Adventures in random graphs: Models, structures and algorithms
|
|
- Basil Pitts
- 5 years ago
- Views:
Transcription
1 BCAM January Adventures in random graphs: Models, structures and algorithms Armand M. Makowski ECE & ISR/HyNet University of Maryland at College Park armand@isr.umd.edu
2 BCAM January Complex networks Many examples Biology (Genomics, protonomics) Transportation (Communication networks, Internet, roads and railroads) Information systems (World Wide Web) Social networks (Facebook, LinkedIn, etc) Sociology (Friendship networks, sexual contacts) Bibliometrics (Co-authorship networks, references) Ecology (food webs) Energy (Electricity distribution, smart grids) Larger context of Network Science
3 BCAM January Objectives Identify generic structures and properties of networks Mathematical models and their analysis Understand how network structure and processes on networks interact What is new? Very large data sets now easily available! Dynamics of networks vs. dynamics on networks
4 BCAM January Bibliography (I): Random graphs N. Alon and J.H. Spencer, The Probabilistic Method (Second Edition), Wiley-Science Series in Discrete Mathematics and Optimization, John Wiley & Sons, New York (NY) A. D. Barbour, L. Holst and S. Janson, Poisson Approximation, Oxford Studies in Probability 2, Oxford University Press, Oxford (UK), B. Bollobás, Random Graphs, Second Edition, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge (UK), R. Durrett, Random Graph Dynamics, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge (UK), 2007.
5 BCAM January M. Draief and L. Massoulié, Epidemics and Rumours in Complex Networks, Cambridge University Press, Cambridge (UK), D. Dubhashi and A. Panconesi, Concentration of Measure for the Analysis of Randomized algorithms, Cambridge University Press, New York (NY), M. Franceschetti and R. Meester, Random Networks for Communication: From Statistical Physics to Information Systems, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge (UK), S. Janson, T. Luczak and A. Ruciński, Random Graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, R. Meester and R. Roy, Continuum Percolation, Cambridge University Press, Cambridge (UK), 1996.
6 BCAM January M.D. Penrose, Random Geometric Graphs, Oxford Studies in Probability 5, Oxford University Press, New York (NY), 2003.
7 BCAM January Bibliography (II): Survey papers R. Albert and A.-L. Barabási, Statistical mechanics of complex networks, Review of Modern Physics 74 (2002), pp M.E.J. Newman, The structure and function of complex networks, SIAM Review 45 (2003), pp
8 BCAM January Bibliography (III): Complex networks A. Barrat, M. Barthelemy and A. Vespignani, Dynamical Processes on Complex Networks, Cambridge University Press, Cambridge (UK), R. Cohen and S. Havlin, Complex Networks - Structure, Robustness and Function, Cambridge University Press, Cambridge (UK), D. Easley and J. Kleinberg, Networks, Crowds, and Markets: Reasoning About a Highly Connected World, Cambridge University Press, Cambridge (UK) (2010). M.O. Jackson, Social and Economic Networks, Princeton University Press, Princeton (NJ), 2008.
9 BCAM January M.E.J. Newman, A.-L. Barabási and D.J. Watts (Editors), The Structure and Dynamics of Networks, Princeton University Press, Princeton (NJ), 2006.
10 BCAM January LECTURE 1
11 BCAM January Basics of graph theory
12 BCAM January What are graphs? With V a finite set, a graph G is an ordered pair (V, E) where elements in V are called vertices/nodes and E is the set of edges/links: E V V E(G) = E Nodes i and j are said to be adjacent, written i j, if e = (i, j) E, i, j V
13 BCAM January Multiple representations for G = (V, E) Set-theoretic Edge variables {ξ ij, i, j V } with 1 if (i, j) E ξ ij = 0 if (i, j) / E Algebraic Adjacency matrix A = (A ij ) with 1 if (i, j) E A ij = 0 if (i, j) / E
14 BCAM January Some terminology Simple graphs vs. multigraphs Directed vs. undirected (i, j) E if and only if (j, i) E No self loops (i, i) / E, i V Here: Simple, undirected graphs with no self loops!
