Graph theory and network analysis. Devika Subramanian Comp 140 Fall 2008

Size: px
Start display at page:

Download "Graph theory and network analysis. Devika Subramanian Comp 140 Fall 2008"

Transcription

1 Graph theory and network analysis Devika Subramanian Comp 140 Fall

2 The bridges of Konigsburg Source: Wikipedia The city of Königsberg in Prussia was set on both sides of the Pregel River, and included two large islands which were connected to each other and the mainland by seven bridges. Leonard Euler posed the following problem: can we find a walk through the city that crosses each bridge once and only once, and begins and ends at the same point? Rules: The islands cannot be reached by any route other than the bridges, and every bridge must have been crossed completely every time (one cannot walk halfway onto the bridge and then turn around to come at it from another side). 2

3 A schematic of the seven bridges problem C b1 b2 b3 b7 A B b4 b5 b6 D 3

4 First paper on graph theory Leonard Euler presented a solution to the St. Petersburg Academy on August 26, 1735 Solutio problematis ad geometriam situs pertinentis (The solution of a problem relating to the geometry of position), Commentarii academiae scientiarum Petropolitanae,

5 Abstract representation A b1 b4 b2 b5 C b7 D b3 b6 B 1. Only land masses and the bridges connecting them matter! 2. Shapes of land masses and lengths of bridges are not relevant. Relative distances between land masses also not relevant. 3. Topological connectivity is the only relevant aspect for solving the problem. 4. The structure shown alongside makes only the relevant factors of the problem explicit. 5

6 Euler s insight When one enters a land mass (that is not the start or the end of the tour) by a bridge, one leaves it by a bridge. If each bridge is to be traversed exactly once, then each land mass that is not the start or the end, needs to have an even number of bridges touching it. Land mass A has five bridges touching it, land masses B, C and D each have three bridges touching them. So a tour that starts and ends on any of these land masses and which crosses each bridge exactly once is not possible. 6

7 Elements of graph theory b1 b2 C b3 Land masses are vertices. Bridges are edges. The problem is represented as an undirected multi-graph. A b7 B The degree of a vertex is the number of edges on it. b4 b5 D b6 all vertexes in this problem have odd degree. Euler s insight: An Eulerian tour in a connected graph is possible only if all vertexes in it have even degree. 7

8 Some definitions A graph G is a pair of sets V and E V is a non-empty set of vertices E is a set of pairs of vertices V = {A,B,C,D,E,F} G={V,E} A B C E={{A,B},{A,D},{B,C},{B,E}, {C,D},{C,E},{E,F}} E F D 8

9 Subgraphs Deleting some vertices or edges from a graph leaves a subgraph. Formally, G =(V,E ) is a subgraph of G = (V,E) if V is a non-empty subset of V E is a subset of E 9

10 A computer scientist reads the paper A 1994 University of Chicago entitled The Social Organization of Sexuality found that on average men have 74% more opposite-gender partners than women. 10

11 Mapping to graph theory Men Women 11

12 Analysis Every edge in this graph connects an M vertex to a W vertex. So the sum of the degrees of the M vertices must equal the sum of the degrees of the W vertices. x M deg(x) = yinw deg(y) 12

13 Analysis contd. x M deg(x) M 1. W = y W deg(y) W 1. M Avg. deg in M Avg. deg in W = W M Avg. deg in M = W.Avg. deg in W M 13

14 Analysis contd. Census Bureau reports W / M is about Therefore, on average men have 3.5% more opposite-gender partners. The University of Chicago study has problematic data. The average number of opposite-gender partners is completely determined by W / M. 14

15 Graph variations Multigraph: more than one edge between a pair of vertices. Directed graph: edges have direction. the edges of a directed graph are ordered pairs of vertices. indegree of a vertex is the number of edges directed into a vertex. outdegree of a vertex is the number of edges directed out of a vertex. 15

16 Problems that map to graphs Social networks: nodes are people, edges represent the is-friends-with relation. Terrorist networks: nodes are terrorist groups/individuals, edges are participatedin-an-incident-with Conflict networks: nodes are countries, edges are cooperate-with or conflict-with 16

