2010-03-29 851-0585-04L – Modelling and Simulating Social Systems with MATLAB Lesson 6 – Graphs (Networks) Anders Johansson and Wenjian Yu (with S. Lozano.

Slides:

Advertisements

Similar presentations

CSE 211 Discrete Mathematics

Advertisements

22C:19 Discrete Math Graphs Fall 2010 Sukumar Ghosh.

22C:19 Discrete Math Graphs Fall 2014 Sukumar Ghosh.

CSE 5243 (AU 14) Graph Basics and a Gentle Introduction to PageRank 1.

Section 14.1 Intro to Graph Theory. Beginnings of Graph Theory Euler’s Konigsberg Bridge Problem (18 th c.)  Can one walk through town and cross all.

Social Network Analysis and Its Applications By Paul Rossman Indiana University of Pennsylvania.

CS 206 Introduction to Computer Science II 03 / 27 / 2009 Instructor: Michael Eckmann.

Management Science 461 Lecture 2b – Shortest Paths September 16, 2008.

Graph Theory. What is Graph Theory? This is the study of structures called ‘graphs’. These graphs are simply a collection of points called ‘vertices’

Mining and Searching Massive Graphs (Networks)

CS 728 Lecture 4 It’s a Small World on the Web. Small World Networks It is a ‘small world’ after all –Billions of people on Earth, yet every pair separated.

Centrality Measures These measure a nodes importance or prominence in the network. The more central a node is in a network the more significant it is to.

Chapter 9 Graph algorithms Lec 21 Dec 1, Sample Graph Problems Path problems. Connectedness problems. Spanning tree problems.

CSE 321 Discrete Structures Winter 2008 Lecture 25 Graph Theory.

How is this going to make us 100K Applications of Graph Theory.

Chapter 11 Graphs and Trees This handout: Terminology of Graphs Eulerian Cycles.

22C:19 Discrete Math Graphs Spring 2014 Sukumar Ghosh.

GRAPH Learning Outcomes Students should be able to:

Chapter 2 Graph Algorithms.

Class 2: Graph theory and basic terminology Learning the language Network Science: Graph Theory 2012 Prof. Albert-László Barabási Dr. Baruch Barzel, Dr.

Graph Theory Topics to be covered:

Social Network Analysis: A Non- Technical Introduction José Luis Molina Universitat Autònoma de Barcelona

Robustness of complex networks English workshop on LEAP May 8, 2012 Kyushu Univ. Yuki YOSHIDA Dept. AEEA. Kyushu Univ.

Structures 7 Decision Maths: Graph Theory, Networks and Algorithms.

COM1721: Freshman Honors Seminar A Random Walk Through Computing Lecture 2: Structure of the Web October 1, 2002.

Vertices and Edges Introduction to Graphs and Networks Mills College Spring 2012.

CSE 326: Data Structures NP Completeness Ben Lerner Summer 2007.

Connected Components Fun with graphs, searching, and queues.

Professor Yashar Ganjali Department of Computer Science University of Toronto

Graphs Topic 21 " Hopefully, you've played around a bit with The Oracle of Bacon at Virginia and discovered how few steps are necessary to link just about.

Data Structures & Algorithms Graphs

Complex Networks: Models Lecture 2 Slides by Panayiotis TsaparasPanayiotis Tsaparas.

The Structure of the Web. Getting to knowing the Web How big is the web and how do you measure it? How many people use the web? How many use search engines?

Topics Paths and Circuits (11.2) A B C D E F G.

Graphs Edge(arc) Vertices can be even or odd or undirected (two-way) Edges can be directed (one-way) This graph is connected. Degree(order) = 3 Odd vertex.

1.5 Graph Theory. Graph Theory The Branch of mathematics in which graphs and networks are used to solve problems.

Discrete Mathematical Structures: Theory and Applications

Graphs and 2-Way Bounding Discrete Structures (CS 173) Madhusudan Parthasarathy, University of Illinois 1 /File:7_bridgesID.png.

Graph theory and networks. Basic definitions  A graph consists of points called vertices (or nodes) and lines called edges (or arcs). Each edge joins.

MAT 2720 Discrete Mathematics Section 8.2 Paths and Cycles

Lecture 11: 9.4 Connectivity Paths in Undirected & Directed Graphs Graph Isomorphisms Counting Paths between Vertices 9.5 Euler and Hamilton Paths Euler.

Chapter 6: Graphs 6.1 Euler Circuits

Social Networking: Large scale Networks

Class 2: Graph Theory IST402. Can one walk across the seven bridges and never cross the same bridge twice? Network Science: Graph Theory THE BRIDGES OF.

Informatics tools in network science

Class 2: Graph Theory IST402.

+ GRAPH Algorithm Dikompilasi dari banyak sumber.

Importance Measures on Nodes Lecture 2 Srinivasan Parthasarathy 1.

1 GRAPH Learning Outcomes Students should be able to: Explain basic terminology of a graph Identify Euler and Hamiltonian cycle Represent graphs using.

