Modeling Human Networks Using Random Graphs
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Optimal Subgraph Structures in Scale-Free Configuration
Submitted to the Annals of Applied Probability OPTIMAL SUBGRAPH STRUCTURES IN SCALE-FREE CONFIGURATION MODELS , By Remco van der Hofstad∗ §, Johan S. H. van , , Leeuwaarden∗ ¶ and Clara Stegehuis∗ k Eindhoven University of Technology§, Tilburg University¶ and Twente Universityk Subgraphs reveal information about the geometry and function- alities of complex networks. For scale-free networks with unbounded degree fluctuations, we obtain the asymptotics of the number of times a small connected graph occurs as a subgraph or as an induced sub- graph. We obtain these results by analyzing the configuration model with degree exponent τ (2, 3) and introducing a novel class of op- ∈ timization problems. For any given subgraph, the unique optimizer describes the degrees of the vertices that together span the subgraph. We find that subgraphs typically occur between vertices with specific degree ranges. In this way, we can count and characterize all sub- graphs. We refrain from double counting in the case of multi-edges, essentially counting the subgraphs in the erased configuration model. 1. Introduction Scale-free networks often have degree distributions that follow power laws with exponent τ (2, 3) [1, 11, 22, 34]. Many net- ∈ works have been reported to satisfy these conditions, including metabolic networks, the internet and social networks. Scale-free networks come with the presence of hubs, i.e., vertices of extremely high degrees. Another property of real-world scale-free networks is that the clustering coefficient (the probability that two uniformly chosen neighbors of a vertex are neighbors themselves) decreases with the vertex degree [4, 10, 24, 32, 34], again following a power law. -
Detecting Statistically Significant Communities
1 Detecting Statistically Significant Communities Zengyou He, Hao Liang, Zheng Chen, Can Zhao, Yan Liu Abstract—Community detection is a key data analysis problem across different fields. During the past decades, numerous algorithms have been proposed to address this issue. However, most work on community detection does not address the issue of statistical significance. Although some research efforts have been made towards mining statistically significant communities, deriving an analytical solution of p-value for one community under the configuration model is still a challenging mission that remains unsolved. The configuration model is a widely used random graph model in community detection, in which the degree of each node is preserved in the generated random networks. To partially fulfill this void, we present a tight upper bound on the p-value of a single community under the configuration model, which can be used for quantifying the statistical significance of each community analytically. Meanwhile, we present a local search method to detect statistically significant communities in an iterative manner. Experimental results demonstrate that our method is comparable with the competing methods on detecting statistically significant communities. Index Terms—Community Detection, Random Graphs, Configuration Model, Statistical Significance. F 1 INTRODUCTION ETWORKS are widely used for modeling the structure function that is able to always achieve the best performance N of complex systems in many fields, such as biology, in all possible scenarios. engineering, and social science. Within the networks, some However, most of these objective functions (metrics) do vertices with similar properties will form communities that not address the issue of statistical significance of commu- have more internal connections and less external links. -
Modern Discrete Probability I
Preliminaries Some fundamental models A few more useful facts about... Modern Discrete Probability I - Introduction Stochastic processes on graphs: models and questions Sebastien´ Roch UW–Madison Mathematics September 6, 2017 Sebastien´ Roch, UW–Madison Modern Discrete Probability – Models and Questions Preliminaries Review of graph theory Some fundamental models Review of Markov chain theory A few more useful facts about... 1 Preliminaries Review of graph theory Review of Markov chain theory 2 Some fundamental models Random walks on graphs Percolation Some random graph models Markov random fields Interacting particles on finite graphs 3 A few more useful facts about... ...graphs ...Markov chains ...other things Sebastien´ Roch, UW–Madison Modern Discrete Probability – Models and Questions Preliminaries Review of graph theory Some fundamental models Review of Markov chain theory A few more useful facts about... Graphs Definition (Undirected graph) An undirected graph (or graph for short) is a pair G = (V ; E) where V is the set of vertices (or nodes, sites) and E ⊆ ffu; vg : u; v 2 V g; is the set of edges (or bonds). The V is either finite or countably infinite. Edges of the form fug are called loops. We do not allow E to be a multiset. We occasionally write V (G) and E(G) for the vertices and edges of G. Sebastien´ Roch, UW–Madison Modern Discrete Probability – Models and Questions Preliminaries Review of graph theory Some fundamental models Review of Markov chain theory A few more useful facts about... An example: the Petersen graph Sebastien´ Roch, UW–Madison Modern Discrete Probability – Models and Questions Preliminaries Review of graph theory Some fundamental models Review of Markov chain theory A few more useful facts about.. -