17 2 weeks prior to Desert Storm 17

18 The SHSU database A human curated database of global terrorist incidents from 1/22/1990 to 12/31/ ,199 incidents 1257 groups Very detailed information on incidents (e.g. weapons used, fatalities, etc) and some information on the groups. (c) Devika Subramanian

19 Pre-Bali network Palestine groups Kashmir groups Columbia Al Qaeda US terror groups (KKK etc) Irish groups Philippines, Indonesian groups Hamas (c) Devika Subramanian

20 Post Bali network Bangladesh Al Qaeda All the rest are fragments of networks from previous slide US environmental Terror groups Splintering of the terror network into smaller, more decentralized pieces (c) Devika Subramanian

21 More problems The web: each vertex is a page, directed edges between vertices represent hyperlinks Algorithm to compute hubs and authorities to determine page rank in Google Modeling the spread of infection in a community: vertices are people, and edges represent contact between them. Routing messages on the Internet: vertices are end hosts and routers, edges denote vertices that are directly linked. 21

V. Adamchik 1. Graph Theory. Victor Adamchik. Fall of 2005

V. 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 information

IE 680 Special Topics in Production Systems: Networks, Routing and Logistics*

IE 680 Special Topics in Production Systems: Networks, Routing and Logistics* IE 680 Special Topics in Production Systems: Networks, Routing and Logistics* Rakesh Nagi Department of Industrial Engineering University at Buffalo (SUNY) *Lecture notes from Network Flows by Ahuja, Magnanti

More information

Social Media Mining. Graph Essentials

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

More information

Euler, Mei-Ko Kwan, Königsberg, and a Chinese Postman

Euler, Mei-Ko Kwan, Königsberg, and a Chinese Postman Documenta Math. 43 Euler, Mei-Ko Kwan, Königsberg, and a Chinese Postman Martin Grötschel and Ya-xiang Yuan 2010 Mathematics Subject Classification: 00-02, 01A05, 05C38, 90-03 Keywords and Phrases: Eulerian

More information

Graph Theory. Euler tours and Chinese postmen. John Quinn. Week 5

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

More information

Graphs and Network Flows IE411 Lecture 1

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

More information

The origins of graph theory are humble, even frivolous. Biggs, E. K. Lloyd, and R. J. Wilson)

The origins of graph theory are humble, even frivolous. Biggs, E. K. Lloyd, and R. J. Wilson) Chapter 11 Graph Theory The origins of graph theory are humble, even frivolous. Biggs, E. K. Lloyd, and R. J. Wilson) (N. Let us start with a formal definition of what is a graph. Definition 72. A graph

More information

CSE 20: Discrete Mathematics for Computer Science. Prof. Miles Jones. Today s Topics: Graphs. The Internet graph

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!

More information

Euler Paths and Euler Circuits

Euler Paths and Euler Circuits Euler Paths and Euler Circuits An Euler path is a path that uses every edge of a graph exactly once. An Euler circuit is a circuit that uses every edge of a graph exactly once. An Euler path starts and

More information

Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs

Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs Kousha Etessami U. of Edinburgh, UK Kousha Etessami (U. of Edinburgh, UK) Discrete Mathematics (Chapter 6) 1 / 13 Overview Graphs and Graph

More information

Graph definition Degree, in, out degree, oriented graph. Complete, regular, bipartite graph. Graph representation, connectivity, adjacency.

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.

More information

Complex Networks Analysis: Clustering Methods

Complex 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 graph-clustering methods and their applications

More information

1 Basic Definitions and Concepts in Graph Theory

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

More information

Graph Theory Origin and Seven Bridges of Königsberg -Rhishikesh

Graph Theory Origin and Seven Bridges of Königsberg -Rhishikesh Graph Theory Origin and Seven Bridges of Königsberg -Rhishikesh Graph Theory: Graph theory can be defined as the study of graphs; Graphs are mathematical structures used to model pair-wise relations between

More information

Section Summary. Introduction to Graphs Graph Taxonomy Graph Models

Section Summary. Introduction to Graphs Graph Taxonomy Graph Models Chapter 10 Chapter Summary Graphs and Graph Models Graph Terminology and Special Types of Graphs Representing Graphs and Graph Isomorphism Connectivity Euler and Hamiltonian Graphs Shortest-Path Problems

More information

MGF 1107 CH 15 LECTURE NOTES Denson. Section 15.1

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.

