Graph Theory and Complex Networks: An Introduction. Chapter 06: Network analysis


 Cody Woods
 3 years ago
 Views:
Transcription
1 Graph Theory and Complex Networks: An Introduction Maarten van Steen VU Amsterdam, Dept. Computer Science Room R4.0, Chapter 06: Network analysis Version: April 8, 04
2 / 3 Contents Chapter Description 0: Introduction History, background 0: Foundations Basic terminology and properties of graphs 03: Extensions Directed & weighted graphs, colorings 04: Network traversal Walking through graphs (cf. traveling) 05: Trees Graphs without cycles; routing algorithms 06: Network analysis Basic metrics for analyzing large graphs 07: Random networks Introduction modeling realworld networks 08: Computer networks The Internet & WWW seen as a huge graph 09: Social networks Communities seen as graphs
3 Network analysis Introduction Observation In realworld situations, graphs (or networks) may become very large, making it difficult to (visually) discover properties we need network analysis tools. Vertex degrees: Consider the distribution of degrees: how many vertices have high degrees versus the number of vertices with low degrees. Distance statistics: Focus on where vertices are positioned in the network: far away from each other, central in the network, etc. Clustering: To what extent are my neighbors also adjacent to each other? Centrality: Are there vertices that are more important than others? 3 / 3
4 Network analysis 6. Vertex degree Vertex degree Question Can you visually observe real (nonisomorphic) differences? 4 / 3
5 5 / 3 Network analysis 6. Vertex degree Vertex degree: Histogram (a): (b): n = 00, m = 300 n = 00, m = (a) (b)
6 6 / 3 Network analysis 6. Vertex degree Vertex degree: Ranked histogram
7 7 / 3 Network analysis 6. Distance statistics Distance statistics Definition G is connected, d(u,v) is distance between vertices u and v: the length of a shortest path between u and v. Eccentricity ε(u): Radius rad(g): Diameter diam(g): max{d(u, v) v V (G)} min{ε(u) u V (G)} max{d(u, v) u, v V (G)} Note Note that these definitions apply to directed as well as undirected graphs.
8 8 / 3 Network analysis 6. Distance statistics Path lengths Definition G is connected with vertex V ; d(u) is average length of shortest paths from u to any other vertex v: d(u) def = V d(u,v) v V,v u The average path length d(g): d(g) def = V u V d(u) = V V d(u,v) u,v V,u v
9 Network analysis 6. Distance statistics Path lengths Definition The characteristic path length is the median over all d(u). Note The median over n nondecreasing values x,x,...,x n : n odd x (n+)/ n even (x n/ + x n/+ )/ The median separates the higher values from the lower values into two equallysized subsets. Example {3,4,4,6,0,6,} [0,,3,4,4,6,6] M = x (7+)/ = x 4 = 4 9 / 3
10 0 / 3 Network analysis 6. Distance statistics Example distance statistics Vertex ε(u) v u d(u,v) d(u)
11 Network analysis 6.3 Clustering coefficient Clustering coefficient Observation Many networks show a high degree of clustering: my neighbors are each other s neighbors. Note An extreme case is formed by having all my neighbors be adjacent to each other neighbors form a complete graph. Question What is the other extreme case? / 3
12 Network analysis 6.3 Clustering coefficient Clustering coefficient Definition G is simple, connected, undirected. Vertex v V (G) with neighborset N(v). Let n v = N(v). Note: max. number of edges between neighbors is ( n v ). Let m v is number of edges in subgraph induced by N(v): m v = E(G[N(v)]). Clustering coefficient cc(v): cc(v) def = { mv / ( n v ) = m v n v (n v ) if δ(v) > undefined otherwise / 3
13 Network analysis 6.3 Clustering coefficient Clustering coefficient Definition G is simple, connected and undirected. Let V def = {v V (G) δ(v) > }. Clustering coefficient CC(G) for G: CC(G) def = V v V cc(v) 3 / 3
