1 / 138

Mining Billion-node Graphs: Patterns, Generators and Tools

Mining Billion-node Graphs: Patterns, Generators and Tools. Christos Faloutsos CMU (on sabbatical at google). 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

denis
Download Presentation

Mining Billion-node Graphs: Patterns, Generators and Tools

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Mining Billion-node Graphs: Patterns, Generators and Tools Christos Faloutsos CMU (on sabbatical at google)

  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 + Google)

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

  4. Graphs - why should we care? Internet Map [lumeta.com] Food Web [Martinez ’91] • Social networks • (orkut, linkedIn …) • twitter Friendship Network [Moody ’01] C. Faloutsos (CMU + Google)

  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 + Google)

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

  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 + Google)

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

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

  10. Problem #1 - network and graph mining • What does the Internet look like? • What 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 + Google)

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

  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 + Google)

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

  14. -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 + Google)

  15. 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 + Google)

  16. 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 + Google)

  17. But: How about graphs from other domains? C. Faloutsos (CMU + Google)

  18. 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 + Google)

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

  20. 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 + Google)

  21. 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 + Google)

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

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

  24. Triangle Law: #S.3 [Tsourakakis ICDM 2008] HEP-TH ASN Epinions X-axis: # of participating triangles Y: count (~ pdf) C. Faloutsos (CMU + Google)

  25. Triangle Law: #S.3 [Tsourakakis ICDM 2008] HEP-TH ASN Epinions X-axis: # of participating triangles Y: count (~ pdf) C. Faloutsos (CMU + Google)

  26. 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 + Google)

  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? C. Faloutsos (CMU + Google)

  28. 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 (S2) , we only need the top few eigenvalues! C. Faloutsos (CMU + Google)

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

  30. 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 + Google)

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

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

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

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

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

  36. 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 + Google)

  37. 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 + Google)

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

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

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

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

  42. 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 + Google)

  43. Bipartite Communities! patents from same inventor(s) `cut-and-paste’ bibliography! magnified bipartite community C. Faloutsos (CMU + Google)

  44. 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 + Google)

  45. 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 + Google)

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

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

  48. 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 + Google)

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

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

More Related