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. -
Percolation Thresholds for Robust Network Connectivity
Percolation Thresholds for Robust Network Connectivity Arman Mohseni-Kabir1, Mihir Pant2, Don Towsley3, Saikat Guha4, Ananthram Swami5 1. Physics Department, University of Massachusetts Amherst, [email protected] 2. Massachusetts Institute of Technology 3. College of Information and Computer Sciences, University of Massachusetts Amherst 4. College of Optical Sciences, University of Arizona 5. Army Research Laboratory Abstract Communication networks, power grids, and transportation networks are all examples of networks whose performance depends on reliable connectivity of their underlying network components even in the presence of usual network dynamics due to mobility, node or edge failures, and varying traffic loads. Percolation theory quantifies the threshold value of a local control parameter such as a node occupation (resp., deletion) probability or an edge activation (resp., removal) probability above (resp., below) which there exists a giant connected component (GCC), a connected component comprising of a number of occupied nodes and active edges whose size is proportional to the size of the network itself. Any pair of occupied nodes in the GCC is connected via at least one path comprised of active edges and occupied nodes. The mere existence of the GCC itself does not guarantee that the long-range connectivity would be robust, e.g., to random link or node failures due to network dynamics. In this paper, we explore new percolation thresholds that guarantee not only spanning network connectivity, but also robustness. We define and analyze four measures of robust network connectivity, explore their interrelationships, and numerically evaluate the respective robust percolation thresholds arXiv:2006.14496v1 [physics.soc-ph] 25 Jun 2020 for the 2D square lattice. -
Markov Chain Monte Carlo Estimation of Exponential Random Graph Models
Markov Chain Monte Carlo Estimation of Exponential Random Graph Models Tom A.B. Snijders ICS, Department of Statistics and Measurement Theory University of Groningen ∗ April 19, 2002 ∗Author's address: Tom A.B. Snijders, Grote Kruisstraat 2/1, 9712 TS Groningen, The Netherlands, email <[email protected]>. I am grateful to Paul Snijders for programming the JAVA applet used in this article. In the revision of this article, I profited from discussions with Pip Pattison and Garry Robins, and from comments made by a referee. This paper is formatted in landscape to improve on-screen readability. It is read best by opening Acrobat Reader in a full screen window. Note that in Acrobat Reader, the entire screen can be used for viewing by pressing Ctrl-L; the usual screen is returned when pressing Esc; it is possible to zoom in or zoom out by pressing Ctrl-- or Ctrl-=, respectively. The sign in the upper right corners links to the page viewed previously. ( Tom A.B. Snijders 2 MCMC estimation for exponential random graphs ( Abstract bility to move from one region to another. In such situations, convergence to the target distribution is extremely slow. To This paper is about estimating the parameters of the exponential be useful, MCMC algorithms must be able to make transitions random graph model, also known as the p∗ model, using frequen- from a given graph to a very different graph. It is proposed to tist Markov chain Monte Carlo (MCMC) methods. The exponen- include transitions to the graph complement as updating steps tial random graph model is simulated using Gibbs or Metropolis- to improve the speed of convergence to the target distribution. -
Exploring Healing Strategies for Random Boolean Networks
Exploring Healing Strategies for Random Boolean Networks Christian Darabos 1, Alex Healing 2, Tim Johann 3, Amitabh Trehan 4, and Am´elieV´eron 5 1 Information Systems Department, University of Lausanne, Switzerland 2 Pervasive ICT Research Centre, British Telecommunications, UK 3 EML Research gGmbH, Heidelberg, Germany 4 Department of Computer Science, University of New Mexico, Albuquerque, USA 5 Division of Bioinformatics, Institute for Evolution and Biodiversity, The Westphalian Wilhelms University of Muenster, Germany. [email protected], [email protected], [email protected], [email protected], [email protected] Abstract. Real-world systems are often exposed to failures where those studied theoretically are not: neuron cells in the brain can die or fail, re- sources in a peer-to-peer network can break down or become corrupt, species