14 Network analysis 6.3 Clustering coefficient Clustering coefficient: triangles Definition A triangle is a complete (sub)graph with exactly 3 vertices. A triple is a (sub)graph with exactly 3 vertices and edges. Definition G is simple and connected with n (G) distinct triangles and n Λ (G) distinct triples. The network transitivity τ(g) def = n (G)/n Λ (G). Notation A triple at v: v is incident to both edges ( in the middle ). n Λ (v) : number of triples at v. 4 / 3
15 5 / 3 Network analysis 6.3 Clustering coefficient Clustering coefficient: example Vertex: cc: /3 0 /3 undefined /3 n Λ : Vertex N() = {,5,7}; E(G[N()]) = 5,7 cc() = 3 Triples at : G[{,,5}],G[{,,7}],G[{5,,7}]
16 6 / 3 Network analysis 6.3 Clustering coefficient Clustering coefficient versus transitivity Observation Let n (v) be the number of triangles of which v is member cc(v) = n (v) n Λ (v) n Λ (v) = ( δ(v) ) n (G) = 3 v V n (v) (Note: V = {v V δ(v) > })
17 7 / 3 Network analysis 6.3 Clustering coefficient Clustering coefficient versus transitivity x x x v v v 3... v n v k v v v 3... v n y G k = G[{x,y,v,v,...,v k }] : cc(u) = y if u = v,...,v k ) = k k(k+) = k+ if u = x or u = y k ( k+ CC(G k ) = k + ( k + + k ) = k + k + 4 k + 3k + lim k CC(G k) = y
18 8 / 3 Network analysis 6.3 Clustering coefficient Clustering coefficient versus transitivity x v v v 3... v n G k = G[{x,y,v,v,...,v k }] { if u = v,...,v k n Λ (u) = ) ( = k+ ) if u = x,y ( δ(u) y τ(g k ) = n (G k ) n Λ (u) = k = k(k + ) + k k + lim τ(g k) = 0 k
19 Network analysis 6.4 Centrality Centrality Issue Are there any vertices that are more important than the others? Definition G is (strongly) connected. The center C(G) is the set of vertices with minimal eccentricity: Intuition C(G) def = {v V (G) ε(v) = rad(g)} At the center means at minimal distance to the farthest node. 9 / 3
20 0 / 3 Network analysis 6.4 Centrality Vertex centrality Definition G is (strongly) connected. The (eccentricity based) vertex centrality c E (u) of u: c E (u) def = ε(u) Intuition The higher the centrality, the closer to the center of a graph.
21 Network analysis 6.4 Centrality Closeness Definition G is (strongly) connected. The closeness c C (u) of u: c C (u) def = v V (G) d(u,v) Intuition How close is a vertex to all other nodes? / 3
22 / 3 Network analysis 6.4 Centrality Centrality: example Vertex: ε(u) d(u, ) c C (u):
23 Network analysis 6.4 Centrality Betweenness Intuition Important vertices are those whose removal significantly increases the distance between other vertices. Example: cut vertices. Definition G is simple and (strongly) connected. S(x,y) is set of shortest paths between x and y. S(x,u,y) S(x,y) paths that pass through u. Betweenness centrality c B (u) of u: c B (u) def = x y u S(x,u,y) S(x, y) 3 / 3
Graph Theory and Complex Networks: An Introduction. Chapter 06: Network analysis. Contents. Introduction. Maarten van Steen. Version: April 28, 2014
Graph Theory and Complex Networks: An Introduction Maarten van Steen VU Amsterdam, Dept. Computer Science Room R.0, steen@cs.vu.nl Chapter 0: Version: April 8, 0 / Contents Chapter Description 0: Introduction
More informationSocial Media Mining. Graph Essentials
Graph Essentials Graph Basics Measures Graph and Essentials Metrics 2 2 Nodes and Edges A network is a graph nodes, actors, or vertices (plural of vertex) Connections, edges or ties Edge Node Measures
More informationGraph Theory and Complex Networks: An Introduction. Chapter 08: Computer networks
Graph Theory and Complex Networks: An Introduction Maarten van Steen VU Amsterdam, Dept. Computer Science Room R4.20, steen@cs.vu.nl Chapter 08: Computer networks Version: March 3, 2011 2 / 53 Contents
More informationGraph Theory. Introduction. Distance in Graphs. Trees. Isabela Drămnesc UVT. Computer Science Department, West University of Timişoara, Romania
Graph Theory Introduction. Distance in Graphs. Trees Isabela Drămnesc UVT Computer Science Department, West University of Timişoara, Romania November 2016 Isabela Drămnesc UVT Graph Theory and Combinatorics
More informationGRAPH THEORY and APPLICATIONS. Trees
GRAPH THEORY and APPLICATIONS Trees Properties Tree: a connected graph with no cycle (acyclic) Forest: a graph with no cycle Paths are trees. Star: A tree consisting of one vertex adjacent to all the others.
