# Graph Theory and Complex Networks: An Introduction. Chapter 06: Network analysis. Contents. Introduction. Maarten van Steen. Version: April 28, 2014

Save this PDF as:

Size: px
Start display at page:

Download "Graph Theory and Complex Networks: An Introduction. Chapter 06: Network analysis. Contents. Introduction. Maarten van Steen. Version: April 28, 2014"

## Transcription

1 Graph Theory and Complex Networks: An Introduction Maarten van Steen VU Amsterdam, Dept. Computer Science Room R.0, Chapter 0: Version: April 8, 0 / Contents Chapter Description 0: Introduction History, background 0: Foundations Basic terminology and properties of graphs 0: Extensions Directed & weighted graphs, colorings 0: Network traversal Walking through graphs (cf. traveling) 0: Trees Graphs without cycles; routing algorithms 0: Basic metrics for analyzing large graphs 07: Random networks Introduction modeling real-world networks 08: Computer networks The Internet & WWW seen as a huge graph 09: Social networks Communities seen as graphs / / Introduction Observation In real-world 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? / /

2 . Vertex degree. Vertex degree Vertex degree Question Can you visually observe real (nonisomorphic) differences? / /. Vertex degree. Vertex degree Vertex degree: Histogram (a): (b): n = 00, m = 00 n = 00, m = (a) (b) / /. Vertex degree. Vertex degree Vertex degree: Ranked histogram / /

3 . Distance statistics. Distance statistics Distance statistics 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. 7 / 7 /. Distance statistics. Distance statistics Path lengths G is connected with vertex V ; d(u) is average length of shortest paths from u to any other vertex v: The average path length d(g): d(u) def = V d(u, v) v V,v u d(g) def = V d(u) = V u V V d(u, v) u,v V,u v 8 / 8 /. Distance statistics. Distance statistics Path lengths 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 equally-sized subsets. Example {,,,,0,,} [0,,,,,,] M = x (7+)/ = x = 9 / 9 /

4 . Distance statistics. Distance statistics Example distance statistics 7 Vertex 7 ε(u) v u d(u,v) d(u) / 0 / 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? / / Clustering coefficient 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 / /

5 Clustering coefficient 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) / / Clustering coefficient: triangles A triangle is a complete (sub)graph with exactly vertices. A triple is a (sub)graph with exactly vertices and edges. 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. / / Clustering coefficient: example 7 Vertex: 7 cc: / 0 / undefined / n Λ : 0 Vertex N() = {,,7}; E(G[N()]) =,7 cc() = Triples at : G[{,,}],G[{,,7}],G[{,,7}] / /

6 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) = v V n (v) (Note: V = {v V δ(v) > }) / / Clustering coefficient versus transitivity x x x v v v... v n v k v v v... v n y G k = G[{x,y,v,v,...,v k }] : cc(u) = if u = v,...,v k k ( k+ ) = k k(k+) = k+ if u = x or u = y y CC(G k ) = k + ( k + + k ) = k + k + k + k + lim k CC(G k) = y 7 / 7 / Clustering coefficient versus transitivity x v v v... v n G k = G[{x,y,v,v,...,v k }] { if u = v,...,v k n Λ (u) = ) ( = k+ ) if u = x,y τ(g k ) = n (G k ) n Λ (u) = ( δ(u) y k = k(k + ) + k k + lim τ(g k) = 0 k 8 / 8 /

7 Centrality Issue Are there any vertices that are more important than the others? G is (strongly) connected. The center C(G) is the set of vertices with minimal eccentricity: C(G) def = {v V (G) ε(v) = rad(g)} Intuition At the center means at minimal distance to the farthest node. 9 / 9 / Vertex centrality 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. 0 / 0 / Closeness 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? / /

8 Centrality: example 7 Vertex: 7 ε(u) d(u, ) c C (u): / / Betweenness Intuition Important vertices are those whose removal significantly increases the distance between other vertices. Example: cut vertices. 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 S(x,u,y) = x y u S(x, y) / /