can disappear from and environment, and so forth. In all cases, for the system to keep running as it did before the failure occurred, or to survive at all, some kind of healing strategy may need to be applied. As an example of such a system subjected to failure and subsequently heal, we study Random Boolean Networks. Deletion of a node in the network was considered a failure if it affected the functional output and we in- vestigated healing strategies that would allow the system to heal either fully or partially. More precisely, our main strategy involves allowing the nodes directly affected by the node deletion (failure) to iteratively rewire in order to achieve healing. We found that such a simple method was ef- fective when dealing with small networks and single-point failures. -
Percolation Threshold Results on Erdos-Rényi Graphs
arXiv: arXiv:0000.0000 Percolation Threshold Results on Erd}os-R´enyi Graphs: an Empirical Process Approach Michael J. Kane Abstract: Keywords and phrases: threshold, directed percolation, stochastic ap- proximation, empirical processes. 1. Introduction Random graphs and discrete random processes provide a general approach to discovering properties and characteristics of random graphs and randomized al- gorithms. The approach generally works by defining an algorithm on a random graph or a randomized algorithm. Then, expected changes for each step of the process are used to propose a limiting differential equation and a large deviation theorem is used to show that the process and the differential equation are close in some sense. In this way a connection is established between the resulting process's stochastic behavior and the dynamics of a deterministic, asymptotic approximation using a differential equation. This approach is generally referred to as stochastic approximation and provides a powerful tool for understanding the asymptotic behavior of a large class of processes defined on random graphs. However, little work has been done in the area of random graph research to investigate the weak limit behavior of these processes before the asymptotic be- havior overwhelms the random component of the process. This context is par- ticularly relevant to researchers studying news propagation in social networks, sensor networks, and epidemiological outbreaks. In each of these applications, investigators may deal graphs containing tens to hundreds of vertices and be interested not only in expected behavior over time but also error estimates. This paper investigates the connectivity of graphs, with emphasis on Erd}os- R´enyi graphs, near the percolation threshold when the number of vertices is not asymptotically large. -
Probability on Graphs Random Processes on Graphs and Lattices
Probability on Graphs Random Processes on Graphs and Lattices GEOFFREY GRIMMETT Statistical Laboratory University of Cambridge c G. R. Grimmett 1/4/10, 17/11/10, 5/7/12 Geoffrey Grimmett Statistical Laboratory Centre for Mathematical Sciences University of Cambridge Wilberforce Road Cambridge CB3 0WB United Kingdom 2000 MSC: (Primary) 60K35, 82B20, (Secondary) 05C80, 82B43, 82C22 With 44 Figures c G. R. Grimmett 1/4/10, 17/11/10, 5/7/12 Contents Preface ix 1 Random walks on graphs 1 1.1 RandomwalksandreversibleMarkovchains 1 1.2 Electrical networks 3 1.3 Flowsandenergy 8 1.4 Recurrenceandresistance 11 1.5 Polya's theorem 14 1.6 Graphtheory 16 1.7 Exercises 18 2 Uniform spanning tree 21 2.1 De®nition 21 2.2 Wilson's algorithm 23 2.3 Weak limits on lattices 28 2.4 Uniform forest 31 2.5 Schramm±LownerevolutionsÈ 32 2.6 Exercises 37 3 Percolation and self-avoiding walk 39 3.1 Percolationandphasetransition 39 3.2 Self-avoiding walks 42 3.3 Coupledpercolation 45 3.4 Orientedpercolation 45 3.5 Exercises 48 4 Association and in¯uence 50 4.1 Holley inequality 50 4.2 FKGinequality 53 4.3 BK inequality 54 4.4 Hoeffdinginequality 56 c G. R. Grimmett 1/4/10, 17/11/10, 5/7/12 vi Contents 4.5 In¯uenceforproductmeasures 58 4.6 Proofsofin¯uencetheorems 63 4.7 Russo'sformulaandsharpthresholds 75 4.8 Exercises 78 5 Further percolation 81 5.1 Subcritical phase 81 5.2 Supercritical phase 86 5.3 Uniquenessofthein®nitecluster 92 5.4 Phase transition 95 5.5 Openpathsinannuli 99 5.6 The critical probability in two dimensions 103 5.7 Cardy's formula 110 5.8 The -
3 a Few More Good Inequalities, Martingale Variety 1 3.1 From