More informationGraphs and Network Flows IE411 Lecture 1
Graphs and Network Flows IE411 Lecture 1 Dr. Ted Ralphs IE411 Lecture 1 1 References for Today s Lecture Required reading Sections 17.1, 19.1 References AMO Chapter 1 and Section 2.1 and 2.2 IE411 Lecture
More informationDistance Degree Sequences for Network Analysis
Universität Konstanz Computer & Information Science Algorithmics Group 15 Mar 2005 based on Palmer, Gibbons, and Faloutsos: ANF A Fast and Scalable Tool for Data Mining in Massive Graphs, SIGKDD 02. Motivation
More informationSocial Media Mining. Network Measures
Klout Measures and Metrics 22 Why Do We Need Measures? Who are the central figures (influential individuals) in the network? What interaction patterns are common in friends? Who are the likeminded users
More informationCourse on Social Network Analysis Graphs and Networks
Course on Social Network Analysis Graphs and Networks Vladimir Batagelj University of Ljubljana Slovenia V. Batagelj: Social Network Analysis / Graphs and Networks 1 Outline 1 Graph...............................
More informationAsking Hard Graph Questions. Paul Burkhardt. February 3, 2014
Beyond Watson: Predictive Analytics and Big Data U.S. National Security Agency Research Directorate  R6 Technical Report February 3, 2014 300 years before Watson there was Euler! The first (Jeopardy!)
More informationDefinition. A graph is a collection of vertices, and edges between them. They are often represented by a drawing:
1. GRAPHS AND COLORINGS Definition. A graph is a collection of vertices, and edges between them. They are often represented by a drawing: 3 vertices 3 edges 4 vertices 4 edges 4 vertices 6 edges A graph
More informationBOUNDARY EDGE DOMINATION IN GRAPHS
BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 04874, ISSN (o) 04955 www.imvibl.org /JOURNALS / BULLETIN Vol. 5(015), 19704 Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA
More informationGraph definition Degree, in, out degree, oriented graph. Complete, regular, bipartite graph. Graph representation, connectivity, adjacency.
Mária Markošová Graph definition Degree, in, out degree, oriented graph. Complete, regular, bipartite graph. Graph representation, connectivity, adjacency. Isomorphism of graphs. Paths, cycles, trials.
More informationMinimum Spanning Trees
Minimum Spanning Trees Algorithms and 18.304 Presentation Outline 1 Graph Terminology Minimum Spanning Trees 2 3 Outline Graph Terminology Minimum Spanning Trees 1 Graph Terminology Minimum Spanning Trees
More informationCMSC 451: Graph Properties, DFS, BFS, etc.
CMSC 451: Graph Properties, DFS, BFS, etc. Slides By: Carl Kingsford Department of Computer Science University of Maryland, College Park Based on Chapter 3 of Algorithm Design by Kleinberg & Tardos. Graphs
More informationDO NOT REDISTRIBUTE THIS SOLUTION FILE
Professor Kindred Math 04 Graph Theory Homework 7 Solutions April 3, 03 Introduction to Graph Theory, West Section 5. 0, variation of 5, 39 Section 5. 9 Section 5.3 3, 8, 3 Section 7. Problems you should
More informationSolutions to Exercises 8
Discrete Mathematics Lent 2009 MA210 Solutions to Exercises 8 (1) Suppose that G is a graph in which every vertex has degree at least k, where k 1, and in which every cycle contains at least 4 vertices.
More informationIntroduction to Graph Mining
Introduction to Graph Mining What is a graph? A graph G = (V,E) is a set of vertices V and a set (possibly empty) E of pairs of vertices e 1 = (v 1, v 2 ), where e 1 E and v 1, v 2 V. Edges may contain
More informationA discussion of Statistical Mechanics of Complex Networks P. Part I
A discussion of Statistical Mechanics of Complex Networks Part I Review of Modern Physics, Vol. 74, 2002 Small Word Networks Clustering Coefficient ScaleFree Networks ErdösRényi model cover only parts
More informationMGF 1107 CH 15 LECTURE NOTES Denson. Section 15.1
1 Section 15.1 Consider the house plan below. This graph represents the house. Consider the mail route below. This graph represents the mail route. 2 Definitions 1. Graph a structure that describes relationships.
