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Mining Billion-node Graphs: Patterns, Generators and Tools

Mining Billion-node Graphs: Patterns, Generators and Tools. Christos Faloutsos CMU. Our goal:. Open source system for mining huge graphs: PEGASUS project (PEta GrAph mining System) www.cs.cmu.edu/~pegasus code and papers. Outline. Introduction – Motivation Problem#1: Patterns in graphs

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Mining Billion-node Graphs: Patterns, Generators and Tools

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  1. Mining Billion-node Graphs: Patterns, Generators and Tools Christos Faloutsos CMU

  2. Our goal: Open source system for mining huge graphs: PEGASUS project (PEta GrAph mining System) • www.cs.cmu.edu/~pegasus • code and papers C. Faloutsos (CMU)

  3. Outline • Introduction – Motivation • Problem#1: Patterns in graphs • Problem#2: Tools • Problem#3: Scalability • Conclusions C. Faloutsos (CMU)

  4. Graphs - why should we care? Internet Map [lumeta.com] Food Web [Martinez ’91] Protein Interactions [genomebiology.com] Friendship Network [Moody ’01] C. Faloutsos (CMU)

  5. T1 D1 ... ... DN TM Graphs - why should we care? • IR: bi-partite graphs (doc-terms) • web: hyper-text graph • ... and more: C. Faloutsos (CMU)

  6. Graphs - why should we care? • network of companies & board-of-directors members • ‘viral’ marketing • web-log (‘blog’) news propagation • computer network security: email/IP traffic and anomaly detection • .... C. Faloutsos (CMU)

  7. Outline • Introduction – Motivation • Problem#1: Patterns in graphs • Static graphs • Weighted graphs • Time evolving graphs • Problem#2: Tools • Problem#3: Scalability • Conclusions C. Faloutsos (CMU)

  8. Problem #1 - network and graph mining • How does the Internet look like? • How does FaceBook look like? • What is ‘normal’/‘abnormal’? • which patterns/laws hold? C. Faloutsos (CMU)

  9. Problem #1 - network and graph mining • How does the Internet look like? • How does FaceBook look like? • What is ‘normal’/‘abnormal’? • which patterns/laws hold? • To spot anomalies (rarities), we have to discover patterns C. Faloutsos (CMU)

  10. Problem #1 - network and graph mining • How does the Internet look like? • How does FaceBook look like? • What is ‘normal’/‘abnormal’? • which patterns/laws hold? • To spot anomalies (rarities), we have to discover patterns • Large datasets reveal patterns/anomalies that may be invisible otherwise… C. Faloutsos (CMU)

  11. Graph mining • Are real graphs random? C. Faloutsos (CMU)

  12. Laws and patterns • Are real graphs random? • A: NO!! • Diameter • in- and out- degree distributions • other (surprising) patterns • So, let’s look at the data C. Faloutsos (CMU)

  13. -0.82 Solution# S.1 • Power law in the degree distribution [SIGCOMM99] internet domains att.com log(degree) ibm.com log(rank) C. Faloutsos (CMU)

  14. Solution# S.2: Eigen Exponent E • A2: power law in the eigenvalues of the adjacency matrix Eigenvalue Exponent = slope E = -0.48 May 2001 Rank of decreasing eigenvalue C. Faloutsos (CMU)

  15. Solution# S.2: Eigen Exponent E • [Mihail, Papadimitriou ’02]: slope is ½ of rank exponent Eigenvalue Exponent = slope E = -0.48 May 2001 Rank of decreasing eigenvalue C. Faloutsos (CMU)

  16. But: How about graphs from other domains? C. Faloutsos (CMU)

  17. users sites More power laws: • web hit counts [w/ A. Montgomery] Web Site Traffic Count (log scale) Zipf ``ebay’’ in-degree (log scale) C. Faloutsos (CMU)

  18. epinions.com • who-trusts-whom [Richardson + Domingos, KDD 2001] count trusts-2000-people user (out) degree C. Faloutsos (CMU)

  19. And numerous more • # of sexual contacts • Income [Pareto] –’80-20 distribution’ • Duration of downloads [Bestavros+] • Duration of UNIX jobs (‘mice and elephants’) • Size of files of a user • … • ‘Black swans’ C. Faloutsos (CMU)

  20. Outline • Introduction – Motivation • Problem#1: Patterns in graphs • Static graphs • degree, diameter, eigen, • triangles • cliques • Weighted graphs • Time evolving graphs • Problem#2: Tools C. Faloutsos (CMU)

  21. Solution# S.3: Triangle ‘Laws’ Real social networks have a lot of triangles C. Faloutsos (CMU)

  22. Solution# S.3: Triangle ‘Laws’ Real social networks have a lot of triangles Friends of friends are friends Any patterns? C. Faloutsos (CMU)

  23. Triangle Law: #S.3 [Tsourakakis ICDM 2008] HEP-TH ASN X-axis: # of Triangles a node participates in Y-axis: count of such nodes Epinions C. Faloutsos (CMU)

  24. Triangle Law: #S.3 [Tsourakakis ICDM 2008] HEP-TH ASN X-axis: # of Triangles a node participates in Y-axis: count of such nodes Epinions C. Faloutsos (CMU)

  25. Triangle Law: #S.4 [Tsourakakis ICDM 2008] Reuters SN X-axis: degree Y-axis: mean # triangles n friends -> ~n1.6 triangles Epinions C. Faloutsos (CMU)