3 A few more good inequalities, martingale variety1 3.1 From independence to martingales...............1 3.2 Hoeffding’s inequality for martingales.............4 3.3 Bennett's inequality for martingales..............8 3.4 *Concentration of random polynomials............. 10 3.5 *Proof of the Kim-Vu inequality................ 14 3.6 Problems............................. 18 3.7 Notes............................... 18 Printed: 8 October 2015 version: 8Oct2015 Mini-empirical printed: 8 October 2015 c David Pollard x3.1 From independence to martingales 1 Chapter 3 A few more good inequalities, martingale variety Section 3.1 introduces the method for bounding tail probabilities using mo- ment generating functions. Section 3.2 discusses the Hoeffding inequality, both for sums of independent bounded random variables and for martingales with bounded increments. Section 3.3 discusses the Bennett inequality, both for sums of indepen- dent random variables that are bounded above by a constant and their martingale analogs. *Section3.4 presents an extended application of a martingale version of the Bennett inequality to derive a version of the Kim-Vu inequality for poly- nomials in independent, bounded random variables. 3.1 From independence to martingales BasicMG::S:intro Throughout this Chapter f(Si; Fi): i = 1; : : : ; ng is a martingale on some probability space (Ω; F; P). That is, we have a sub-sigma fields F0 ⊆ F1 ⊆ · · · ⊆ Fn ⊆ F and integrable, Fi-measurable random variables Si for which PFi−1 Si = Si−1 almost surely. Equivalently, the martingale differ- ences ξi := Si − Si−1 are integrable, Fi-measurable, and PFi−1 ξi = 0 almost surely, for i = 1; : : : ; n. In that case Si = S0 + ξ1 + ··· + ξi = Si−1 + ξi. -
Random Boolean Networks As a Toy Model for the Brain
UNIVERSITY OF GENEVA SCIENCE FACULTY VRIJE UNIVERSITEIT OF AMSTERDAM PHYSICS SECTION Random Boolean Networks as a toy model for the brain MASTER THESIS presented at the science faculty of the University of Geneva for obtaining the Master in Theoretical Physics by Chlo´eB´eguin Supervisor (VU): Pr. Greg J Stephens Co-Supervisor (UNIGE): Pr. J´er^ome Kasparian July 2017 Contents Introduction1 1 Biology, physics and the brain4 1.1 Biological description of the brain................4 1.2 Criticality in the brain......................8 1.2.1 Physics reminder..................... 10 1.2.2 Experimental evidences.................. 15 2 Models of neural networks 20 2.1 Classes of models......................... 21 2.1.1 Spiking models...................... 21 2.1.2 Rate-based models.................... 23 2.1.3 Attractor networks.................... 24 2.1.4 Links between the classes of models........... 25 2.2 Random Boolean Networks.................... 28 2.2.1 General definition..................... 28 2.2.2 Kauffman network.................... 30 2.2.3 Hopfield network..................... 31 2.2.4 Towards a RBN for the brain.............. 32 2.2.5 The model......................... 33 3 Characterisation of RBNs 34 3.1 Attractors............................. 34 3.2 Damage spreading........................ 36 3.3 Canonical specific heat...................... 37 4 Results 40 4.1 One population with Gaussian weights............. 40 4.2 Dale's principle and balance of inhibition - excitation..... 46 4.3 Lognormal distribution of the weights.............. 51 4.4 Discussion............................. 55 i 5 Conclusion 58 Bibliography 60 Acknowledgements 66 A Python Code 67 A.1 Dynamics............................. 67 A.2 Attractor search.......................... 69 A.3 Hamming Distance........................ 73 A.4 Canonical specific heat..................... -
An Introduction to Exponential Random Graph (P*) Models for Social Networks
Social Networks 29 (2007) 173–191 An introduction to exponential random graph (p*) models for social networks Garry Robins ∗, Pip Pattison, Yuval Kalish, Dean Lusher Department of Psychology, School of Behavioural Science, University of Melbourne, Vic. 3010, Australia Abstract This article provides an introductory summary to the formulation and application of exponential random graph models for social networks. The possible ties among nodes of a network are regarded as random variables, and assumptions about dependencies among these random tie variables determine the general form of the exponential random graph model for the network. Examples of different dependence assumptions and their associated models are given, including Bernoulli, dyad-independent and Markov random graph models. The incorporation of actor attributes in social selection models is also reviewed. Newer, more complex dependence assumptions are briefly outlined. Estimation procedures are discussed, including new methods for Monte Carlo maximum likelihood estimation. We foreshadow the discussion taken up in other papers in this special edition: that the homogeneous Markov random graph models of Frank and Strauss [Frank, O., Strauss, D., 1986. Markov graphs. Journal of the American Statistical Association 81, 832–842] are not appropriate for many observed networks, whereas the new model specifications of Snijders et al. [Snijders, T.A.B., Pattison, P., Robins, G.L., Handock, M. New specifications for exponential random graph models. Sociological Methodology, in press] offer substantial improvement. © 2006 Elsevier B.V. All rights reserved. Keywords: Exponential random graph models; Statistical models for social networks; p* models In recent years, there has been growing interest in exponential random graph models for social networks, commonly called the p* class of models (Frank and Strauss, 1986; Pattison and Wasserman, 1999; Robins et al., 1999; Wasserman and Pattison, 1996).