More informationSmall Maximal Independent Sets and Faster Exact Graph Coloring
Small Maximal Independent Sets and Faster Exact Graph Coloring David Eppstein Univ. of California, Irvine Dept. of Information and Computer Science The Exact Graph Coloring Problem: Given an undirected
More informationV. Adamchik 1. Graph Theory. Victor Adamchik. Fall of 2005
V. Adamchik 1 Graph Theory Victor Adamchik Fall of 2005 Plan 1. Basic Vocabulary 2. Regular graph 3. Connectivity 4. Representing Graphs Introduction A.Aho and J.Ulman acknowledge that Fundamentally, computer
More informationGraph Mining and Social Network Analysis
Graph Mining and Social Network Analysis Data Mining and Text Mining (UIC 583 @ Politecnico di Milano) References Jiawei Han and Micheline Kamber, "Data Mining: Concepts and Techniques", The Morgan Kaufmann
More informationPart 2: Community Detection
Chapter 8: Graph Data Part 2: Community Detection Based on Leskovec, Rajaraman, Ullman 2014: Mining of Massive Datasets Big Data Management and Analytics Outline Community Detection  Social networks 
More informationOn the independence number of graphs with maximum degree 3
On the independence number of graphs with maximum degree 3 Iyad A. Kanj Fenghui Zhang Abstract Let G be an undirected graph with maximum degree at most 3 such that G does not contain any of the three graphs
More informationGRAPH THEORY LECTURE 4: TREES
GRAPH THEORY LECTURE 4: TREES Abstract. 3.1 presents some standard characterizations and properties of trees. 3.2 presents several different types of trees. 3.7 develops a counting method based on a bijection
More informationChapter 2 Paths and Searching
Chapter 2 Paths and Searching Section 2.1 Distance Almost every day you face a problem: You must leave your home and go to school. If you are like me, you are usually a little late, so you want to take
More informationA MEASURE OF GLOBAL EFFICIENCY IN NETWORKS. Aysun Aytac 1, Betul Atay 2. Faculty of Science Ege University 35100, Bornova, Izmir, TURKEY
International Journal of Pure and Applied Mathematics Volume 03 No. 05, 670 ISSN: 38080 (printed version); ISSN: 343395 (online version) url: http://www.ijpam.eu doi: http://dx.doi.org/0.73/ijpam.v03i.5
More informationChapter 4. Trees. 4.1 Basics
Chapter 4 Trees 4.1 Basics A tree is a connected graph with no cycles. A forest is a collection of trees. A vertex of degree one, particularly in a tree, is called a leaf. Trees arise in a variety of applications.
More informationMultilevel analysis of an interaction network between individuals in a mailinglist
2050Her/Teleco 62/34 14/03/07 13:48 Page 320 320 pp. 320344 Multilevel analysis of an interaction network between individuals in a mailinglist Rémi DORAT 1, 2, 3 Matthieu LATAPY 1, Bernard CONEIN
More informationOn Total Domination in Graphs
University of Houston  Downtown Senior Project  Fall 2012 On Total Domination in Graphs Author: David Amos Advisor: Dr. Ermelinda DeLaViña Senior Project Committee: Dr. Sergiy Koshkin Dr. Ryan Pepper
More informationGraph Theory. Euler tours and Chinese postmen. John Quinn. Week 5
Graph Theory Euler tours and Chinese postmen John Quinn Week 5 Recap: connectivity Connectivity and edgeconnectivity of a graph Blocks Kruskal s algorithm Königsberg, Prussia The Seven Bridges of Königsberg
More informationCSL851: Algorithmic Graph Theory Semester I Lecture 4: August 5
CSL851: Algorithmic Graph Theory Semester I 201314 Lecture 4: August 5 Lecturer: Naveen Garg Scribes: Utkarsh Ohm Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer: These notes have not
More informationSEMITOTAL AND TOTAL BLOCKCUTVERTEX GRAPH
CHAPTER 3 SEMITOTAL AND TOTAL BLOCKCUTVERTEX GRAPH ABSTRACT This chapter begins with the notion of block distances in graphs. Using block distance we defined the central tendencies of a block, like Bradius
More informationHomework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS
Homework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS 1.1.25 Prove that the Petersen graph has no cycle of length 7. Solution: There are 10 vertices in the Petersen graph G. Assume there is a cycle C
More informationGraph. Consider a graph, G in Fig Then the vertex V and edge E can be represented as:
Graph A graph G consist of 1. Set of vertices V (called nodes), (V = {v1, v2, v3, v4...}) and 2. Set of edges E (i.e., E {e1, e2, e3...cm} A graph can be represents as G = (V, E), where V is a finite and
More informationA Study of Sufficient Conditions for Hamiltonian Cycles
DeLeon 1 A Study of Sufficient Conditions for Hamiltonian Cycles Melissa DeLeon Department of Mathematics and Computer Science Seton Hall University South Orange, New Jersey 07079, U.S.A. ABSTRACT A graph