15 BCAM January Types of graphs The empty graph Complete graphs Trees/forests A subgraph H = (W, F ) of G = (V, E) is a graph with vertex set W such that Cliques (Complete subgraphs) W V and F = E (W W )
16 BCAM January Labeled vs. unlabeled graphs A graph automorphism of G = (V, E) is any one-to-one mapping σ : V V that preserves the graph structure, namely (σ(i), σ(j)) E if and only if (i, j) E Group Aut(G) of graph automorphisms of G
17 BCAM January Of interest Connectivity and k-connectivity (with k 1) Number and size of components Isolated nodes Degree of a node: degree distribution/average degree, maximal/minimal degree Distance between nodes (in terms of number of hops): Shortest path, diameter, eccentricity, radius Small graph containment (e.g., triangles, trees, cliques, etc.) Clustering Centrality: Degree, closeness, in-betweenness
18 BCAM January For i, j in V, l ij = Shortest path length between nodes i and j in the graph G = (V, E) Convention: l ij = if nodes i and j belong to different components and l ii = 0. Average distance l Avg = 1 V ( V 1) i V j V l ij Diameter d(g) = max (l ij, i, j V )
19 BCAM January Eccentricity Ec(i) = max (l ij, j V ), i V Radius rad(g) = min (Ec(i), i V ) l Avg d(g) and rad(g) d(g) 2 rad(g)
20 BCAM January Centrality Q: How central is a node? Closeness centrality g(i) = 1 j V l, i V ij Betweenness centrality b(i) = k i, k j, σ kj (i) σ kj, i V
21 BCAM January Clustering Clustering coefficient of node i j i, k i, j k C(i) = ξ ijξ ik ξ kj j i, k i, j k ξ ijξ ik Average clustering coefficient C Avg = 1 n C(i) i V C = 3 Number of fully connected triples Number of triples
22 BCAM January Random graphs
23 BCAM January Random graphs? G(V ) Collection of all (simple free of self-loops undirected) graphs with vertex set V. Definition Given a probability triple (Ω, F, P), a random graph is simply a graph-valued rv G : Ω G(V ). Modeling We need only specify the pmf {P [G = G], G G(V )}. Many, many ways to do that!
24 BCAM January Equivalent representations for G Set-theoretic Link assignment rvs {ξ ij, i, j V } with 1 if (i, j) E(G) ξ ij = 0 if (i, j) / E(G) Algebraic Random adjacency matrix A = (A ij ) with 1 if (i, j) E(G) A ij = 0 if (i, j) / E(G)
25 BCAM January Why random graphs? Useful models in many applications to capture binary relationships between participating entities Because V ( V 1) G(V ) = 2 2 V A very large number! there is a need to identify/discover typicality! Scaling laws Zero-one laws as V becomes large, e.g., V V n = {1,..., n} (n )
26 BCAM January Ménagerie of random graphs Erdős-Renyi graphs G(n; m) Erdős-Renyi graphs G(n; p) Generalized Erdős-Renyi graphs Geometric random models/disk models Intrinsic fitness and threshold random models Random intersection graphs Growth models: Preferential attachment, copying Small worlds Exponential random graphs Etc
27 BCAM January Erdős-Renyi graphs G(n; m) With 1 m ( ) n 2 = n(n 1), 2 the pmf on G(V n ) is specified by u(n; m) 1 if E(G) = 2m P [G(n; m) = G] = 0 if E(G) 2m where u(n; m) = ( n(n 1) ) 2 m
28 BCAM January Uniform selection over the collection of all graphs on the vertex set {1,..., n} with exactly m edges
29 BCAM January Erdős-Renyi graphs G(n; p) With 0 p 1, the link assignment rvs {χ ij (p), 1 i < j n} are i.i.d. {0, 1}-valued rvs with P [χ ij (p) = 1] = 1 P [χ ij (p) = 0] = p, 1 i < j n For every G in G(V ), P [G(n; p) = G] = p EG 2 (1 p) n(n 1) 2 EG 2
30 BCAM January Related to, but easier to implement than G(n; m) Similar behavior/results under the matching condition E(G(n; m)) = E [ E(G(n; p)) ], namely m = n(n 1) p 2
31 BCAM January Generalized Erdős-Renyi graphs With 0 p ij 1, 1 i < j n the link assignment rvs {χ ij (p), 1 i < j n} are mutually independent {0, 1}-valued rvs with P [χ ij (p ij ) = 1] = 1 P [χ ij (p ij ) = 0] = p ij, 1 i < j n An important case: With positive weights w 1,..., w n, p ij = w iw j W with W = w w n
32 BCAM January Geometric random graphs (d 1) With random locations in R d at X 1,..., X n, the link assignment rvs {χ ij (ρ), 1 i < j n} are given by χ ij (ρ) = 1 [ X i X j ρ ], 1 i < j n where ρ > 0.
33 BCAM January Usually, the rvs X 1,..., X n are taken to be i.i.d. rvs uniformly distributed over some compact subset Γ R d Even then, not so obvious to write P [G(n; ρ) = G], G G(V ). since the rvs {χ ij (ρ), 1 i < j n} are no more i.i.d. rvs For d = 2, long history for modeling wireless networks (known as the disk model) where ρ interpretated as transmission range
34 BCAM January Threshold random graphs Given R + -valued i.i.d. rvs W 1,..., W n with absolutely continuous probability distribution function F, for some θ > 0 i j if and only if W i + W j > θ Generalizations: i j if and only if R(W i, W j ) > θ for some symmetric mapping R : R 2 + R +.
