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GE498-ECI, Lecture 8: Connectivity Everywhere; Graph Theory 101

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GE498-ECI, Lecture 8: Connectivity Everywhere; Graph Theory 101

GE498-ECI, Lecture 8: Connectivity Everywhere; Graph Theory 101

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    GE498-ECI, Lecture 8: Connectivity Everywhere; Graph Theory 101 GE498-ECI, Lecture 8: Connectivity Everywhere; Graph Theory 101 Presentation Transcript

    • Enterprise Collaboration and Innovation Support Systems (GE/IE 498 ECI): Connectivity Everywhere; Graph Theory 101 Xavier Llorà Illinois Genetic Algorithms Lab & National Center for Supercomputing Applications University of Illinois at Urbana-Champaign xllora@uiuc.edu
    • Where did we leave it? • Giving structure to information exchange • Markup languages • XML basics • The semantic web (RDF) GE/IE 498 ECI - Spring 2007 Xavier Llorà © Spring 2007 2
    • What’s the plan for today? • Graphs as structure • Basic concepts • Types of graphs • Basic concepts of graph analysis GE/IE 498 ECI - Spring 2007 Xavier Llorà © Spring 2007 3
    • Königsberg and Mr. Euler Leonhard Euler was a Swiss mathematician • (April 15, 1707 – September 7, 1783) His paper in 1736 is considered the first paper on • graph theory People say Mr. Euler spent their Sundays walking • around, trying to find a starting point so that he could walk about Königsberg, cross each bridge exactly once, and then return to their starting point. Mr. Euler was a mathematician so he ask himself questions like the following • ones: – Could I find a path that allow me to do so? Moreover, is that possible at all? GE/IE 498 ECI - Spring 2007 Xavier Llorà © Spring 2007 4
    • Welcome to Königsberg http://www-gap.dcs.st-and.ac.uk/%7Ehistory/HistTopics/Topology`_in_mathematics.html GE/IE 498 ECI - Spring 2007 Xavier Llorà © Spring 2007 5
    • The formalized Königsberg GE/IE 498 ECI - Spring 2007 Xavier Llorà © Spring 2007 6
    • What is a graph? A graph is a set of vertex and a set of edges • G=<V,E> Vertices are usually labeled • – V = {A,B,C,D} The set of edges • – Direction is important (directed graph) • E={<A,B>,<B,C>,<C,B>} • <X,Y>: X = head, Y = tail • <X,Y> ≠ <Y,X> unless X=Y – Direction is not important (undirected graph) • E = { (A,B), (B,C), (C,B) } • (X,Y): X = head, Y = tail • (X,Y) = (Y,X) (unless we have a multigraph) GE/IE 498 ECI - Spring 2007 Xavier Llorà © Spring 2007 7
    • Some basic examples A A B D B D C C Undirected graph: Directed graph: G=<V,E> G=<V,E> V={A,B,C,D} V={A,B,C,D} E={(A,B),(B,C),(C,D),(A,D),(B,D)} E={<A,B>,<C,B>,<A,D>,<D,B>,<D,C>} GE/IE 498 ECI - Spring 2007 Xavier Llorà © Spring 2007 8
    • Königsberg is a multigraph GE/IE 498 ECI - Spring 2007 Xavier Llorà © Spring 2007 9
    • Mixed graph = directed + undirected A E B D F H C G G=<V,E> V={A,B,C,D,E,F,G,H} E={(A,B),(B,C),(C,D),(A,D),(B,D), <E,F>,<G,F>,<E,H>,<H,F>,<H,G>,<A,E>,(C,G)} GE/IE 498 ECI - Spring 2007 Xavier Llorà © Spring 2007 10
    • Information on the edges Edges may be labeled • Edges may contain weight (weighted graph) • A 3 1 2 B D 1.1 3.6 C GE/IE 498 ECI - Spring 2007 Xavier Llorà © Spring 2007 11
    • Degree of a vertex Undirected graphs • – The degree of a node is the number of edges involving that node Directed graphs • – Indegree: number of edges that arrive (head) to a node – Outdegree: number of edges departing (tail) from a node 2 0,2 A A 3 3,0 1,2 3 B D B D 3 1,1 C C GE/IE 498 ECI - Spring 2007 Xavier Llorà © Spring 2007 12