More information

most influential member(s) of a social network; key infrastructure nodes; in an urban network; super-spreaders of disease;...

most influential member(s) of a social network; key infrastructure nodes; in an urban network; super-spreaders of disease;... Ranking in Networks http://en.wikipedia.org/wiki/centralityi Question: Given a communication network N, how to discover important nodes? How to define the importance of members of the network? The answer

More information

Course on Social Network Analysis Graphs and Networks

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...............................

More information

Planar Graph and Trees

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

More information

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

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

More information

About the Tutorial. Audience. Prerequisites. Disclaimer & Copyright

About the Tutorial. Audience. Prerequisites. Disclaimer & Copyright 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,

More information

CMPSCI611: Approximating MAX-CUT Lecture 20

CMPSCI611: Approximating MAX-CUT Lecture 20 CMPSCI611: Approximating MAX-CUT Lecture 20 For the next two lectures we ll be seeing examples of approximation algorithms for interesting NP-hard problems. Today we consider MAX-CUT, which we proved to

More information

Asking Hard Graph Questions. Paul Burkhardt. February 3, 2014

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!)

More information

GRAPHS Representation :

GRAPHS Representation : Graph consists of a non empty set of points called vertices and set of edges that link vertices. Speaking more formally: Definition: A graph G= (V, E) consists of a set V={v 1, v 2...,v n } of n >1 vertices

More information

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 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 information

Outline. NP-completeness. When is a problem easy? When is a problem hard? Today. Euler Circuits

Outline. NP-completeness. When is a problem easy? When is a problem hard? Today. Euler Circuits Outline NP-completeness Examples of Easy vs. Hard problems Euler circuit vs. Hamiltonian circuit Shortest Path vs. Longest Path 2-pairs sum vs. general Subset Sum Reducing one problem to another Clique

More information

Simple Graphs Degrees, Isomorphism, Paths

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

More information

Class One: Degree Sequences

Class One: Degree Sequences Class One: Degree Sequences For our purposes a graph is a just a bunch of points, called vertices, together with lines or curves, called edges, joining certain pairs of vertices. Three small examples of

More information

CMSC 451: Graph Properties, DFS, BFS, etc.

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

More information

/ Approximation Algorithms Lecturer: Michael Dinitz Topic: Steiner Tree and TSP Date: 01/29/15 Scribe: Katie Henry

/ 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 information

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

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

More information

Long questions answer Advanced Mathematics for Computer Application If P= , find BT. 19. If B = 1 0, find 2B and -3B.

Long questions answer Advanced Mathematics for Computer Application If P= , find BT. 19. If B = 1 0, find 2B and -3B. Unit-1: Matrix Algebra Short questions answer 1. What is Matrix? 2. Define the following terms : a) Elements matrix b) Row matrix c) Column matrix d) Diagonal matrix e) Scalar matrix f) Unit matrix OR

More information

136 CHAPTER 4. INDUCTION, GRAPHS AND TREES

136 CHAPTER 4. INDUCTION, GRAPHS AND TREES 136 TER 4. INDUCTION, GRHS ND TREES 4.3 Graphs In this chapter we introduce a fundamental structural idea of discrete mathematics, that of a graph. Many situations in the applications of discrete mathematics

More information

Data Structures in Java. Session 16 Instructor: Bert Huang

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

More information

Graph Theory and Complex Networks: An Introduction. Chapter 08: Computer networks

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

More information

Brain Teasers 2 Teacher Resources Classroom Activities. Math Puzzles

Brain Teasers 2 Teacher Resources Classroom Activities. Math Puzzles Brain Teasers 2 Teacher Resources Classroom Activities Math Puzzles Teacher s Notes: Like the puzzles "Four Equations" and "Disorder" in Brain Teasers 2, these puzzles help students develop the ability