35 BCAM January Random intersection graphs Given a finite set W {1,..., W } of features, with random subsets K 1,..., K n of W, i j if and only if K i K j Co-authorship networks, random key distribution schemes, classification/clustering
36 BCAM January Growth models with rules {G t, t = 0, 1,...} and V t+1 V t G t+1 (G t, V t+1 ) Preferential attachment Copying Scale-free networks
37 BCAM January Small worlds Between randomness and order Shortcuts Short paths but high clustering Milgram s experiment and six degrees of separation
38 BCAM January Exponential random graphs Models favored by sociologists and statisticians Graph analog of exponential families often used in statistical modeling Related to Markov random fields
39 BCAM January With I parameters θ = (θ 1,..., θ I ) and a set of I observables (statistics) we postulate with normalization constant u i : G(V ) R +, P [G = G] = ep I i=1 θ iu i (G), G G(V ) Z(θ) Z(θ) = G G(V ) e P I i=1 θ iu i (G)
40 BCAM January In sum Many different ways to specify the pmf on G(V ) Local description vs. global representation Static vs. dynamic Application-dependent mechanisms
Adventures in random graphs: Models, structures and algorithms
BCAM January 2011 1 Adventures in random graphs: Models, structures and algorithms Armand M. Makowski ECE & ISR/HyNet University of Maryland at College Park armand@isr.umd.edu BCAM January 2011 2 LECTURE
More informationNetwork models: random graphs
Network models: random graphs Leonid E. Zhukov School of Data Analysis and Artificial Intelligence Department of Computer Science National Research University Higher School of Economics Structural Analysis
More informationA zero-one law for the existence of triangles in random key graphs
THE INSTITUTE FOR SYSTEMS RESEARCH ISR TECHNICAL REPORT 2011-11 A zero-one law for the existence of triangles in random key graphs Osman Yagan and Armand M. Makowski ISR develops, applies and teaches advanced
More information6.207/14.15: Networks Lecture 7: Search on Networks: Navigation and Web Search
6.207/14.15: Networks Lecture 7: Search on Networks: Navigation and Web Search Daron Acemoglu and Asu Ozdaglar MIT September 30, 2009 1 Networks: Lecture 7 Outline Navigation (or decentralized search)
More informationDecision Making and Social Networks
Decision Making and Social Networks Lecture 4: Models of Network Growth Umberto Grandi Summer 2013 Overview In the previous lecture: We got acquainted with graphs and networks We saw lots of definitions:
More informationData Mining and Analysis: Fundamental Concepts and Algorithms
Data Mining and Analysis: Fundamental Concepts and Algorithms dataminingbook.info Mohammed J. Zaki 1 Wagner Meira Jr. 2 1 Department of Computer Science Rensselaer Polytechnic Institute, Troy, NY, USA
More informationNetworks as a tool for Complex systems
Complex Networs Networ is a structure of N nodes and 2M lins (or M edges) Called also graph in Mathematics Many examples of networs Internet: nodes represent computers lins the connecting cables Social
More informationDesigning securely connected wireless sensor networks in the presence of unreliable links
Designing securely connected wireless sensor networks in the presence of unreliable links Osman Yağan and Armand M. Makowski Department of Electrical and Computer Engineering and the Institute for Systems
More information6.207/14.15: Networks Lecture 12: Generalized Random Graphs
6.207/14.15: Networks Lecture 12: Generalized Random Graphs 1 Outline Small-world model Growing random networks Power-law degree distributions: Rich-Get-Richer effects Models: Uniform attachment model
More informationSharp threshold functions for random intersection graphs via a coupling method.
Sharp threshold functions for random intersection graphs via a coupling method. Katarzyna Rybarczyk Faculty of Mathematics and Computer Science, Adam Mickiewicz University, 60 769 Poznań, Poland kryba@amu.edu.pl
More informationChaos, Complexity, and Inference (36-462)
Chaos, Complexity, and Inference (36-462) Lecture 21 Cosma Shalizi 3 April 2008 Models of Networks, with Origin Myths Erdős-Rényi Encore Erdős-Rényi with Node Types Watts-Strogatz Small World Graphs Exponential-Family
More information1 Mechanistic and generative models of network structure
1 Mechanistic and generative models of network structure There are many models of network structure, and these largely can be divided into two classes: mechanistic models and generative or probabilistic
More informationChaos, Complexity, and Inference (36-462)
Chaos, Complexity, and Inference (36-462) Lecture 21: More Networks: Models and Origin Myths Cosma Shalizi 31 March 2009 New Assignment: Implement Butterfly Mode in R Real Agenda: Models of Networks, with
More informationRandom Geometric Graphs
Random Geometric Graphs Mathew D. Penrose University of Bath, UK Networks: stochastic models for populations and epidemics ICMS, Edinburgh September 2011 1 MODELS of RANDOM GRAPHS Erdos-Renyi G(n, p):
More information1 Complex Networks - A Brief Overview
Power-law Degree Distributions 1 Complex Networks - A Brief Overview Complex networks occur in many social, technological and scientific settings. Examples of complex networks include World Wide Web, Internet,
