xvi CONTENTS
10.5 Finding Overlapping Communities . . . . . . . . . . . . . . . . . 381
10.5.1 The Nature o f Communities . . . . . . . . . . . . . . . . . 381
10.5.2 Maximum-Likelihood Estimation . . . . . . . . . . . . . . 381
10.5.3 The Affiliation-Graph Model . . . . . . . . . . . . . . . . 383
10.5.4 Discrete Optimization of Community Assignments . . . . 385
10.5.5 Avoiding the Use of Discrete Member ship Changes . . . . 386
10.5.6 Exercises for Section 10.5 . . . . . . . . . . . . . . . . . . 388
10.6 Simrank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
10.6.1 Random Walkers on a Social Graph . . . . . . . . . . . . 389
10.6.2 Random Walks with Restart . . . . . . . . . . . . . . . . 390
10.6.3 Exercises for Section 10.6 . . . . . . . . . . . . . . . . . . 393
10.7 Counting Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . 393
10.7.1 Why Count Triangles? . . . . . . . . . . . . . . . . . . . . 393
10.7.2 An Algor ithm for Finding Triangles . . . . . . . . . . . . 394
10.7.3 Optimality of the Triangle-Finding Algorithm . . . . . . . 395
10.7.4 Finding Triangles Using MapReduce . . . . . . . . . . . . 396
10.7.5 Using Fewer Reduce Tasks . . . . . . . . . . . . . . . . . . 397
10.7.6 Exercises for Section 10.7 . . . . . . . . . . . . . . . . . . 398
10.8 Neighborhood Properties of Graphs . . . . . . . . . . . . . . . . . 399
10.8.1 Directed Graphs and Neighborhoods . . . . . . . . . . . . 400
10.8.2 The Diameter o f a Graph . . . . . . . . . . . . . . . . . . 40 1
10.8.3 Transitive C losure and Reachability . . . . . . . . . . . . 402
10.8.4 Reachability Via MapReduce . . . . . . . . . . . . . . . . 403
10.8.5 Seminaive Evaluation . . . . . . . . . . . . . . . . . . . . 404
10.8.6 Linear Transitive Closure . . . . . . . . . . . . . . . . . . 405
10.8.7 Transitive C losure by Recursive Doubling . . . . . . . . . 405
10.8.8 Smart Transitive Closure . . . . . . . . . . . . . . . . . . 407
10.8.9 Comparison of Methods . . . . . . . . . . . . . . . . . . . 409
10.8.10 Transitive Closure by Graph Reduction . . . . . . . . . . 411
10.8.11 Approximating the Sizes of Neighborhoods . . . . . . . . 413
10.8.12 Exercises for Section 10.8 . . . . . . . . . . . . . . . . . . 414
10.9 Summary of Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . 416
10.10References for Chapter 10 . . . . . . . . . . . . . . . . . . . . . . 420
11 Dimensionality Reduction 423
11.1 Eigenvalues and Eigenvectors of Symmetric Matrices . . . . . . . 424
11.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 424
11.1.2 Computing Eigenvalues and Eigenvectors . . . . . . . . . 425
11.1.3 Finding Eigenpairs by Power Iteration . . . . . . . . . . . 426
11.1.4 The Matr ix of Eigenvectors . . . . . . . . . . . . . . . . . 429
11.1.5 Exercises for Section 11.1 . . . . . . . . . . . . . . . . . . 429
11.2 Principal-Component Analysis . . . . . . . . . . . . . . . . . . . 430
11.2.1 An Illustrative Example . . . . . . . . . . . . . . . . . . . 431
11.2.2 Using Eigenvectors for Dimensionality Reduction . . . . . 434
11.2.3 The Matr ix of Distances . . . . . . . . . . . . . . . . . . . 435
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