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Part I: Introductory Materials Introduction to Graph Theory

This introduction to graph theory covers directed and undirected graphs, types of graphs, labeled graphs, graph isomorphism and automorphism, graph representation formats, families of graphs, the clique problem, complexity theory, classic graph theory problems, and graph mining problems.

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Part I: Introductory Materials Introduction to Graph Theory

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  1. Part I: Introductory MaterialsIntroduction to Graph Theory Dr. Nagiza F. Samatova Department of Computer Science North Carolina State University and Computer Science and Mathematics Division Oak Ridge National Laboratory

  2. Directed Graphs Graph with 7 nodes and 16 edges Nodes / Vertices Undirected Edges

  3. Types of Graphs • Undirected vs. Directed • Attributed/Labeled (e.g., vertex, edge) vs. Unlabeled • Weighted vs. Unweighted • General vs. Bipartite (Multipartite) • Trees (no cycles) • Hypergraphs • Simple vs. w/ loops vs. w/ multi-edges

  4. Labeled Graphs and Induced Subgraphs Bold: A subgraph induced by vertices b, c and d Labeled graph w/ loops

  5. Graph Isomorphism (A) (B) (C) C Which graphs are isomorphic?

  6. Graph Automorphism Automorphism is isomorphism that preserves the labels. (A) (B) (C) B Which graphs are automorphic?

  7. Directed Vertex degree, in-degree, out-degree t h tail head In-degree of the vertex is the number of in-coming edges Out-degree of the vertex is the number of out-going edges Degree of the vertex is the number of edges (both in- & out-degree) 7

  8. Graph Representation and Formats • Adjacency Matrix (vertex vs. vertex) • Incidence Matrix (vertex vs. edge) • Sparse vs. Dense Matrices • DIMACS file format • In R: igraph object

  9. Adjacency Matrix Representation Representation is NOTunique. Algorithms can be order-sensitive. Src: “Introduction to Data Mining” by Kumar et al

  10. Families of Graphs • Cliques • Path and simple path • Cycle • Tree • Connected graphs Read the book chapter for definitions and examples.

  11. Clique Complete Graph, or Clique Each pair of vertices is connected.

  12. Clique: a complete subgraph Maximal Clique: a clique cannot be enlarged by adding any more vertices Maximum Clique: the largest maximal clique in the graph The CLIQUE Problem Maximum Clique of Size 5

  13. Indeed it does! Does this graph contain a 4-clique? But, if it had not, what evidence would have been needed?

  14. Problem: Decision, Optimization or Search Problem Optimization Decision Search Enumeration (self-reduction) Parameter k  max/min All solutions Actual solution “Yes”-”No” • Which problem is harder to solve? • If we solve Decision problem, can we use it for the others? Formulate each version for the CLIQUE problem.

  15. Refresher: Class P and Class NP Definition: P (NP) is the class of languages/problems that are decidable in polynomial time on a (non-)deterministic single-tape Turing machine. Class NP P ???? non-polynomial Non-deterministic polynomial Polynomially verifiable

  16. “forget about it” P ∑ … … NP P PSPACE 2 P vs. NP The Classic Complexity Theory View: “easy” “hard” “About ten years ago some computer scientists came by and said they heard we have some really cool problems. They showed that the problems are NP-complete and went away!”

  17. Classical Graph Theory ProblemsCSC505:Algorithms, CSC707 :Complexity Theory, CSC5??:Graph Theory • Longest Path • Maximum Clique • Minimum Vertex Cover • Hamiltonian Path/Cycle • Traveling Salesman (TSP) • Maximum Independent Set • Minimum Dominating Set • Graph/Subgraph Isomorphism • Maximum Common Subgraph • … NP-hard Problems

  18. Graph Mining ProblemsCSC 422/522 and Our Book • Clustering + Maximal Clique Enumeration • Classification • Association Rule Mining +Frequent Subgraph Mining • Anomaly Detection • Similarity/Dissimilarity/Distance Measures • Graph-based Dimension Reduction • Link Analysis • … Many graph mining problems have to deal with classical graph problems as part of its data mining pipeline.

  19. Dealing with Computational Intractability • Exact Algorithms: • Small graph problems • Small parameters to graph problems • Special classes of graphs (e.g., bounded tree-width) • Approximation Polynomial-Time Algorithms (O(nc)) • Guaranteed error-bar on the solution • Heuristic Polynomial-Time Algorithms • No guarantee on the quality of the solution • Low degree polynomial solutions Our focus

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