More informationSpectral Graph Theory Tools. Analysis of Complex Networks
Spectral Graph Theory Tools for the Department of Mathematics and Computer Science Emory University Atlanta, GA 30322, USA Acknowledgments Christine Klymko (Emory) Ernesto Estrada (Strathclyde, UK) Support:
More informationEfficient routing in Poisson small-world networks
Efficient routing in Poisson small-world networks M. Draief and A. Ganesh Abstract In recent work, Jon Kleinberg considered a small-world network model consisting of a d-dimensional lattice augmented with
More informationShlomo Havlin } Anomalous Transport in Scale-free Networks, López, et al,prl (2005) Bar-Ilan University. Reuven Cohen Tomer Kalisky Shay Carmi
Anomalous Transport in Complex Networs Reuven Cohen Tomer Kalisy Shay Carmi Edoardo Lopez Gene Stanley Shlomo Havlin } } Bar-Ilan University Boston University Anomalous Transport in Scale-free Networs,
More informationHyperbolic metric spaces and their applications to large complex real world networks II
Hyperbolic metric spaces Hyperbolic metric spaces and their applications to large complex real world networks II Gabriel H. Tucci Bell Laboratories Alcatel-Lucent May 17, 2010 Tucci Hyperbolic metric spaces
More informationSparse Linear Algebra Issues Arising in the Analysis of Complex Networks
Sparse Linear Algebra Issues Arising in the Analysis of Complex Networks Department of Mathematics and Computer Science Emory University Atlanta, GA 30322, USA Acknowledgments Christine Klymko (Emory)
More informationSelf Similar (Scale Free, Power Law) Networks (I)
Self Similar (Scale Free, Power Law) Networks (I) E6083: lecture 4 Prof. Predrag R. Jelenković Dept. of Electrical Engineering Columbia University, NY 10027, USA {predrag}@ee.columbia.edu February 7, 2007
More informationMini course on Complex Networks
Mini course on Complex Networks Massimo Ostilli 1 1 UFSC, Florianopolis, Brazil September 2017 Dep. de Fisica Organization of The Mini Course Day 1: Basic Topology of Equilibrium Networks Day 2: Percolation
More informationSocial Networks- Stanley Milgram (1967)
Complex Networs Networ is a structure of N nodes and 2M lins (or M edges) Called also graph in Mathematics Many examples of networs Internet: nodes represent computers lins the connecting cables Social
More informationA LINE GRAPH as a model of a social network
A LINE GRAPH as a model of a social networ Małgorzata Krawczy, Lev Muchni, Anna Mańa-Krasoń, Krzysztof Kułaowsi AGH Kraów Stern School of Business of NY University outline - ideas, definitions, milestones
More informationThe Spreading of Epidemics in Complex Networks
The Spreading of Epidemics in Complex Networks Xiangyu Song PHY 563 Term Paper, Department of Physics, UIUC May 8, 2017 Abstract The spreading of epidemics in complex networks has been extensively studied
More informationThe expansion of random regular graphs
The expansion of random regular graphs David Ellis Introduction Our aim is now to show that for any d 3, almost all d-regular graphs on {1, 2,..., n} have edge-expansion ratio at least c d d (if nd is
More informationOn the Quality of Wireless Network Connectivity
Globecom 2012 - Ad Hoc and Sensor Networking Symposium On the Quality of Wireless Network Connectivity Soura Dasgupta Department of Electrical and Computer Engineering The University of Iowa Guoqiang Mao
More informationErdős-Rényi random graph
Erdős-Rényi random graph introduction to network analysis (ina) Lovro Šubelj University of Ljubljana spring 2016/17 graph models graph model is ensemble of random graphs algorithm for random graphs of
More informationRANDOM GRAPHS. Joel Spencer ICTP 12
RANDOM GRAPHS Joel Spencer ICTP 12 Graph Theory Preliminaries A graph G, formally speaking, is a pair (V (G), E(G)) where the elements v V (G) are called vertices and the elements of E(G), called edges,
More informationRandom Geometric Graphs
Random Geometric Graphs Mathew D. Penrose University of Bath, UK SAMBa Symposium (Talk 2) University of Bath November 2014 1 MODELS of RANDOM GRAPHS Erdos-Renyi G(n, p): start with the complete n-graph,
More informationarxiv: v1 [math.st] 1 Nov 2017
ASSESSING THE RELIABILITY POLYNOMIAL BASED ON PERCOLATION THEORY arxiv:1711.00303v1 [math.st] 1 Nov 2017 SAJADI, FARKHONDEH A. Department of Statistics, University of Isfahan, Isfahan 81744,Iran; f.sajadi@sci.ui.ac.ir
More informationAlmost giant clusters for percolation on large trees
for percolation on large trees Institut für Mathematik Universität Zürich Erdős-Rényi random graph model in supercritical regime G n = complete graph with n vertices Bond percolation with parameter p(n)
More informationA New Random Graph Model with Self-Optimizing Nodes: Connectivity and Diameter
A New Random Graph Model with Self-Optimizing Nodes: Connectivity and Diameter Richard J. La and Maya Kabkab Abstract We introduce a new random graph model. In our model, n, n 2, vertices choose a subset
More informationNote on the structure of Kruskal s Algorithm
Note on the structure of Kruskal s Algorithm Nicolas Broutin Luc Devroye Erin McLeish November 15, 2007 Abstract We study the merging process when Kruskal s algorithm is run with random graphs as inputs.