More information

Introduction to Networks and Business Intelligence

Introduction to Networks and Business Intelligence Introduction to Networks and Business Intelligence Prof. Dr. Daning Hu Department of Informatics University of Zurich Sep 17th, 2015 Outline Network Science A Random History Network Analysis Network Topological

More information

2.3 Scheduling jobs on identical parallel machines

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

More information

Network Analysis Basics and applications to online data

Network Analysis Basics and applications to online data Network Analysis Basics and applications to online data Katherine Ognyanova University of Southern California Prepared for the Annenberg Program for Online Communities, 2010. Relational data Node (actor,

More information

Graph Theory for Articulated Bodies

Graph Theory for Articulated Bodies Graph Theory for Articulated Bodies Alba Perez-Gracia Department of Mechanical Engineering, Idaho State University Articulated Bodies A set of rigid bodies (links) joined by joints that allow relative

More information

Extremal Wiener Index of Trees with All Degrees Odd

Extremal Wiener Index of Trees with All Degrees Odd MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 70 (2013) 287-292 ISSN 0340-6253 Extremal Wiener Index of Trees with All Degrees Odd Hong Lin School of

More information

Basic Notions on Graphs. Planar Graphs and Vertex Colourings. Joe Ryan. Presented by

Basic Notions on Graphs. Planar Graphs and Vertex Colourings. Joe Ryan. Presented by Basic Notions on Graphs Planar Graphs and Vertex Colourings Presented by Joe Ryan School of Electrical Engineering and Computer Science University of Newcastle, Australia Planar graphs Graphs may be drawn

More information

Minimum Spanning Trees

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

More information

Graph. Consider a graph, G in Fig Then the vertex V and edge E can be represented as:

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

More information

Homework 15 Solutions

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

More information

Some questions... Graphs

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

More information

Networks and Paths. The study of networks in mathematics began in the middle 1700 s with a famous puzzle called the Seven Bridges of Konigsburg.

Networks and Paths. The study of networks in mathematics began in the middle 1700 s with a famous puzzle called the Seven Bridges of Konigsburg. ame: Day: etworks and Paths Try This: For each figure,, and, draw a path that traces every line and curve exactly once, without lifting your pencil.... Figures,, and above are examples of ETWORKS. network

More information

General Network Analysis: Graph-theoretic. COMP572 Fall 2009

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

More information

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

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

More information

Chapter 4. Trees. 4.1 Basics

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.

More information

Homework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS

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

More information

DATA ANALYSIS II. Matrix Algorithms

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

More information

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. 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 information

Social Network Mining

Social Network Mining Social Network Mining Data Mining November 11, 2013 Frank Takes (ftakes@liacs.nl) LIACS, Universiteit Leiden Overview Social Network Analysis Graph Mining Online Social Networks Friendship Graph Semantics

More information

GRAPH THEORY and APPLICATIONS. Trees

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.

More information

STATE SPACES & BLIND SEARCH. M. Anthony Kapolka III Wilkes University CS Online

STATE SPACES & BLIND SEARCH. M. Anthony Kapolka III Wilkes University CS Online STATE SPACES & BLIND SEARCH M. Anthony Kapolka III Wilkes University CS 340 - Online Issues in Search Knowledge Representation Topology (of State Space) Algorithms Forward or Backward The Farmer, Fox,

More information

Graph/Network Visualization

Graph/Network Visualization Graph/Network Visualization Data model: graph structures (relations, knowledge) and networks. Applications: Telecommunication systems, Internet and WWW, Retailers distribution networks knowledge representation

More information

Graphs. Graph G=(V,E): representation

Graphs. Graph G=(V,E): representation Graphs G = (V,E) V the vertices of the graph {v 1, v 2,..., v n } E the edges; E a subset of V x V A cost function cij is the cost/ weight of the edge (v i, v j ) Graph G=(V,E): representation 1. Adjacency

More information

6.042/18.062J Mathematics for Computer Science October 3, 2006 Tom Leighton and Ronitt Rubinfeld. Graph Theory III

