1. Random Dot Product Graphs Ed Scheinerman Applied Mathematics & Statistics Johns Hopkins University IPAM Intelligent Extraction of Information from Graphs & High Dimensional Data July 26, 2005

2. Coconspirators Libby Beer John Conroy (IDA) Paul Hand (Columbia) Miro Kraetzl (DSTO) Christine Nickel Carey Priebe Kim Tucker Stephen Young (Georgia Tech)

3. Overview Mathematical context Modeling networks Random dot product model The inverse problem

4. Mathematical Context

5. Graphs I Have Loved Interval graphs & intersection graphs Random graphs Random intersection graphs Threshold graphs & dot product graphs

6. Interval Graphs

7. Intersection Graphs {1} {1,2} {2}

8. Random Graphs Erdös-Rényi style… p1 – p Randomness is “in” the edges. Vertices are “dumb” placeholders.

9. Random Intersection Graphs Assign random sets to vertices. Two vertices are adjacent iff their sets intersect. Randomness is “in” the vertices. Edges reflect relationships between vertices.

10. Threshold Graphs 0.5 0.6 0.8 0.3

11. Dot Product Graphs [1 0] [2 0] [1 1] [0 1] Fractional intersection graphs

12. Communication Networks

13. Physical Networks Telephone Local area network Power grid Internet

14. Social Networks Alice Bob A B 2003-4-10

15. Social Network Graphs Vertices (Actors)Edges (Dyads) TelephonesCalls Email addressesMessages ComputersIP Packets Human beingsAcquaintance AcademiciansCoauthorship

16. Example: Email at HP 485 employees 185,000 emails Social network (who emails whom) identified 7 “communities”, validated by interviews with employees.

17. Properties of Social Networks Clustering Low diameter Power law

18. Properties of Social Networks Clustering Low diameter Power law a b c

19. Properties of Social Networks Clustering Low diameter Power law “Six degrees of separation”

20. Properties of Social Networks Clustering Low diameter Power law log d log N(d) Degree Histogram

21. Degree Histogram Example 1 2838 vertices degree Number of vertices

22. Degree Histogram Example 2 16142 vertices degree Number of vertices

23. Random Graph Models Goal: Simple and realistic random graph models of social networks.

24. Erdös-Rényi? Low diameter! No clustering: P[a~c]=P[a~c|a~b~c]. No power-law degree distribution. Not a good model.

25. Model by Fan Chung et al Consider only those graphs with with all such graphs equally likely.

26. People as Vectors Sports Politics Movies Graph theory

27. Shared Interests Alice and Bob are more likely to communicate when they have more shared interests.

28. Selecting the Function

31. Random Dot Product Graphs, I

32. Generalize Erdös-Rényi

33. Generalize Intersection Graphs

34. Whence the Vectors? Vectors are given in advance. Vectors chosen (iid) from some distribution.

35. Random Dot Product Graphs, II Step 1: Pick the vectors  Given by fiat.  Chosen from iid a distribution. Step 2: For all i<j  Let p=f(x i x j ).  Insert an edge from i to j with probability p.

36. Megageneralization Generalization of:  Intersection graphs (ordinary & random)  Threshold graphs  Dot product graphs  Erdös-Rényi random graphs Randomness is “in” both the vertices and the edges. P[a~b] independent of P[c~d] when a,b,c,d are distinct.

37. Results in Dimension 1

38. Probability/Number of Edges

39. Clustering

40. Power Law

41. Power Law Example

42. Isolated Vertices Thus, the graph is not connected, but…

43. “Mostly” Connected “Giant” connected component A “few” isolated vertices

44. Six Degrees of Separation Diameter ≤ 6

45. Attached Attached pair Diameter ≤ 6 Proof Outline Diameter = 2 Isolated

46. Diameter ≤ 6 Proof Outline

47. Graphs to Vectors The Inverse Problem

48. Given Graphs, Find Vectors Given: A graph, or a series of graphs, on a common vertex set. Problem: Find vectors to assign to vertices that “best” model the graph(s).

49. Maximum Likelihood Method Feasible in dimension 1. Awful d>1. Nice results for f(t) = t / (1+t).

50. Gram Matrix Approach

51. Wrong Best Solution

52. Real Problem

53. Iterative Algorithm

54. Convergence

55. iteration diagonal entries

56. Convergence iteration diagonal entries

57. Convergence iteration diagonal entries

58. Convergence iteration diagonal entries

59. Convergence iteration diagonal entries

60. Convergence iteration diagonal entries

61. Enron example

62. Applications Network Change/Anomaly Detection Clustering

63. Change/Anomaly Detection

65. Graph Clustering

67. Synthetic Lethality Graphs Vertices are genes in yeast Edge between u and v iff  Deleting one of u or v does not kill, but  Deleting both is lethal.

68. SL Graph Status Yeast has about 6000 genes. Full graph known on 126 “query” genes (about 1300 edges). Partial graph known on 1000 “library” genes.

70. What Next?

71. Random Dot Product Graphs Extension to higher dimension  Cube  Unit ball intersect positive orthant Small world measures: clustering coefficient Other random graph properties

72. Vector Estimation MLE method  Computationally efficient?  More useful? Eigenvalue method  Understand convergence  Prove that it globally minimizes  Extension to missing data Validate against real data

73. Network Evolution Communication influences interests:

74. Rapid Generation Can we generate a sparse random dot product graph with n vertices and m edges in time O(n+m)? Partial answer: Yes, but.

75. The End