More informationStatistical Inference for Networks. Peter Bickel
Statistical Inference for Networks 4th Lehmann Symposium, Rice University, May 2011 Peter Bickel Statistics Dept. UC Berkeley (Joint work with Aiyou Chen, Google, E. Levina, U. Mich, S. Bhattacharyya,
More informationA prolific construction of strongly regular graphs with the n-e.c. property
A prolific construction of strongly regular graphs with the n-e.c. property Peter J. Cameron and Dudley Stark School of Mathematical Sciences Queen Mary, University of London London E1 4NS, U.K. Abstract
More informationParameterizing Exponential Family Models for Random Graphs: Current Methods and New Directions
Carter T. Butts p. 1/2 Parameterizing Exponential Family Models for Random Graphs: Current Methods and New Directions Carter T. Butts Department of Sociology and Institute for Mathematical Behavioral Sciences
More informationComplex networks: an introduction
Alain Barrat Complex networks: an introduction CPT, Marseille, France ISI, Turin, Italy http://www.cpt.univ-mrs.fr/~barrat http://cxnets.googlepages.com Plan of the lecture I. INTRODUCTION II. I. Networks:
More informationPermutations and Combinations
Permutations and Combinations Permutations Definition: Let S be a set with n elements A permutation of S is an ordered list (arrangement) of its elements For r = 1,..., n an r-permutation of S is an ordered
More informationConnectivity of inhomogeneous random key graphs intersecting inhomogeneous Erdős-Rényi graphs
Connectivity of inhomogeneous random key graphs intersecting inhomogeneous Erdős-Rényi graphs Rashad Eletreby and Osman Yağan Department of Electrical and Computer Engineering and CyLab, Carnegie Mellon
More informationLecture 1: Graphs, Adjacency Matrices, Graph Laplacian
Lecture 1: Graphs, Adjacency Matrices, Graph Laplacian Radu Balan January 31, 2017 G = (V, E) An undirected graph G is given by two pieces of information: a set of vertices V and a set of edges E, G =
More informationThe Beginning of Graph Theory. Theory and Applications of Complex Networks. Eulerian paths. Graph Theory. Class Three. College of the Atlantic
Theory and Applications of Complex Networs 1 Theory and Applications of Complex Networs 2 Theory and Applications of Complex Networs Class Three The Beginning of Graph Theory Leonhard Euler wonders, can
More informationEFFICIENT ROUTEING IN POISSON SMALL-WORLD NETWORKS
J. Appl. Prob. 43, 678 686 (2006) Printed in Israel Applied Probability Trust 2006 EFFICIENT ROUTEING IN POISSON SMALL-WORLD NETWORKS M. DRAIEF, University of Cambridge A. GANESH, Microsoft Research Abstract
More informationRandom graphs: Random geometric graphs
Random graphs: Random geometric graphs Mathew Penrose (University of Bath) Oberwolfach workshop Stochastic analysis for Poisson point processes February 2013 athew Penrose (Bath), Oberwolfach February
More informationEpidemics and information spreading
Epidemics and information spreading Leonid E. Zhukov School of Data Analysis and Artificial Intelligence Department of Computer Science National Research University Higher School of Economics Social Network
More informationNetwork Analysis and Modeling
lecture 0: what are networks and how do we talk about them? 2017 Aaron Clauset 003 052 002 001 Aaron Clauset @aaronclauset Assistant Professor of Computer Science University of Colorado Boulder External
More informationLecture 1 and 2: Introduction and Graph theory basics. Spring EE 194, Networked estimation and control (Prof. Khan) January 23, 2012
Lecture 1 and 2: Introduction and Graph theory basics Spring 2012 - EE 194, Networked estimation and control (Prof. Khan) January 23, 2012 Spring 2012: EE-194-02 Networked estimation and control Schedule
More informationECS 289 F / MAE 298, Lecture 15 May 20, Diffusion, Cascades and Influence
ECS 289 F / MAE 298, Lecture 15 May 20, 2014 Diffusion, Cascades and Influence Diffusion and cascades in networks (Nodes in one of two states) Viruses (human and computer) contact processes epidemic thresholds
More informationGroups of vertices and Core-periphery structure. By: Ralucca Gera, Applied math department, Naval Postgraduate School Monterey, CA, USA
Groups of vertices and Core-periphery structure By: Ralucca Gera, Applied math department, Naval Postgraduate School Monterey, CA, USA Mostly observed real networks have: Why? Heavy tail (powerlaw most
More informationRandom Graphs. 7.1 Introduction
7 Random Graphs 7.1 Introduction The theory of random graphs began in the late 1950s with the seminal paper by Erdös and Rényi [?]. In contrast to percolation theory, which emerged from efforts to model
More informationComparative analysis of transport communication networks and q-type statistics
Comparative analysis of transport communication networs and -type statistics B. R. Gadjiev and T. B. Progulova International University for Nature, Society and Man, 9 Universitetsaya Street, 498 Dubna,
More informationOn the Average Pairwise Connectivity of Wireless Multihop Networks
On the Average Pairwise Connectivity of Wireless Multihop Networks Fangting Sun and Mark Shayman Department of Electrical and Computer Engineering University of Maryland, College Park, MD 2742 {ftsun,
More informationEfficient Network Structures with Separable Heterogeneous Connection Costs
Efficient Network Structures with Separable Heterogeneous Connection Costs Babak Heydari a, Mohsen Mosleh a, Kia Dalili b arxiv:1504.06634v3 [q-fin.ec] 11 Dec 2015 a School of Systems and Enterprises,
More informationarxiv:cond-mat/ v1 28 Feb 2005
How to calculate the main characteristics of random uncorrelated networks Agata Fronczak, Piotr Fronczak and Janusz A. Hołyst arxiv:cond-mat/0502663 v1 28 Feb 2005 Faculty of Physics and Center of Excellence
More informationNumerical evaluation of the upper critical dimension of percolation in scale-free networks
umerical evaluation of the upper critical dimension of percolation in scale-free networks Zhenhua Wu, 1 Cecilia Lagorio, 2 Lidia A. Braunstein, 1,2 Reuven Cohen, 3 Shlomo Havlin, 3 and H. Eugene Stanley
More informationMóstoles, Spain. Keywords: complex networks, dual graph, line graph, line digraph.