6.042/18.062J Mathematics for Computer Science October 3, 2006 Tom Leighton and Ronitt Rubinfeld. Graph Theory III 6.04/8.06J Mathematics for Computer Science October 3, 006 Tom Leighton and Ronitt Rubinfeld Lecture Notes Graph Theory III Draft: please check back in a couple of days for a modified version of these

More information

Graph Theory Problems and Solutions

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

More information

Systems and Algorithms for Big Data Analytics

Systems and Algorithms for Big Data Analytics Systems and Algorithms for Big Data Analytics YAN, Da Email: yanda@cse.cuhk.edu.hk My Research Graph Data Distributed Graph Processing Spatial Data Spatial Query Processing Uncertain Data Querying & Mining

More information

Distributed Computing over Communication Networks: Maximal Independent Set

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.

More information

Theorem A graph T is a tree if, and only if, every two distinct vertices of T are joined by a unique path.

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

More information

GRAPHS. Definitions. The Graph ADT. Data structures for graphs PVD LAX STL HNL DFW FTL. Graphs

GRAPHS. Definitions. The Graph ADT. Data structures for graphs PVD LAX STL HNL DFW FTL. Graphs GRAPHS Definitions The Graph ADT Data structures for graphs PVD HNL LAX DFW STL FTL 1 What is a Graph? Agraph G = (V,E) is composed of: V: set of vertices E: set of edges connecting the vertices in V Anedge

More information

Practical Graph Mining with R. 5. Link Analysis

Practical 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 information

Coloring Eulerian triangulations of the projective plane

Coloring Eulerian triangulations of the projective plane Coloring Eulerian triangulations of the projective plane Bojan Mohar 1 Department of Mathematics, University of Ljubljana, 1111 Ljubljana, Slovenia bojan.mohar@uni-lj.si Abstract A simple characterization

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) E6893 Big Data Analytics Lecture 10: Linked Big Data Graph Computing (II) Ching-Yung Lin, Ph.D. Adjunct Professor, Dept. of Electrical Engineering and Computer Science Mgr., Dept. of Network Science and

More information

SCAN: A Structural Clustering Algorithm for Networks

SCAN: A Structural Clustering Algorithm for Networks SCAN: A Structural Clustering Algorithm for Networks Xiaowei Xu, Nurcan Yuruk, Zhidan Feng (University of Arkansas at Little Rock) Thomas A. J. Schweiger (Acxiom Corporation) Networks scaling: #edges connected

More information

THE PROBLEM WORMS (1) WORMS (2) THE PROBLEM OF WORM PROPAGATION/PREVENTION THE MINIMUM VERTEX COVER PROBLEM

THE PROBLEM WORMS (1) WORMS (2) THE PROBLEM OF WORM PROPAGATION/PREVENTION THE MINIMUM VERTEX COVER PROBLEM 1 THE PROBLEM OF WORM PROPAGATION/PREVENTION I.E. THE MINIMUM VERTEX COVER PROBLEM Prof. Tiziana Calamoneri Network Algorithms A.y. 2014/15 2 THE PROBLEM WORMS (1)! A computer worm is a standalone malware

More information

Introduction to Graph Theory

Introduction to Graph Theory Introduction to Graph Theory Allen Dickson October 2006 1 The Königsberg Bridge Problem The city of Königsberg was located on the Pregel river in Prussia. The river divided the city into four separate

More information

Planarity Planarity

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?

More information

Max Flow, Min Cut, and Matchings (Solution)

Max Flow, Min Cut, and Matchings (Solution) Max Flow, Min Cut, and Matchings (Solution) 1. The figure below shows a flow network on which an s-t flow is shown. The capacity of each edge appears as a label next to the edge, and the numbers in boxes

More information

Lecture Notes on GRAPH THEORY Tero Harju

Lecture Notes on GRAPH THEORY Tero Harju Lecture Notes on GRAPH THEORY Tero Harju Department of Mathematics University of Turku FIN-20014 Turku, Finland e-mail: harju@utu.fi 1994 2011 Contents 1 Introduction..........................................................