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

Graph Theory and Complex Networks: An Introduction Maarten van Steen VU Amsterdam, Dept. Computer Science Room R4.0, steen@cs.vu.nl Chapter 06: Network analysis Version: April 8, 04 / 3 Contents Chapter

### Social 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

### Graph 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

### Graph 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

### Graphs 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

### Graph 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.

### Definition. 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

### Solutions 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.

### Part 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 -

### GRAPH 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.

### Course 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...............................

### Social 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 like-minded users

### CMSC 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

### Chapter 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

### Distance 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

### 1 Digraphs. Definition 1

1 Digraphs Definition 1 Adigraphordirected graphgisatriplecomprisedofavertex set V(G), edge set E(G), and a function assigning each edge an ordered pair of vertices (tail, head); these vertices together

### Minimum 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

### Homework 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

### A 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, 6-70 ISSN: 3-8080 (printed version); ISSN: 34-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/0.73/ijpam.v03i.5

### Asking 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!)

### Graph 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 edge-connectivity of a graph Blocks Kruskal s algorithm Königsberg, Prussia The Seven Bridges of Königsberg

### Homework 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

### GRAPH 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

### BOUNDARY EDGE DOMINATION IN GRAPHS

BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 0-4874, ISSN (o) 0-4955 www.imvibl.org /JOURNALS / BULLETIN Vol. 5(015), 197-04 Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA

### On 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

### Trees 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

### Chapter 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.

### The TSP is a special case of VRP, which means VRP is NP-hard.

49 6. ROUTING PROBLEMS 6.1. VEHICLE ROUTING PROBLEMS Vehicle Routing Problem, VRP: Customers i=1,...,n with demands of a product must be served using a fleet of vehicles for the deliveries. The vehicles,

### CSE 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!

### Some questions... Graphs

Uni Innsbruck Informatik - 1 Uni Innsbruck Informatik - 2 Some questions... Peer-to to-peer Systems Analysis of unstructured P2P systems How scalable is Gnutella? How robust is Gnutella? Why does FreeNet

### Midterm 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

### Lecture 9. 1 Introduction. 2 Random Walks in Graphs. 1.1 How To Explore a Graph? CS-621 Theory Gems October 17, 2012

CS-62 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

### CIS 700: algorithms for Big Data

CIS 700: algorithms for Big Data Lecture 6: Graph Sketching Slides at http://grigory.us/big-data-class.html Grigory Yaroslavtsev http://grigory.us Sketching Graphs? We know how to sketch vectors: v Mv

### 2.3 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

### Graph. 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

### On 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

### Linear 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

### MGF 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.

### Simple Graphs Degrees, Isomorphism, Paths

Mathematics for Computer Science MIT 6.042J/18.062J Simple Graphs Degrees, Isomorphism, Types of Graphs Simple Graph this week Multi-Graph Directed Graph next week Albert R Meyer, March 10, 2010 lec 6W.1

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

### Graph 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

### What cannot be measured on the Internet? Yvonne-Anne Pignolet, Stefan Schmid, G. Trédan. Misleading stars

: What cannot be measured on the Internet? Yvonne-Anne Pignolet, Stefan Schmid, Gilles Tredan How accurate are network maps? Why? To develop/adapt protocols to Internet PaDIS, RMTP To understand the impact

### Extracting 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,

### DO NOT RE-DISTRIBUTE 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

### On minimal forbidden subgraphs for the class of EDM-graphs

Also available at http://amc-journaleu ISSN 8-966 printed edn, ISSN 8-97 electronic edn ARS MATHEMATICA CONTEMPORANEA 9 0 6 On minimal forbidden subgraphs for the class of EDM-graphs Gašper Jaklič FMF