Int. J. Complex Systems in Science vol. 1(2) (2011), pp. 100 106 Line graphs for directed and undirected networks: An structural and analytical comparison Regino Criado 1, Julio Flores 1, Alejandro García
More informationA continuous time model for network evolution. Gail Gilboa-Freedman, Refael Hassin. January 2, 2011
A continuous time model for network evolution Gail Gilboa-Freedman, Refael Hassin January 2, 2011 Abstract The study of networks is in great focus of many branches of science. We suggest a novel approach
More informationA = A U. U [n] P(A U ). n 1. 2 k(n k). k. k=1
Lecture I jacques@ucsd.edu Notation: Throughout, P denotes probability and E denotes expectation. Denote (X) (r) = X(X 1)... (X r + 1) and let G n,p denote the Erdős-Rényi model of random graphs. 10 Random
More informationNetworks: Lectures 9 & 10 Random graphs
Networks: Lectures 9 & 10 Random graphs Heather A Harrington Mathematical Institute University of Oxford HT 2017 What you re in for Week 1: Introduction and basic concepts Week 2: Small worlds Week 3:
More informationTHE DECK RATIO AND SELF-REPAIRING GRAPHS
THE DECK RATIO AND SELF-REPAIRING GRAPHS YULIA BUGAEV AND FELIX GOLDBERG Abstract. In this paper we define, for a graph invariant ψ, the ψ(g) deck ratio of ψ by D ψ (G) = P ψ(g v). We give generic upper
More informationCourse Notes. Part IV. Probabilistic Combinatorics. Algorithms
Course Notes Part IV Probabilistic Combinatorics and Algorithms J. A. Verstraete Department of Mathematics University of California San Diego 9500 Gilman Drive La Jolla California 92037-0112 jacques@ucsd.edu
More informationThe Hierarchical Product of Graphs
The Hierarchical Product of Graphs Lali Barrière Francesc Comellas Cristina Dalfó Miquel Àngel Fiol Universitat Politècnica de Catalunya - DMA4 April 8, 2008 Outline 1 Introduction 2 Graphs and matrices
More informationCombining Geographic and Network Analysis: The GoMore Rideshare Network. Kate Lyndegaard
Combining Geographic and Network Analysis: The GoMore Rideshare Network Kate Lyndegaard 10.15.2014 Outline 1. Motivation 2. What is network analysis? 3. The project objective 4. The GoMore network 5. The
More informationAverage Distance, Diameter, and Clustering in Social Networks with Homophily
arxiv:0810.2603v1 [physics.soc-ph] 15 Oct 2008 Average Distance, Diameter, and Clustering in Social Networks with Homophily Matthew O. Jackson September 24, 2008 Abstract I examine a random network model
More informationNetworks. Can (John) Bruce Keck Founda7on Biotechnology Lab Bioinforma7cs Resource
Networks Can (John) Bruce Keck Founda7on Biotechnology Lab Bioinforma7cs Resource Networks in biology Protein-Protein Interaction Network of Yeast Transcriptional regulatory network of E.coli Experimental
More informationSmall subgraphs of random regular graphs
Discrete Mathematics 307 (2007 1961 1967 Note Small subgraphs of random regular graphs Jeong Han Kim a,b, Benny Sudakov c,1,vanvu d,2 a Theory Group, Microsoft Research, Redmond, WA 98052, USA b Department
More informationRANDOM GRAPHS BY JOEL SPENCER. as n.
