out graphs have received renewed of Kn ≥ 2 for all n is sufficient to ensure that H(n; µ, Kn) interest for designing securely connected sensor networks contains a giant component of size n − O(1) whp. Next, to model [7], anonymity preserving cryptocurrency networks [9], and settings where nodes can fail or get compromised, we investigate fully decentralized learning networks with differential privacy the size of the largest connected sub-network in H(n; µ, Kn) when dn nodes are selected uniformly at random and removed from guarantees [10]. the network. We show that if dn = O(1), a giant component In the context of wireless sensor networks (WSNs), random of size n − O(1) persists for all Kn ≥ 2 whp. Further, when K-out graphs have been studied [7], [11]–[13] to model dn = o(n) nodes are removed from H(n; µ, Kn), the remaining the random pairwise key predistribution scheme [14]. Along nodes contain a giant component of size n(1 − o(1)) whp for all with the original key predistribution scheme proposed by Kn ≥ 2. We present numerical results to demonstrate the size of the largest connected component when the number of nodes Escheanuer and Gligor [15], the random pairwise scheme is finite. is one of the most widely recognized security protocols for Index Terms—Network analysis and control, random graphs, sensor networks. The random pairwise scheme is implemented connectivity, ad-hoc networks in two phases. In the first phase, each sensor node is paired offline with K distinct nodes chosen uniformly at random among all other sensor nodes. Next, a unique pairwise key I. INTRODUCTION is inserted in the memory of each of the paired sensors. After A. Background deployment, two sensor nodes can communicate securely only arXiv:2009.01610v2 [cs.IT] 29 Jul 2021 The field of random graphs lays the mathematical foun- if they have at least one key in common. The random pairwise dations for modeling and analyzing the complex patterns of key predistribution scheme induces a random K-out graph on interconnections typical to real-world networks including com- the set of the participating sensor nodes [7], and therefore the munication networks [2], social networks [3], and biological connectivity properties of random K-out graphs are of interest networks [4]. A class of random graphs called the random in key management schemes for establishing secure WSNs. K-out graph is one of the earliest known models in the In recent years, the rapid proliferation of affordable sensing literature [5], [6]. The random K-out graph comprising n and mobile devices has led to advances in applications such as remote-sensing, environmental surveillance [16], and decen- This work has been supported in part by the National Science Foundation through grant CCF #1617934, David H. Barakat and LaVerne Owen-Barakat tralized learning [17]. Such emerging applications increasingly Carnegie Institute of Technology Dean’s Fellowship, Pennsylvania Infras- rely on integrating heterogeneous agents that have different tructure Technology Alliance (PITA), CyLab@IoT, and Cylab Presidential capabilities and (security and connectivity) requirements. Con- Fellowship at Carnegie Mellon University. M. Sood and O. Yagan˘ are with Department of Electrical and sequently, the analysis of heterogeneous variants of classical Computer Engineering and CyLab, Carnegie Mellon University, random graph models has emerged as an active area of Pittsburgh, PA, 15213 USA. Email: [email protected], research. [18]–[23], [24]–[28]. A heterogeneous variant of [email protected] A preliminary version of this work was presented at the 59th IEEE random K-out graph known as the inhomogeneous random Conference of Decision and Control (CDC) 2020 [1]. K-out graph was proposed recently to model heterogeneous 2 B C B C instance, if the power available for transmission is limited, picks C picks B, D, F the underlying physical network may not be dense enough to guarantee global connectivity with key predistribution schemes A A [30]. In such resource-constrained environments where it is F picks B F picks B, E, F infeasible to establish a connected network, the goal is to establish a large connected sub-network spanning almost the E E picks B, D, F picks F D D entire network as efficiently as possible, i.e., by using the least amount of resources (links). [31]. For example, if a sensor Fig. 1. An inhomogeneous random K-out graph with 6 nodes. Nodes A; C and E are type-2 and the rest (B; D; F ) are type-1. Each type-1 node selects 1 node network is designed to monitor the temperature of a field, it uniformly at random and each type-2 node selects Kn = 3 nodes uniformly at may suffice to aggregate readings from a majority of sensors in random. An undirected edge is drawn between two nodes if at least one selects the field [32]. Formally, the notion of connected sub-networks the other. is described by connected components; see Definition 3.1 for a precise characterization. With this in mind, the first networks secured by random pairwise key predistribution question which we address is when Kn is bounded (i.e., schemes [19]. Kn = O(1)), how many nodes are contained in the largest The inhomogeneous random K-out graph is constructed on connected component of H(n; µ, Kn)? In the literature on a set of n nodes as follows. First, each node is independently random graphs, this is often studied in terms of the existence classified as type-1 with probability µ and type-2 with proba- and size of the giant component, defined as a connected sub- bility 1−µ, where 0 < µ < 1. The type of the node determines network comprising Ω(n) nodes; see [33] for a classical study the number of selections made by it, viz., each type-1 node on the size of the giant component of Erdos-R˝ enyi´ graphs. selects one node uniformly at random from all other nodes and Not only is it important to design networks that bear large each type-2 node selects Kn ≥ 2 distinct nodes uniformly connected components, but it is also crucial to ensure these at random from all other nodes; see Figure 1. The notation large components are robust to node failures. Sensor networks Kn indicates that the number of selections made by type-2 are often deployed in hostile environments e.g., battlefield nodes scales as a function of the number of nodes (n). Since monitoring and environmental surveillance, which makes the homogeneous random K-out graphs, where all nodes make the nodes susceptible to operational failures and adversarial cap- same number of selections, achieve connectivity whp for any tures. Therefore, it is of interest to analyze the properties of K ≥ 2 [5], [7], the interesting case arises when we allow networks induced by random K-out graphs when some nodes type-1 nodes to select just 1 edge; see Section II. Further, are dishonest, have failed, or have been captured from the the analysis in this paper also generalizes to inhomogeneous network. From a modeling perspective, this translates to ana- random K-out graphs with an arbitrary number of node types; lyzing the connectivity properties of inhomogeneous random see Appendix. Throughout our discussion, we let H(n; µ, Kn) K-out graphs when a subset of nodes are selected uniformly denote the inhomogeneous random K-out graph on n nodes at random and deleted.
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]