More informationTrees and Fundamental Circuits
Trees and Fundamental Circuits Tree A connected graph without any circuits. o must have at least one vertex. o definition implies that it must be a simple graph. o only finite trees are being considered
More informationSimple Graphs Degrees, Isomorphism, Paths
Mathematics for Computer Science MIT 6.042J/18.062J Simple Graphs Degrees, Isomorphism, Types of Graphs Simple Graph this week MultiGraph Directed Graph next week Albert R Meyer, March 10, 2010 lec 6W.1
More information2.3 Scheduling jobs on identical parallel machines
2.3 Scheduling jobs on identical parallel machines There are jobs to be processed, and there are identical machines (running in parallel) to which each job may be assigned Each job = 1,,, must be processed
More informationHomework 15 Solutions
PROBLEM ONE (Trees) Homework 15 Solutions 1. Recall the definition of a tree: a tree is a connected, undirected graph which has no cycles. Which of the following definitions are equivalent to this definition
More informationGraph Algorithms using MapReduce
Graph Algorithms using MapReduce Graphs are ubiquitous in modern society. Some examples: The hyperlink structure of the web 1/7 Graph Algorithms using MapReduce Graphs are ubiquitous in modern society.
More informationGeneral Network Analysis: Graphtheoretic. COMP572 Fall 2009
General Network Analysis: Graphtheoretic Techniques COMP572 Fall 2009 Networks (aka Graphs) A network is a set of vertices, or nodes, and edges that connect pairs of vertices Example: a network with 5
More informationWhat cannot be measured on the Internet? YvonneAnne Pignolet, Stefan Schmid, G. Trédan. Misleading stars
: What cannot be measured on the Internet? YvonneAnne Pignolet, Stefan Schmid, Gilles Tredan How accurate are network maps? Why? To develop/adapt protocols to Internet PaDIS, RMTP To understand the impact
More informationSome questions... Graphs
Uni Innsbruck Informatik  1 Uni Innsbruck Informatik  2 Some questions... Peerto topeer Systems Analysis of unstructured P2P systems How scalable is Gnutella? How robust is Gnutella? Why does FreeNet
More informationChapter 11. 11.1 Load Balancing. Approximation Algorithms. Load Balancing. Load Balancing on 2 Machines. Load Balancing: Greedy Scheduling
Approximation Algorithms Chapter Approximation Algorithms Q. Suppose I need to solve an NPhard problem. What should I do? A. Theory says you're unlikely to find a polytime algorithm. Must sacrifice one
More informationDistributed Computing over Communication Networks: Maximal Independent Set
Distributed Computing over Communication Networks: Maximal Independent Set What is a MIS? MIS An independent set (IS) of an undirected graph is a subset U of nodes such that no two nodes in U are adjacent.
More informationMidterm Practice Problems
6.042/8.062J Mathematics for Computer Science October 2, 200 Tom Leighton, Marten van Dijk, and Brooke Cowan Midterm Practice Problems Problem. [0 points] In problem set you showed that the nand operator
More informationA 2factor in which each cycle has long length in clawfree graphs
A factor in which each cycle has long length in clawfree graphs Roman Čada Shuya Chiba Kiyoshi Yoshimoto 3 Department of Mathematics University of West Bohemia and Institute of Theoretical Computer Science
More informationTracers Placement for IP Traceback against DDoS Attacks
Tracers Placement for IP Traceback against DDoS Attacks ChunHsin Wang, ChangWu Yu, ChiuKuo Liang, KunMin Yu, Wen Ouyang, ChingHsien Hsu, and YuGuang Chen Department of Computer Science and Information
More informationGraph models for the Web and the Internet. Elias Koutsoupias University of Athens and UCLA. Crete, July 2003
Graph models for the Web and the Internet Elias Koutsoupias University of Athens and UCLA Crete, July 2003 Outline of the lecture Small world phenomenon The shape of the Web graph Searching and navigation
More informationChapter 6: Graph Theory
Chapter 6: Graph Theory Graph theory deals with routing and network problems and if it is possible to find a best route, whether that means the least expensive, least amount of time or the least distance.
More information/ Approximation Algorithms Lecturer: Michael Dinitz Topic: Steiner Tree and TSP Date: 01/29/15 Scribe: Katie Henry
600.469 / 600.669 Approximation Algorithms Lecturer: Michael Dinitz Topic: Steiner Tree and TSP Date: 01/29/15 Scribe: Katie Henry 2.1 Steiner Tree Definition 2.1.1 In the Steiner Tree problem the input
More informationWhy graph clustering is useful?