RANDOM GRAHS BY JOEL SENCER Notation 1. We say fn gn as n if fx gx as n. Notation 2. We say fn gn as n if fn gn 1 as n Notation 3. We say fn = ogn as n if fn gn 0 as n Notation 4. n k := nn 1n 2... n k
More informationFIRST ORDER SENTENCES ON G(n, p), ZERO-ONE LAWS, ALMOST SURE AND COMPLETE THEORIES ON SPARSE RANDOM GRAPHS
FIRST ORDER SENTENCES ON G(n, p), ZERO-ONE LAWS, ALMOST SURE AND COMPLETE THEORIES ON SPARSE RANDOM GRAPHS MOUMANTI PODDER 1. First order theory on G(n, p) We start with a very simple property of G(n,
More informationLecture II: Matrix Functions in Network Science, Part 1
Lecture II: Matrix Functions in Network Science, Part 1 Michele Benzi Department of Mathematics and Computer Science Emory University Atlanta, Georgia, USA Summer School on Theory and Computation of Matrix
More informationLecture 06 01/31/ Proofs for emergence of giant component
M375T/M396C: Topics in Complex Networks Spring 2013 Lecture 06 01/31/13 Lecturer: Ravi Srinivasan Scribe: Tianran Geng 6.1 Proofs for emergence of giant component We now sketch the main ideas underlying
More informationEquivalence of the random intersection graph and G(n, p)
Equivalence of the random intersection graph and Gn, p arxiv:0910.5311v1 [math.co] 28 Oct 2009 Katarzyna Rybarczyk Faculty of Mathematics and Computer Science, Adam Mickiewicz University, 60 769 Poznań,
More information< k 2n. 2 1 (n 2). + (1 p) s) N (n < 1
List of Problems jacques@ucsd.edu Those question with a star next to them are considered slightly more challenging. Problems 9, 11, and 19 from the book The probabilistic method, by Alon and Spencer. Question
More informationLecture 2: Rumor Spreading (2)
Alea Meeting, Munich, February 2016 Lecture 2: Rumor Spreading (2) Benjamin Doerr, LIX, École Polytechnique, Paris-Saclay Outline: Reminder last lecture and an answer to Kosta s question Definition: Rumor
More informationLecture VI Introduction to complex networks. Santo Fortunato
Lecture VI Introduction to complex networks Santo Fortunato Plan of the course I. Networks: definitions, characteristics, basic concepts in graph theory II. III. IV. Real world networks: basic properties
More informationBasic graph theory 18.S995 - L26.
Basic graph theory 18.S995 - L26 dunkel@math.mit.edu http://java.dzone.com/articles/algorithm-week-graphs-and no cycles Isomorphic graphs f(a) = 1 f(b) = 6 f(c) = 8 f(d) = 3 f(g) = 5 f(h) = 2 f(i) = 4
More informationProbabilistic Methods in Combinatorics Lecture 6
Probabilistic Methods in Combinatorics Lecture 6 Linyuan Lu University of South Carolina Mathematical Sciences Center at Tsinghua University November 16, 2011 December 30, 2011 Balance graphs H has v vertices
More informationThe Hierarchical Product of Graphs
The Hierarchical Product of Graphs Lali Barrière Francesc Comellas Cristina Dalfó Miquel Àngel Fiol Universitat Politècnica de Catalunya - DMA4 March 22, 2007 Outline 1 Introduction 2 The hierarchical
More informationInfinite Limits of Other Random Graphs
Infinite Limits of Other Random Graphs Richard Elwes, University of Leeds 29 July 2016 Erdős - Rényi Graphs Definition (Erdős & Rényi, 1960) Fix p (0, 1) and n N {ω}. Form a random graph G n (p) with n
More informationVertex colorings of graphs without short odd cycles
Vertex colorings of graphs without short odd cycles Andrzej Dudek and Reshma Ramadurai Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 1513, USA {adudek,rramadur}@andrew.cmu.edu
More information6.207/14.15: Networks Lecture 3: Erdös-Renyi graphs and Branching processes
6.207/14.15: Networks Lecture 3: Erdös-Renyi graphs and Branching processes Daron Acemoglu and Asu Ozdaglar MIT September 16, 2009 1 Outline Erdös-Renyi random graph model Branching processes Phase transitions
More informationSelf-organized scale-free networks
Self-organized scale-free networks Kwangho Park and Ying-Cheng Lai Departments of Electrical Engineering, Arizona State University, Tempe, Arizona 85287, USA Nong Ye Department of Industrial Engineering,
More informationCombinatorial Rigidity and the Molecular Conjecture
Page 1 of 65 Combinatorial Rigidity and the Molecular Conjecture Brigitte Servatius Worcester Polytechnic Institute The Proof of the Product Rule To derivate a product as defined The diff rence quotient
More informationSpectra of adjacency matrices of random geometric graphs
Spectra of adjacency matrices of random geometric graphs Paul Blackwell, Mark Edmondson-Jones and Jonathan Jordan University of Sheffield 22nd December 2006 Abstract We investigate the spectral properties
More informationAn evolving network model with community structure
INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL J. Phys. A: Math. Gen. 38 (2005) 9741 9749 doi:10.1088/0305-4470/38/45/002 An evolving network model with community structure
More informationSubhypergraph counts in extremal and random hypergraphs and the fractional q-independence
Subhypergraph counts in extremal and random hypergraphs and the fractional q-independence Andrzej Dudek adudek@emory.edu Andrzej Ruciński rucinski@amu.edu.pl June 21, 2008 Joanna Polcyn joaska@amu.edu.pl
More informationCoupling Scale-Free and Classical Random Graphs
Internet Mathematics Vol. 1, No. 2: 215-225 Coupling Scale-Free and Classical Random Graphs Béla Bollobás and Oliver Riordan Abstract. Recently many new scale-free random graph models have been introduced,
More informationarxiv:cond-mat/ v1 [cond-mat.dis-nn] 24 Mar 2005
APS/123-QED Scale-Free Networks Emerging from Weighted Random Graphs Tomer Kalisky, 1, Sameet Sreenivasan, 2 Lidia A. Braunstein, 2,3 arxiv:cond-mat/0503598v1 [cond-mat.dis-nn] 24 Mar 2005 Sergey V. Buldyrev,
More informationDeterministic Decentralized Search in Random Graphs
Deterministic Decentralized Search in Random Graphs Esteban Arcaute 1,, Ning Chen 2,, Ravi Kumar 3, David Liben-Nowell 4,, Mohammad Mahdian 3, Hamid Nazerzadeh 1,, and Ying Xu 1, 1 Stanford University.