More information

Sum of Degrees of Vertices Theorem

Sum of Degrees of Vertices Theorem Sum of Degrees of Vertices Theorem Theorem (Sum of Degrees of Vertices Theorem) Suppose a graph has n vertices with degrees d 1, d 2, d 3,...,d n. Add together all degrees to get a new number d 1 + d 2

More information

Graph Theory: Penn State Math 485 Lecture Notes. Christopher Griffin 2011-2012

Graph Theory: Penn State Math 485 Lecture Notes. Christopher Griffin 2011-2012 Graph Theory: Penn State Math 485 Lecture Notes Version 1.4..1 Christopher Griffin 011-01 Licensed under a Creative Commons Attribution-Noncommercial-Share Alike.0 United States License With Contributions

More information

USE OF EIGENVALUES AND EIGENVECTORS TO ANALYZE BIPARTIVITY OF NETWORK GRAPHS

USE OF EIGENVALUES AND EIGENVECTORS TO ANALYZE BIPARTIVITY OF NETWORK GRAPHS USE OF EIGENVALUES AND EIGENVECTORS TO ANALYZE BIPARTIVITY OF NETWORK GRAPHS Natarajan Meghanathan Jackson State University, 1400 Lynch St, Jackson, MS, USA natarajan.meghanathan@jsums.edu ABSTRACT This

More information

Graph Theory Lecture 3: Sum of Degrees Formulas, Planar Graphs, and Euler s Theorem Spring 2014 Morgan Schreffler Office: POT 902

Graph Theory Lecture 3: Sum of Degrees Formulas, Planar Graphs, and Euler s Theorem Spring 2014 Morgan Schreffler Office: POT 902 Graph Theory Lecture 3: Sum of Degrees Formulas, Planar Graphs, and Euler s Theorem Spring 2014 Morgan Schreffler Office: POT 902 http://www.ms.uky.edu/~mschreffler Different Graphs, Similar Properties

More information

Social Media Mining. Network Measures

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

More information

Random graphs with a given degree sequence

Random 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 information

NP-Completeness. CptS 223 Advanced Data Structures. Larry Holder School of Electrical Engineering and Computer Science Washington State University

NP-Completeness. CptS 223 Advanced Data Structures. Larry Holder School of Electrical Engineering and Computer Science Washington State University NP-Completeness CptS 223 Advanced Data Structures Larry Holder School of Electrical Engineering and Computer Science Washington State University 1 Hard Graph Problems Hard means no known solutions with

More information

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

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

More information

In the following we will only consider undirected networks.

In the following we will only consider undirected networks. Roles in Networks Roles in Networks Motivation for work: Let topology define network roles. Work by Kleinberg on directed graphs, used topology to define two types of roles: authorities and hubs. (Each

More information

Types of Degrees in Bipolar Fuzzy Graphs

Types of Degrees in Bipolar Fuzzy Graphs pplied Mathematical Sciences, Vol. 7, 2013, no. 98, 4857-4866 HIKRI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2013.37389 Types of Degrees in Bipolar Fuzzy Graphs Basheer hamed Mohideen Department

More information

Protein Protein Interaction Networks

Protein Protein Interaction Networks Functional Pattern Mining from Genome Scale Protein Protein Interaction Networks Young-Rae Cho, Ph.D. Assistant Professor Department of Computer Science Baylor University it My Definition of Bioinformatics

More information

Midterm Practice Problems

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

More information

Lecture notes from Foundations of Markov chain Monte Carlo methods University of Chicago, Spring 2002 Lecture 1, March 29, 2002

Lecture notes from Foundations of Markov chain Monte Carlo methods University of Chicago, Spring 2002 Lecture 1, March 29, 2002 Lecture notes from Foundations of Markov chain Monte Carlo methods University of Chicago, Spring 2002 Lecture 1, March 29, 2002 Eric Vigoda Scribe: Varsha Dani & Tom Hayes 1.1 Introduction The aim of this

More information

3. Eulerian and Hamiltonian Graphs

3. Eulerian and Hamiltonian Graphs 3. Eulerian and Hamiltonian Graphs There are many games and puzzles which can be analysed by graph theoretic concepts. In fact, the two early discoveries which led to the existence of graphs arose from