Graph Clustering Why graph clustering is useful? Distance matrices are graphs as useful as any other clustering Identification of communities in social networks Webpage clustering for better data management
More informationCombinatorics: The Fine Art of Counting
Combinatorics: The Fine Art of Counting Week 9 Lecture Notes Graph Theory For completeness I have included the definitions from last week s lecture which we will be using in today s lecture along with
More informationCSE 20: Discrete Mathematics for Computer Science. Prof. Miles Jones. Today s Topics: Graphs. The Internet graph
Today s Topics: CSE 0: Discrete Mathematics for Computer Science Prof. Miles Jones. Graphs. Some theorems on graphs. Eulerian graphs Graphs! Model relations between pairs of objects The Internet graph!
More informationNetwork/Graph Theory. What is a Network? What is network theory? Graphbased representations. Friendship Network. What makes a problem graphlike?
What is a Network? Network/Graph Theory Network = graph Informally a graph is a set of nodes joined by a set of lines or arrows. 1 1 2 3 2 3 4 5 6 4 5 6 Graphbased representations Representing a problem
More informationSeminar. Path planning using Voronoi diagrams and BSplines. Stefano Martina stefano.martina@stud.unifi.it
Seminar Path planning using Voronoi diagrams and BSplines Stefano Martina stefano.martina@stud.unifi.it 23 may 2016 This work is licensed under a Creative Commons AttributionShareAlike 4.0 International
More informationMining Social Network Graphs
Mining Social Network Graphs Debapriyo Majumdar Data Mining Fall 2014 Indian Statistical Institute Kolkata November 13, 17, 2014 Social Network No introduc+on required Really? We s7ll need to understand
More informationExtracting Information from Social Networks
Extracting Information from Social Networks Aggregating site information to get trends 1 Not limited to social networks Examples Google search logs: flu outbreaks We Feel Fine Bullying 2 Bullying Xu, Jun,
More informationCIS 700: algorithms for Big Data
CIS 700: algorithms for Big Data Lecture 6: Graph Sketching Slides at http://grigory.us/bigdataclass.html Grigory Yaroslavtsev http://grigory.us Sketching Graphs? We know how to sketch vectors: v Mv
More informationLecture 9. 1 Introduction. 2 Random Walks in Graphs. 1.1 How To Explore a Graph? CS621 Theory Gems October 17, 2012
CS62 Theory Gems October 7, 202 Lecture 9 Lecturer: Aleksander Mądry Scribes: Dorina Thanou, Xiaowen Dong Introduction Over the next couple of lectures, our focus will be on graphs. Graphs are one of
More informationPlanar Tree Transformation: Results and Counterexample
Planar Tree Transformation: Results and Counterexample Selim G Akl, Kamrul Islam, and Henk Meijer School of Computing, Queen s University Kingston, Ontario, Canada K7L 3N6 Abstract We consider the problem
More informationRecap. Type of graphs Connectivity/Giant component Diameter Clustering coefficient Betweenness Centrality Degree distributions
Recap Type of graphs Connectivity/Giant component Diameter Clustering coefficient Betweenness Centrality Degree distributions Degree Distribution N k is the number of nodes with degree k P(k) is the probability
More informationAbout the Tutorial. Audience. Prerequisites. Disclaimer & Copyright
About the Tutorial This tutorial offers a brief introduction to the fundamentals of graph theory. Written in a readerfriendly style, it covers the types of graphs, their properties, trees, graph traversability,
More informationData Structures in Java. Session 16 Instructor: Bert Huang
Data Structures in Java Session 16 Instructor: Bert Huang http://www.cs.columbia.edu/~bert/courses/3134 Announcements Homework 4 due next class Remaining grades: hw4, hw5, hw6 25% Final exam 30% Midterm
More informationFinding and counting given length cycles
Finding and counting given length cycles Noga Alon Raphael Yuster Uri Zwick Abstract We present an assortment of methods for finding and counting simple cycles of a given length in directed and undirected
More informationTheorem A graph T is a tree if, and only if, every two distinct vertices of T are joined by a unique path.