### CSL851: Algorithmic Graph Theory Semester I Lecture 1: July 24

CSL851: Algorithmic Graph Theory Semester I 2013-2014 Lecture 1: July 24 Lecturer: Naveen Garg Scribes: Suyash Roongta Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer: These notes have

### Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010. Chapter 7: Digraphs

MCS-236: 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

### / 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

### Chapter 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.

### Planarity 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?

### Graph 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

### A 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 Scale-Free Networks Erdös-Rényi model cover only parts

### Small 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

### Introduction 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

### Graph 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

### SEMITOTAL AND TOTAL BLOCK-CUTVERTEX GRAPH

CHAPTER 3 SEMITOTAL AND TOTAL BLOCK-CUTVERTEX 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 B-radius

### Outline 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

### Planar 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

### A maximum degree theorem for diameter-2-critical graphs

Cent. Eur. J. Math. 1(1) 01 188-1889 DOI: 10.78/s11533-01-09-3 Central European Journal of Mathematics A maximum degree theorem for diameter--critical graphs Research Article Teresa W. Haynes 1,, Michael

### COLORED 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

### 2. (a) Explain the strassen s matrix multiplication. (b) Write deletion algorithm, of Binary search tree. [8+8]

Code No: R05220502 Set No. 1 1. (a) Describe the performance analysis in detail. (b) Show that f 1 (n)+f 2 (n) = 0(max(g 1 (n), g 2 (n)) where f 1 (n) = 0(g 1 (n)) and f 2 (n) = 0(g 2 (n)). [8+8] 2. (a)

### Multi-level analysis of an interaction network between individuals in a mailing-list

2050-Her/Teleco 62/3-4 14/03/07 13:48 Page 320 320 pp. 320-344 Multi-level analysis of an interaction network between individuals in a mailing-list Rémi DORAT 1, 2, 3 Matthieu LATAPY 1, Bernard CONEIN

### B490 Mining the Big Data. 2 Clustering

B490 Mining the Big Data 2 Clustering Qin Zhang 1-1 Motivations Group together similar documents/webpages/images/people/proteins/products One of the most important problems in machine learning, pattern

About the Tutorial This tutorial offers a brief introduction to the fundamentals of graph theory. Written in a reader-friendly style, it covers the types of graphs, their properties, trees, graph traversability,

### Connectivity and cuts

Math 104, Graph Theory February 19, 2013 Measure of connectivity How connected are each of these graphs? > increasing connectivity > I G 1 is a tree, so it is a connected graph w/minimum # of edges. Every

### Recap. 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

### Graph Algorithms using Map-Reduce

Graph Algorithms using Map-Reduce Graphs are ubiquitous in modern society. Some examples: The hyperlink structure of the web 1/7 Graph Algorithms using Map-Reduce Graphs are ubiquitous in modern society.

### Combinatorics: 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

### Approximation Algorithms. Scheduling. Approximation algorithms. Scheduling jobs on a single machine

Approximation algorithms Approximation Algorithms Fast. Cheap. Reliable. Choose two. NP-hard problems: choose 2 of optimal polynomial time all instances Approximation algorithms. Trade-off between time

### 1. Relevant standard graph theory

Color identical pairs in 4-chromatic graphs Asbjørn Brændeland I argue that, given a 4-chromatic graph G and a pair of vertices {u, v} in G, if the color of u equals the color of v in every 4-coloring

### Planar Graph and Trees

Dr. Nahid Sultana December 16, 2012 Tree Spanning Trees Minimum Spanning Trees Maps and Regions Eulers Formula Nonplanar graph Dual Maps and the Four Color Theorem Tree Spanning Trees Minimum Spanning

### CSE 373 Analysis of Algorithms Date: Nov. 1, Solution to HW3. Total Points: 100

CSE 373 Analysis of Algorithms Date: Nov. 1, 2016 Steven Skiena Problem 1 5-4 [5] Solution to HW3 Total Points: 100 Let T be the BFS-tree of the graph G. For any e in G and e T, we have to show that e