More informationpreferential attachment
preferential attachment introduction to network analysis (ina) Lovro Šubelj University of Ljubljana spring 2018/19 preferential attachment graph models are ensembles of random graphs generative models
More informationTreewidth of Erdős-Rényi Random Graphs, Random Intersection Graphs, and Scale-Free Random Graphs
Treewidth of Erdős-Rényi Random Graphs, Random Intersection Graphs, and Scale-Free Random Graphs Yong Gao Department of Computer Science, Irving K. Barber School of Arts and Sciences University of British
More informationWeak convergence in Probability Theory A summer excursion! Day 3
BCAM June 2013 1 Weak convergence in Probability Theory A summer excursion! Day 3 Armand M. Makowski ECE & ISR/HyNet University of Maryland at College Park armand@isr.umd.edu BCAM June 2013 2 Day 1: Basic
More informationErzsébet Ravasz Advisor: Albert-László Barabási
Hierarchical Networks Erzsébet Ravasz Advisor: Albert-László Barabási Introduction to networks How to model complex networks? Clustering and hierarchy Hierarchical organization of cellular metabolism The
More informationModeling of Growing Networks with Directional Attachment and Communities
Modeling of Growing Networks with Directional Attachment and Communities Masahiro KIMURA, Kazumi SAITO, Naonori UEDA NTT Communication Science Laboratories 2-4 Hikaridai, Seika-cho, Kyoto 619-0237, Japan
More informationON THE NP-COMPLETENESS OF SOME GRAPH CLUSTER MEASURES
ON THE NP-COMPLETENESS OF SOME GRAPH CLUSTER MEASURES JIŘÍ ŠÍMA AND SATU ELISA SCHAEFFER Academy of Sciences of the Czech Republic Helsinki University of Technology, Finland elisa.schaeffer@tkk.fi SOFSEM
More informationStochastic Models in Computer Science A Tutorial
Stochastic Models in Computer Science A Tutorial Dr. Snehanshu Saha Department of Computer Science PESIT BSC, Bengaluru WCI 2015 - August 10 to August 13 1 Introduction 2 Random Variable 3 Introduction
More informationCS224W: Social and Information Network Analysis Jure Leskovec, Stanford University
CS224W: Social and Information Network Analysis Jure Leskovec, Stanford University http://cs224w.stanford.edu Intro sessions to SNAP C++ and SNAP.PY: SNAP.PY: Friday 9/27, 4:5 5:30pm in Gates B03 SNAP
More informationRecent Progress in Complex Network Analysis. Properties of Random Intersection Graphs
Recent Progress in Complex Network Analysis. Properties of Random Intersection Graphs Mindaugas Bloznelis, Erhard Godehardt, Jerzy Jaworski, Valentas Kurauskas, Katarzyna Rybarczyk Adam Mickiewicz University,
More informationModeling, Analysis, and Control of Information Propagation in Multi-layer and Multiplex Networks. Osman Yağan
Modeling, Analysis, and Control of Information Propagation in Multi-layer and Multiplex Networks Osman Yağan Department of ECE Carnegie Mellon University Joint work with Y. Zhuang and V. Gligor (CMU) Alex
More informationCritical phenomena in complex networks
Critical phenomena in complex networks S. N. Dorogovtsev* and A. V. Goltsev Departamento de Física, Universidade de Aveiro, 3810-193 Aveiro, Portugal and A. F. Ioffe Physico-Technical Institute, 194021
More informationDesign and characterization of chemical space networks
Design and characterization of chemical space networks Martin Vogt B-IT Life Science Informatics Rheinische Friedrich-Wilhelms-University Bonn 16 August 2015 Network representations of chemical spaces
More information