STARTER: CAN YOU FIND A WAY OF CROSSING ALL THE BRIDGES EXACTLY ONCE? Here’s what this question would look like drawn as a graph.

Introduction to Graph & Network Theory Thinking About Networks: From Metabolism to the Genome to Social Conflict Summer Workshop for Teachers June 27 th.

Hiroki Sayama NECSI Summer School 2008 Week 2: Complex Systems Modeling and Networks Network Models Hiroki Sayama

L – Modeling and Simulating Social Systems with MATLAB

Structural Properties of Networks: Introduction

Applications of graph theory in complex systems research

Network analysis.

Structural Properties of Networks: Introduction

Discrete Maths 9. Graphs Objective

Genome Assembly.

Why Social Graphs Are Different Communities Finding Triangles

Department of Computer Science University of York

Graph Operations And Representation

Connectivity Section 10.4.

Walks, Paths, and Circuits

Konigsberg- in days past.

Network Graph Chuck Alice Vertices Nodes Edges Links Bob.

Discrete Mathematics Lecture 13_14: Graph Theory and Tree

(Social) Networks Analysis II

Euler and Hamilton Paths

Chapter 10 Graphs and Trees

Presentation transcript:

L – Modelling and Simulating Social Systems with MATLAB Lesson 6 – Graphs (Networks) Anders Johansson and Wenjian Yu (with S. Lozano and S. Wehrli) © ETH Zürich |

A. Johansson & W. Yu / 2 Lesson 6 – Contents  History: Seven Bridges of Königsberg  Definition of graph  Properties of graphs  Examples of graphs  MATLAB implementation of graphs

A. Johansson & W. Yu / 3 Space There are different ways to model the space in which social interaction takes place, e.g.:  No space, e.g. Dynamical Systems  Discrete space, e.g. Cellular Automata  Continuous space (Next week)  Graphs

A. Johansson & W. Yu / 4 Seven Bridges of Königsberg  Graph Theory was born in 1736, when Euler posted the following problem: Is it possible to have a walk in the city of Königsberg, that crosses each of the seven bridges only once?

A. Johansson & W. Yu / 5 Seven Bridges of Königsberg (II)  In order to approach the problem, Euler represented the important information as a graph: Source: wikipedia.org

A. Johansson & W. Yu / 6 Definition of Graph A graph (sometimes called: Network) consists of two entities:  Node N (or: vertex)  Link L:  Edge: Undirected link  Arc: Directed link The graph is defined as G = (N, L) Source: Batagelj

A. Johansson & W. Yu / 7 Properties of Links and Nodes  A link can either be a  Boolean property (connection vs. no connection), or  encoded as a strength (distance, traveling time, etc.)  A node can also contain information.

A. Johansson & W. Yu / 8 Properties of Graphs  A node can be characterized by:  Degree k: Number of connections.  Importance: Degree, Betweenness centrality (number of paths through the node), Closeness, Eigenvector centrality (e.g. PageRank).  A graph can be characterized by:  Degree distribution P(k): Fraction of nodes with k connections.  Diameter: The longest of the shortest paths between all nodes in the graph.

A. Johansson & W. Yu / 9 Degree Distribution  Graphs can be classified by their topology, by measuring the degree-distribution function P(k), of the number of connections k per node:  Random graph: P(k) = binomial distribution  Scale-free graph: P(k) = k -γ Source:

A. Johansson & W. Yu / 10 The Small World Experiment  Graphs are useful for modeling social networks, disease spreading, transportation, and so on …  One of the most famous graph studies is the Small World Experiment (S. Milgram), which shows that the minimum distance between any two persons in the world is almost never longer than through 5 friends.

A. Johansson & W. Yu / 11 Oracle of Bacon  There is a web page finding the path from any actor at any time to the Hollywood actor Kevin Bacon.  It can also be used to find the shortest path between any two actors.

A. Johansson & W. Yu / 12 MATLAB Implementation  A graph can be implemented in MATLAB via its adjacency matrix, i.e. an N x N matrix, defining how N nodes are connected to the other N-1 nodes: N = 10; A = zeros(N, N); A(1,2) = 1; A(10,4) = 1; …

A. Johansson & W. Yu / 13 Graphs  If the nodes are cities and the links define connections and travel times for the SBB network it looks like this: Geneva Basel Bern Zurich

A. Johansson & W. Yu / 14 Graphs  If the nodes are cities and the links define connections and travel times for the SBB network it looks like this: A = [ ; ; ; ]; Geneva Basel A = Bern Zurich

A. Johansson & W. Yu / 15 Graphs  If the nodes are cities and the links define connections and travel times for the SBB network it looks like this: Geneva Bern Basel Zurich 0:57 0:54 0:55 1:41

A. Johansson & W. Yu / 16 Graphs  If the nodes are cities and the links define connections and travel times for the SBB network it looks like this: Geneva Bern Basel Zurich A = :57 0:54 0:55 1:41

A. Johansson & W. Yu / 17 Projects  From next week, Wenjian Yu will take over the course. Therefore, address all future communication only to:  Today, there are no exercises. Instead, you can work on your projects and we will supervise you.