CS8803-NS Network Science Fall 2013

1. CS8803-NSNetwork ScienceFall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu http://www.cc.gatech.edu/~dovrolis/Courses/NetSci/

2. Disclaimers The following slides include only the figures or videos that we use in class; they do not include detailed explanations, derivations or descriptionscovered in class. Many of the following figures are copied from open sources at the Web. I do not claim any intellectual property for the following material.

3. Outline • What does “network community” mean? • Community detection versus graph partitioning versus hierarchical clustering • Graph partitioning algorithms • Spectral partitioning (Fiedler’s method based on graph Laplacian) • Modularity metric for community detection • Spectral-based modularity optimization • Other methods for modularity optimization • Community detection methods that do not rely on modularity metric • Betweenness-Centrality method • Radicchiet al. method • Hierarchical agglomerative clustering

4. Outline for next week’s class • Variations of the community detection problem • Overlapping communities • Dynamic communities • Link-based communities • Properties of real-world network communities • Applications of community detection • In social networks • In biological networks • In brain networks • In ecological networks • In climate networks

5. Today’s outline (reordered) • What does “network community” mean? • Community detection versus graph partitioning versus hierarchical clustering • Modularity metric for community detection • Spectral-based modularity optimization • Other methods for modularity optimization • Community detection methods that do not rely on modularity metric • Betweenness-Centrality method • Radicchiet al. method • Hierarchical agglomerative clustering • Graph partitioning algorithms • Spectral partitioning (Fiedler’s method based on graph Laplacian)

13. Hierarchical network

14. Today’s outline (reordered) • What does “network community” mean? • Community detection versus graph partitioning versus hierarchical clustering • Modularity metric for community detection • Spectral-based modularity optimization • Other methods for modularity optimization • Community detection methods that do not rely on modularity metric • Betweenness-Centrality method • Radicchiet al. method • Hierarchical agglomerative clustering • Graph partitioning algorithms • Spectral partitioning (Fiedler’s method based on graph Laplacian)

15. Graph partitioning vs Community detection • In graph partitioning, the desired number and size of the partitions is given • E.g., graph bisection in two equal-sized partitions • NP-Hard • In community detection, the number of communities (and their size) results from the method itself • It is a property of the network • The community detection problem is less well-defined than the graph partitioning problem

17. Spectral bisection method for graph partitioning(see last few slides for more details)

18. Graph partitioning vs Hierarchical clustering

19. Community detection vs Hierarchical clustering • Hierarchical clustering comes in two forms: • Divisive algs: top-down • Agglomerative: bottom-up • Key points: • Need a similarity metric for any two nodes • Which metric to use? • How to examine similarity of groups of nodes? • Which horizontal partition gives more insight? • Some clusters are artificial; not “real communities” • Fundamentally, many networks are NOT hierarchical

20. Today’s outline (reordered) • What does “network community” mean? • Community detection versus graph partitioning • Modularity metric for community detection • Spectral-based modularity optimization • Other methods for modularity optimization • Community detection methods that do not rely on modularity metric • Betweenness-Centrality method • Radicchiet al. method • Hierarchical agglomerative clustering • Graph partitioning algorithms • Spectral partitioning (Fiedler’s method based on graph Laplacian)

21. Modularity definition • Fraction of edges between pairs of nodes that belong to the same community RELATIVE TO • Fraction of edges between same pair of nodes if edges were placed randomly (but in a degree-preserving manner)

22. Spectral maximization of modularity (2006)

23. Spectral maximization of modularity (see class notes for detailed derivations)

24. Spectral maximization of modularity (see class notes for detailed derivations)

25. Dividing a community into smaller communities

26. Spectral maximization of modularity (see class notes for detailed derivations)

27. Greedy optimization of modularity (2004)

28. Complexity of Clausetet al.’s method

29. Today’s outline (reordered) • What does “network community” mean? • Community detection versus graph partitioning • Modularity metric for community detection • Spectral-based modularity optimization • Other methods for modularity optimization • Community detection methods that do not rely on modularity metric • Betweenness-Centrality method • Radicchiet al. method • Hierarchical agglomerative clustering • Graph partitioning algorithms • Spectral partitioning (Fiedler’s method based on graph Laplacian)

30. The algorithm of Girvan-Newman

31. The algorithm of Girvan-Newman

32. The algorithm of Radicchi et al.

33. Today’s outline (reordered) • What does “network community” mean? • Community detection versus graph partitioning • Modularity metric for community detection • Spectral-based modularity optimization • Other methods for modularity optimization • Community detection methods that do not rely on modularity metric • Betweenness-Centrality method • Radicchiet al. method • Hierarchical clustering • Graph partitioning algorithms • Spectral partitioning (Fiedler’s method based on graph Laplacian)

34. Hierarchical clustering http://condor.depaul.edu/ntomuro/courses/578/notes/notes-Clustering.html

35. Hierarchical agglomerative clustering http://condor.depaul.edu/ntomuro/courses/578/notes/notes-Clustering.html

36. Hierarchical divisive clustering http://mines.humanoriented.com/classes/2010/fall/csci568/portfolio_exports/mvoget/cluster/cluster.html

37. Node similarity metrics

38. Cluster similarity – 3 approaches

39. Today’s outline (reordered) • What does “network community” mean? • Community detection versus graph partitioning • Modularity metric for community detection • Spectral-based modularity optimization • Other methods for modularity optimization • Community detection methods that do not rely on modularity metric • Betweenness-Centrality method • Radicchiet al. method • Hierarchical agglomerative clustering • Graph partitioning algorithms • Spectral partitioning (Fiedler’s method based on graph Laplacian)

40. Key points(See class notes for detailed derivations) • Define Laplacian of an (undirected, unweighted) graph • Show that all eigenvalues of Laplacian are non-negative • Show that Laplacian has at least one zero eigenvalue • The number of zero eigenvalues is equal to the number of connected components in the graph • The lowest non-zero eigenvalue is called “algebraic connectivity” and it is proportional to the graph’s min cut set

41. Key points (cont’)(See class notes for detailed derivations) • Formulate graph bisection problem as a constrained optimization problem • Show that min cut set is proportional to algebraic connectivity (min non-zero eigenvalue of Laplacian) • Compute corresponding eigenvector (appropriately normalized) • And determine graph partitions based on the values of that eigenvector • For a sparse graph, this method is O(n2) • If the second eigenvector is computed using the orthogonalization or Lanczos method (which is O(m*n))