Chapter 3 Trees Section 3. Fundamental Properties of Trees Suppose your city is planning to construct a rapid rail system. They want to construct the most economical system possible that will meet the
More informationA Sublinear Bipartiteness Tester for Bounded Degree Graphs
A Sublinear Bipartiteness Tester for Bounded Degree Graphs Oded Goldreich Dana Ron February 5, 1998 Abstract We present a sublineartime algorithm for testing whether a bounded degree graph is bipartite
More informationRandom graphs with a given degree sequence
Sourav Chatterjee (NYU) Persi Diaconis (Stanford) Allan Sly (Microsoft) Let G be an undirected simple graph on n vertices. Let d 1,..., d n be the degrees of the vertices of G arranged in descending order.
More informationWhy? A central concept in Computer Science. Algorithms are ubiquitous.
Analysis of Algorithms: A Brief Introduction Why? A central concept in Computer Science. Algorithms are ubiquitous. Using the Internet (sending email, transferring files, use of search engines, online
More informationGraph Theory Notes. Vadim Lozin. Institute of Mathematics University of Warwick
Graph Theory Notes Vadim Lozin Institute of Mathematics University of Warwick 1 Introduction A graph G = (V, E) consists of two sets V and E. The elements of V are called the vertices and the elements
More information! E6893 Big Data Analytics Lecture 10:! Linked Big Data Graph Computing (II)
E6893 Big Data Analytics Lecture 10: Linked Big Data Graph Computing (II) ChingYung Lin, Ph.D. Adjunct Professor, Dept. of Electrical Engineering and Computer Science Mgr., Dept. of Network Science and
More informationGraph Theory Problems and Solutions
raph Theory Problems and Solutions Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles November, 005 Problems. Prove that the sum of the degrees of the vertices of any finite graph is
More informationZachary Monaco Georgia College Olympic Coloring: Go For The Gold
Zachary Monaco Georgia College Olympic Coloring: Go For The Gold Coloring the vertices or edges of a graph leads to a variety of interesting applications in graph theory These applications include various
More informationLink Prediction in Social Networks
CS378 Data Mining Final Project Report Dustin Ho : dsh544 Eric Shrewsberry : eas2389 Link Prediction in Social Networks 1. Introduction Social networks are becoming increasingly more prevalent in the daily
More informationAnalysis of Algorithms, I
Analysis of Algorithms, I CSOR W4231.002 Eleni Drinea Computer Science Department Columbia University Thursday, February 26, 2015 Outline 1 Recap 2 Representing graphs 3 Breadthfirst search (BFS) 4 Applications
More informationHandout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010. Chapter 7: Digraphs
MCS236: Graph Theory Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010 Chapter 7: Digraphs Strong Digraphs Definitions. A digraph is an ordered pair (V, E), where V is the set
More informationGeneralized Induced Factor Problems
Egerváry Research Group on Combinatorial Optimization Technical reports TR200207. Published by the Egrerváry Research Group, Pázmány P. sétány 1/C, H 1117, Budapest, Hungary. Web site: www.cs.elte.hu/egres.
More informationA maximum degree theorem for diameter2critical graphs
Cent. Eur. J. Math. 1(1) 01 1881889 DOI: 10.78/s1153301093 Central European Journal of Mathematics A maximum degree theorem for diametercritical graphs Research Article Teresa W. Haynes 1,, Michael
More informationPractical Graph Mining with R. 5. Link Analysis
Practical Graph Mining with R 5. Link Analysis Outline Link Analysis Concepts Metrics for Analyzing Networks PageRank HITS Link Prediction 2 Link Analysis Concepts Link A relationship between two entities
More informationUSING SPECTRAL RADIUS RATIO FOR NODE DEGREE TO ANALYZE THE EVOLUTION OF SCALE FREE NETWORKS AND SMALLWORLD NETWORKS
USING SPECTRAL RADIUS RATIO FOR NODE DEGREE TO ANALYZE THE EVOLUTION OF SCALE FREE NETWORKS AND SMALLWORLD NETWORKS Natarajan Meghanathan Jackson State University, 1400 Lynch St, Jackson, MS, USA natarajan.meghanathan@jsums.edu
More informationPlanarity Planarity
Planarity 8.1 71 Planarity Up until now, graphs have been completely abstract. In Topological Graph Theory, it matters how the graphs are drawn. Do the edges cross? Are there knots in the graph structure?
More information! Solve problem to optimality. ! Solve problem in polytime. ! Solve arbitrary instances of the problem. !approximation algorithm.