### Theorem 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

### A 2-factor in which each cycle has long length in claw-free graphs

A -factor in which each cycle has long length in claw-free graphs Roman Čada Shuya Chiba Kiyoshi Yoshimoto 3 Department of Mathematics University of West Bohemia and Institute of Theoretical Computer Science

### Why 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

### General Network Analysis: Graph-theoretic. COMP572 Fall 2009

General Network Analysis: Graph-theoretic 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

### Sample Problems in Discrete Mathematics

Sample Problems in Discrete Mathematics This handout lists some sample problems that you should be able to solve as a pre-requisite to Computer Algorithms Try to solve all of them You should also read

### Multi-level analysis of an interaction network between individuals in a mailing-list

Multi-level analysis of an interaction network between individuals in a mailing-list Rémi Dorat,2,3 Matthieu Latapy Bernard Conein 2 Nicolas Auray 3 Abstract It is well known now that most real-world complex

### Network/Graph Theory. What is a Network? What is network theory? Graph-based representations. Friendship Network. What makes a problem graph-like?

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 Graph-based representations Representing a problem

### Zachary 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

### Lesson 3. Algebraic graph theory. Sergio Barbarossa. Rome - February 2010

Lesson 3 Algebraic graph theory Sergio Barbarossa Basic notions Definition: A directed graph (or digraph) composed by a set of vertices and a set of edges We adopt the convention that the information flows

### USING SPECTRAL RADIUS RATIO FOR NODE DEGREE TO ANALYZE THE EVOLUTION OF SCALE- FREE NETWORKS AND SMALL-WORLD NETWORKS

USING SPECTRAL RADIUS RATIO FOR NODE DEGREE TO ANALYZE THE EVOLUTION OF SCALE- FREE NETWORKS AND SMALL-WORLD NETWORKS Natarajan Meghanathan Jackson State University, 1400 Lynch St, Jackson, MS, USA natarajan.meghanathan@jsums.edu

### Lecture Notes on Spanning Trees

Lecture Notes on Spanning Trees 15-122: Principles of Imperative Computation Frank Pfenning Lecture 26 April 26, 2011 1 Introduction In this lecture we introduce graphs. Graphs provide a uniform model

### Tracers Placement for IP Traceback against DDoS Attacks

Tracers Placement for IP Traceback against DDoS Attacks Chun-Hsin Wang, Chang-Wu Yu, Chiu-Kuo Liang, Kun-Min Yu, Wen Ouyang, Ching-Hsien Hsu, and Yu-Guang Chen Department of Computer Science and Information

### Finding 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

### Mining 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

### 1 Basic Definitions and Concepts in Graph Theory

CME 305: Discrete Mathematics and Algorithms 1 Basic Definitions and Concepts in Graph Theory A graph G(V, E) is a set V of vertices and a set E of edges. In an undirected graph, an edge is an unordered

### MapReduce 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:

### Data 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

### Graph 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

### Frans 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

### Distributed 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.

### Dynamic 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.

### Math 4707: Introduction to Combinatorics and Graph Theory

Math 4707: Introduction to Combinatorics and Graph Theory Lecture Addendum, November 3rd and 8th, 200 Counting Closed Walks and Spanning Trees in Graphs via Linear Algebra and Matrices Adjacency Matrices

### DATA ANALYSIS II. Matrix Algorithms

DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where

### Math 443/543 Graph Theory Notes 4: Connector Problems

Math 443/543 Graph Theory Notes 4: Connector Problems David Glickenstein September 19, 2012 1 Trees and the Minimal Connector Problem Here is the problem: Suppose we have a collection of cities which we

### Total colorings of planar graphs with small maximum degree

Total colorings of planar graphs with small maximum degree Bing Wang 1,, Jian-Liang Wu, Si-Feng Tian 1 Department of Mathematics, Zaozhuang University, Shandong, 77160, China School of Mathematics, Shandong