    • Other concepts • A loop: an edge where the to vertex are the same • A simple graph: a graph that does not contain loops or parallel edges • A multigraph: a graph that contains parallel edges • To edges are adjacent if they connect to the same vertex • To vertices are adjacent if they are connected by an edge • A complete graph: a graph that every pair of vertices are adjacent GE/IE 498 ECI - Spring 2007 Xavier Llorà © Spring 2007 13
    • A path in the graph A path through a graph is a traversal of consecutive vertices along • a sequence of edges – The vertex at the end of one edge in the sequence must also be the vertex at the beginning of the next edge in the sequence – The vertices that begin and end the path are termed the initial vertex and terminal vertex, respectively – With the exception of these initial and terminal vertices, each vertex within the path has two neighboring vertices that must also be adjacent to the vertex – The length of the path is the number of edges that are traversed along the path – An elementary path is a path where no vertex appears more than once GE/IE 498 ECI - Spring 2007 Xavier Llorà © Spring 2007 14
    • Some path examples A A B D B D C C P= {A,D,B} is a path P= {A,B,C} P= {A,D,C,B} is a path P= {A,D,A} P= {A,B,C} is not a path P= {D,B,A} P= {A,D,A} is not a path P= {D,B,A} is not a path GE/IE 498 ECI - Spring 2007 Xavier Llorà © Spring 2007 15
    • Cycle • A cycle is a path in which the initial vertex of the path is also the terminal vertex of the path • Removing all the cycles in a path we get an elementary path A P= {A,B,A} P= {A,B,D,A} P= {A,B,C,D,A} P= {{A,B,C,D,A},B,C} B D P= {D,C,B,{A,B,C,D,A},B,C} C GE/IE 498 ECI - Spring 2007 Xavier Llorà © Spring 2007 16
    • Connectedness Undirected graphs • – An undirected graph is considered to be connected if a path exists between all pairs of vertices thus making each of the vertices in a pair reachable from the other – An unconnected graph may be subdivided into what are termed connected subgraphs or connected components of the graph. Directed graphs • – A weakly connected graph is where the direction of the graph is ignored and the connectedness is defined as if the graph was undirected – A unilaterally connected graph is defined as a graph for which at least one vertex of any pair of vertices is reachable from the other – A strongly connected graph is one in which for all pairs of vertices, both vertices are reachable from the other GE/IE 498 ECI - Spring 2007 Xavier Llorà © Spring 2007 17
    • Implementation Graphs can be implemented in several way • 1. Set of nodes, and list of edges 2. Adjacency matrix (triangular matrix for undirected graphs) A B C D E F A A D C B B C D F E E F GE/IE 498 ECI - Spring 2007 Xavier Llorà © Spring 2007 18
    • What will we do in the next lecture? • Human communication • Text as a form of unstructured data • Basic notions of text mining • Common operations in text mining GE/IE 498 ECI - Spring 2007 Xavier Llorà © Spring 2007 19
    • Something to think about Further reading • http://ocw.mit.edu/NR/rdonlyres/Electrical-Engineering-and-Computer-Science/6-042JSpring-2005/FF95BA8F-6457-4592-B473-349A5C1CA277/0/l8_graphs1.pdf http://www.math.lsa.umich.edu/mmss/coursesONLINE/graph/ http://www.cs.usask.ca/resources/tutorials/csconcepts/1999_8/ Homework: • 1. What is the solution to Mr. Euler’s question, and why? 2. Review last lesson’s RDF examples and paint down the underlying graph and write the adjacency matrix. What else can you tell about this graph? GE/IE 498 ECI - Spring 2007 Xavier Llorà © Spring 2007 20