Approximation Algorithms Chapter Approximation Algorithms Q Suppose I need to solve an NPhard problem What should I do? A Theory says you're unlikely to find a polytime algorithm Must sacrifice one of
More informationFrans J.C.T. de Ruiter, Norman L. Biggs Applications of integer programming methods to cages
Frans J.C.T. de Ruiter, Norman L. Biggs Applications of integer programming methods to cages Article (Published version) (Refereed) Original citation: de Ruiter, Frans and Biggs, Norman (2015) Applications
More informationGephi Network Statistics
Gephi Network Statistics Google Summer of Code 2009 Project Proposal Patrick J. McSweeney pjmcswee@syr.edu 1 Introduction My name is Patrick J. McSweeney and I am a fourth year PhD candidate in computer
More informationMapReduce Algorithms. Sergei Vassilvitskii. Saturday, August 25, 12
MapReduce Algorithms A Sense of Scale At web scales... Mail: Billions of messages per day Search: Billions of searches per day Social: Billions of relationships 2 A Sense of Scale At web scales... Mail:
More informationMATHEMATICAL ENGINEERING TECHNICAL REPORTS. An Improved Approximation Algorithm for the Traveling Tournament Problem
MATHEMATICAL ENGINEERING TECHNICAL REPORTS An Improved Approximation Algorithm for the Traveling Tournament Problem Daisuke YAMAGUCHI, Shinji IMAHORI, Ryuhei MIYASHIRO, Tomomi MATSUI METR 2009 42 September
More informationComplex Networks Analysis: Clustering Methods
Complex Networks Analysis: Clustering Methods Nikolai Nefedov Spring 2013 ISI ETH Zurich nefedov@isi.ee.ethz.ch 1 Outline Purpose to give an overview of modern graphclustering methods and their applications
More informationCOLORED GRAPHS AND THEIR PROPERTIES
COLORED GRAPHS AND THEIR PROPERTIES BEN STEVENS 1. Introduction This paper is concerned with the upper bound on the chromatic number for graphs of maximum vertex degree under three different sets of coloring
More informationInet3.0: Internet Topology Generator
Inet3.: Internet Topology Generator Jared Winick Sugih Jamin {jwinick,jamin}@eecs.umich.edu CSETR4562 Abstract In this report we present version 3. of Inet, an Autonomous System (AS) level Internet
More informationOutline 2.1 Graph Isomorphism 2.2 Automorphisms and Symmetry 2.3 Subgraphs, part 1
GRAPH THEORY LECTURE STRUCTURE AND REPRESENTATION PART A Abstract. Chapter focuses on the question of when two graphs are to be regarded as the same, on symmetries, and on subgraphs.. discusses the concept
More information1. Relevant standard graph theory
Color identical pairs in 4chromatic graphs Asbjørn Brændeland I argue that, given a 4chromatic graph G and a pair of vertices {u, v} in G, if the color of u equals the color of v in every 4coloring
More informationDynamic Programming. Applies when the following Principle of Optimality
Dynamic Programming Applies when the following Principle of Optimality holds: In an optimal sequence of decisions or choices, each subsequence must be optimal. Translation: There s a recursive solution.
More information1 Plane and Planar Graphs. Definition 1 A graph G(V,E) is called plane if
Plane and Planar Graphs Definition A graph G(V,E) is called plane if V is a set of points in the plane; E is a set of curves in the plane such that. every curve contains at most two vertices and these
More informationMultilevel analysis of an interaction network between individuals in a mailinglist
Multilevel analysis of an interaction network between individuals in a mailinglist Rémi Dorat,2,3 Matthieu Latapy Bernard Conein 2 Nicolas Auray 3 Abstract It is well known now that most realworld complex
More informationTotal colorings of planar graphs with small maximum degree
Total colorings of planar graphs with small maximum degree Bing Wang 1,, JianLiang Wu, SiFeng Tian 1 Department of Mathematics, Zaozhuang University, Shandong, 77160, China School of Mathematics, Shandong
More informationLecture Notes on Spanning Trees
Lecture Notes on Spanning Trees 15122: Principles of Imperative Computation Frank Pfenning Lecture 26 April 26, 2011 1 Introduction In this lecture we introduce graphs. Graphs provide a uniform model
More informationLinear Algebra and its Applications
Linear Algebra and its Applications 438 2013) 1393 1397 Contents lists available at SciVerse ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa Note on the
More informationOption 1: empirical network analysis. Task: find data, analyze data (and visualize it), then interpret.
Programming project Task Option 1: empirical network analysis. Task: find data, analyze data (and visualize it), then interpret. Obtaining data This project focuses upon cocktail ingredients. Data was
More information