  26. Triangle Law: Computations [Tsourakakis ICDM 2008] details But: triangles are expensive to compute (3-way join; several approx. algos) Q: Can we do that quickly? C. Faloutsos (CMU)

  27. Triangle Law: Computations [Tsourakakis ICDM 2008] details But: triangles are expensive to compute (3-way join; several approx. algos) Q: Can we do that quickly? A: Yes! #triangles = 1/6 Sum ( li3 ) (and, because of skewness, we only need the top few eigenvalues! C. Faloutsos (CMU)

  28. Triangle Law: Computations [Tsourakakis ICDM 2008] details 1000x+ speed-up, >90% accuracy C. Faloutsos (CMU)

  29. EigenSpokes B. Aditya Prakash, Mukund Seshadri, Ashwin Sridharan, Sridhar Machiraju and Christos Faloutsos: EigenSpokes: Surprising Patterns and Scalable Community Chipping in Large Graphs, PAKDD 2010, Hyderabad, India, 21-24 June 2010. C. Faloutsos (CMU)

  30. EigenSpokes • Eigenvectors of adjacency matrix • equivalent to singular vectors (symmetric, undirected graph) C. Faloutsos (CMU)

  31. EigenSpokes details • Eigenvectors of adjacency matrix • equivalent to singular vectors (symmetric, undirected graph) N N C. Faloutsos (CMU)

  32. EigenSpokes 2nd Principal component • EE plot: • Scatter plot of scores of u1 vs u2 • One would expect • Many points @ origin • A few scattered ~randomly u2 u1 1st Principal component C. Faloutsos (CMU)

  33. EigenSpokes • EE plot: • Scatter plot of scores of u1 vs u2 • One would expect • Many points @ origin • A few scattered ~randomly u2 90o u1 C. Faloutsos (CMU)

  34. EigenSpokes - pervasiveness • Present in mobile social graph • across time and space • Patent citation graph C. Faloutsos (CMU)

  35. EigenSpokes - explanation Near-cliques, or near-bipartite-cores, loosely connected C. Faloutsos (CMU)

  36. EigenSpokes - explanation Near-cliques, or near-bipartite-cores, loosely connected C. Faloutsos (CMU)

  37. EigenSpokes - explanation Near-cliques, or near-bipartite-cores, loosely connected So what? • Extract nodes with high scores • high connectivity • Good “communities” spy plot of top 20 nodes C. Faloutsos (CMU)

  38. Bipartite Communities! patents from same inventor(s) cut-and-paste bibliography! magnified bipartite community C. Faloutsos (CMU)

  39. Outline • Introduction – Motivation • Problem#1: Patterns in graphs • Static graphs • degree, diameter, eigen, • triangles • cliques • Weighted graphs • Time evolving graphs • Problem#2: Tools C. Faloutsos (CMU)

  40. Observations on weighted graphs? • A: yes - even more ‘laws’! M. McGlohon, L. Akoglu, and C. Faloutsos Weighted Graphs and Disconnected Components: Patterns and a Generator. SIG-KDD 2008 C. Faloutsos (CMU)

  41. Observation W.1: Fortification Q: How do the weights of nodes relate to degree? C. Faloutsos (CMU)

  42. Observation W.1: Fortification More donors, more $ ? ‘Reagan’ $10 $5 ‘Clinton’ $7 C. Faloutsos (CMU)

  43. Observation W.1: fortification:Snapshot Power Law Weight: super-linear on in-degree exponent ‘iw’:1.01 < iw < 1.26 Orgs-Candidates More donors, even more $ e.g. John Kerry, $10M received, from 1K donors In-weights ($) $10 $5 Edges (# donors) C. Faloutsos (CMU)

  44. Outline • Introduction – Motivation • Problem#1: Patterns in graphs • Static graphs • Weighted graphs • Time evolving graphs • Problem#2: Tools • … C. Faloutsos (CMU)

  45. Problem: Time evolution • with Jure Leskovec (CMU -> Stanford) • and Jon Kleinberg (Cornell – sabb. @ CMU) C. Faloutsos (CMU)

  46. T.1 Evolution of the Diameter • Prior work on Power Law graphs hints atslowly growing diameter: • diameter ~ O(log N) • diameter ~ O(log log N) • What is happening in real data? C. Faloutsos (CMU)

  47. T.1 Evolution of the Diameter • Prior work on Power Law graphs hints atslowly growing diameter: • diameter ~ O(log N) • diameter ~ O(log log N) • What is happening in real data? • Diameter shrinks over time C. Faloutsos (CMU)

  48. T.1 Diameter – “Patents” • Patent citation network • 25 years of data • @1999 • 2.9 M nodes • 16.5 M edges diameter time [years] C. Faloutsos (CMU)

  49. T.2 Temporal Evolution of the Graphs • N(t) … nodes at time t • E(t) … edges at time t • Suppose that N(t+1) = 2 * N(t) • Q: what is your guess for E(t+1) =? 2 * E(t) C. Faloutsos (CMU)

  50. T.2 Temporal Evolution of the Graphs • N(t) … nodes at time t • E(t) … edges at time t • Suppose that N(t+1) = 2 * N(t) • Q: what is your guess for E(t+1) =? 2 * E(t) • A: over-doubled! • But obeying the ``Densification Power Law’’ C. Faloutsos (CMU)

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