Part I: Introductory Materials Introduction 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
Graphs ( , ) = G V E { , ,..., } = V v v v 1 2 n Graph with 7 nodes and 16 edges { ( , ) | , , 1,..., } = = ∈ = E e v v v v V k m k i j i j Nodes / Vertices Undirected Edges Directed ( , ) ( , ) ≠ v v v v ( , ) ( , ) = v v v v i j j i i j j i 2
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 3
Labeled Graphs and Induced Subgraphs Bold: A subgraph induced by Labeled graph w/ loops vertices b, c and d 4
Graph Isomorphism (A) (B) (C) C Which graphs are isomorphic? 5
Graph Automorphism Automorphism is isomorphism that preserves the labels. (A) (B) (C) B Which graphs are automorphic? 6
Vertex degree, in-degree, out-degree t h tail head Directed 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
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 8
Adjacency Matrix Representation Representation is NOT unique . Algorithms can be order-sensitive . A(1) A(2) A(3) A(4) B(5) B(6) B(7) B(8) A(2) A(1) A(1) 1 1 1 0 1 0 0 0 A(2) B (6) B (5) 1 1 0 1 0 1 0 0 A(3) 1 0 1 1 0 0 1 0 A(4) 0 1 1 1 0 0 0 1 B(5) B (7) B (8) 1 0 0 0 1 1 1 0 B(6) 0 1 0 0 1 1 0 1 B(7) 0 0 1 0 1 0 1 1 B(8) A(3) A(4) 0 0 0 1 0 1 1 1 A(1) A(2) A(1) A(2) A(3) A(4) B(5) B(6) B(7) B(8) A(1) 1 1 0 1 0 1 0 0 B (6) B (7) A(2) 1 1 1 0 0 0 1 0 A(3) 0 1 1 1 1 0 0 0 A(4) 1 0 1 1 0 0 0 1 B (5) B (8) B(5) 0 0 1 0 1 0 1 1 B(6) 1 0 0 0 0 1 1 1 B(7) 0 1 0 0 1 1 1 0 A(3) A(4) B(8) 0 0 0 1 1 1 0 1 9 Src: “Introduction to Data Mining” by Kumar et al
Families of Graphs • Cliques • Path and simple path • Cycle • Tree • Connected graphs Read the book chapter for definitions and examples. 10
Complete Graph, or Clique Each pair of vertices is connected. Clique 11
The CLIQUE Problem { , | has a clique of size } = < > CLIQUE G k G k 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 Maxim um Clique of Size 5 12
Does this graph contain a 4-clique? Indeed it does! But, if it had not, what evidence would have been needed? 13
Problem: Decision, Optimization or Search Problem Decision Optimization Search Enumeration (self-reduction) Parameter k � max/min Actual solution All solutions “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. 14
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 ( ) ( ) = U k NP NTIME n = U k P DTIME n k k non-polynomial Non- deterministic polynomial Polynomially verifiable 15
P vs. NP The Classic Complexity Theory View: “forget about it” “easy” P ∑ 2 NP P PSPACE … … “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 16 away!”
Classical Graph Theory Problems CSC505:Algorithms, CSC707 :Complexity Theory, CSC5??:Graph Theory • Longest Path • Maximum Clique • Minimum Vertex Cover • Hamiltonian Path/Cycle • Traveling Salesman (TSP) NP-hard • Maximum Independent Set Problems • Minimum Dominating Set • Graph/Subgraph Isomorphism • Maximum Common Subgraph • … 17
Graph Mining Problems CSC 422/522 and Our Book Many graph mining problems have to deal with classical graph problems as part of its data mining pipeline. • Clustering + Maximal Clique Enumeration • Classification • Association Rule Mining +Frequent Subgraph Mining • Anomaly Detection • Similarity/Dissimilarity/Distance Measures • Graph-based Dimension Reduction • Link Analysis • … 18
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 ( n c )) – Guaranteed error-bar on the solution • Heuristic Polynomial-Time Algorithms Our focus – No guarantee on the quality of the solution – Low degree polynomial solutions 19
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