More information

Chapter 1. Introduction

Chapter 1. Introduction Chapter 1 Introduction Intermodal freight transportation describes the movement of goods in standardized loading units (e.g., containers) by at least two transportation modes (rail, maritime, and road)

More information

SPANNING CACTI FOR STRUCTURALLY CONTROLLABLE NETWORKS NGO THI TU ANH NATIONAL UNIVERSITY OF SINGAPORE

SPANNING CACTI FOR STRUCTURALLY CONTROLLABLE NETWORKS NGO THI TU ANH NATIONAL UNIVERSITY OF SINGAPORE SPANNING CACTI FOR STRUCTURALLY CONTROLLABLE NETWORKS NGO THI TU ANH NATIONAL UNIVERSITY OF SINGAPORE 2012 SPANNING CACTI FOR STRUCTURALLY CONTROLLABLE NETWORKS NGO THI TU ANH (M.Sc., SFU, Russia) A THESIS

More information

Chapter 6: Graph Theory

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.

More information

Network Analysis and Visualization of Staphylococcus aureus. by Russ Gibson

Network Analysis and Visualization of Staphylococcus aureus. by Russ Gibson Network Analysis and Visualization of Staphylococcus aureus by Russ Gibson Network analysis Based on graph theory Probabilistic models (random graphs) developed by Erdős and Rényi in 1959 Theory and tools

More information

Network Analysis. BCH 5101: Analysis of -Omics Data 1/34

Network Analysis. BCH 5101: Analysis of -Omics Data 1/34 Network Analysis BCH 5101: Analysis of -Omics Data 1/34 Network Analysis Graphs as a representation of networks Examples of genome-scale graphs Statistical properties of genome-scale graphs The search

More information

NodeXL for Network analysis Demo/hands-on at NICAR 2012, St Louis, Feb 24. Peter Aldhous, San Francisco Bureau Chief. peter@peteraldhous.

NodeXL for Network analysis Demo/hands-on at NICAR 2012, St Louis, Feb 24. Peter Aldhous, San Francisco Bureau Chief. peter@peteraldhous. NodeXL for Network analysis Demo/hands-on at NICAR 2012, St Louis, Feb 24 Peter Aldhous, San Francisco Bureau Chief peter@peteraldhous.com NodeXL is a template for Microsoft Excel 2007 and 2010, which

More information

Chinese postman problem

Chinese postman problem PTR hinese postman problem Learning objectives fter studying this chapter, you should be able to: understand the hinese postman problem apply an algorithm to solve the problem understand the importance

More information

Lecture 4: The Chromatic Number

Lecture 4: The Chromatic Number Introduction to Graph Theory Instructor: Padraic Bartlett Lecture 4: The Chromatic Number Week 1 Mathcamp 2011 In our discussion of bipartite graphs, we mentioned that one way to classify bipartite graphs

More information

Topological Properties

Topological Properties Advanced Computer Architecture Topological Properties Routing Distance: Number of links on route Node degree: Number of channels per node Network diameter: Longest minimum routing distance between any

More information

Million Dollar Mathematics!

Million Dollar Mathematics! Million Dollar Mathematics! Alissa S. Crans Loyola Marymount University Southern California Undergraduate Math Day University of California, San Diego April 30, 2011 This image is from the Wikipedia article

More information

COT5405 Analysis of Algorithms Homework 3 Solutions

COT5405 Analysis of Algorithms Homework 3 Solutions COT0 Analysis of Algorithms Homework 3 Solutions. Prove or give a counter example: (a) In the textbook, we have two routines for graph traversal - DFS(G) and BFS(G,s) - where G is a graph and s is any

More information

Data Structure [Question Bank]

Data Structure [Question Bank] Unit I (Analysis of Algorithms) 1. What are algorithms and how they are useful? 2. Describe the factor on best algorithms depends on? 3. Differentiate: Correct & Incorrect Algorithms? 4. Write short note:

More information

12 Abstract Data Types

12 Abstract Data Types 12 Abstract Data Types 12.1 Source: Foundations of Computer Science Cengage Learning Objectives After studying this chapter, the student should be able to: Define the concept of an abstract data type (ADT).

More information