1 / 47

RTM: Laws and a Recursive Generator for Weighted Time-Evolving Graphs

RTM: Laws and a Recursive Generator for Weighted Time-Evolving Graphs. Leman Akoglu , Mary McGlohon, Christos Faloutsos Carnegie Mellon University School of Computer Science. Motivation. Graphs are popular! Social, communication, network traffic, call graphs…. …and interesting

taariq
Download Presentation

RTM: Laws and a Recursive Generator for Weighted Time-Evolving Graphs

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. RTM: Laws and a Recursive Generator for Weighted Time-Evolving Graphs Leman Akoglu,Mary McGlohon, Christos Faloutsos Carnegie Mellon University School of Computer Science

  2. Motivation • Graphs are popular! • Social, communication, network traffic, call graphs… • …and interesting • surprising common properties for static and un-weighted graphs • How about weighted graphs? • …and their dynamic properties? • How can we model such graphs? • for simulation studies, what-if scenarios, future prediction, sampling

  3. Outline • Motivation • Related Work - Patterns - Generators - Burstiness • Datasets • Laws and Observations • Proposed graph generator: RTM • (Sketch of proofs) • Experiments • Conclusion

  4. Graph Patterns (I) • Small diameter • 19 for the web [Albert and Barabási, 1999] • 5-6 for the Internet AS topology graph [Faloutsos, Faloutsos, Faloutsos, 1999] • Shrinking diameter [Leskovec et al.‘05] • Power Laws Blog Network diameter y(x) = Ax−γ, A>0, γ>0 time

  5. Graph Patterns (II) • Densification [Leskovec et al.‘05] and Weight [McGlohon et al.‘08] Power-laws • Eigenvalues Power Law [Faloutsos et al.‘99] • Degree Power Law [Richardson and Domingos, ‘01] |W| |srcN| Eigenvalue Count |dstN| In-degree Rank |E| Epinions who-trusts-whom graph Inter-domain Internet graph DBLP Keyword-to-Conference Network

  6. Graph Generators • Erdős-Rényi (ER)model [Erdős, Rényi ‘60] • Small-world model [Watts, Strogatz ‘98] • Preferential Attachment [Barabási, Albert ‘99] • Edge Copying models [Kumar et al.’99], [Kleinberg et al.’99], • Forest Fire model [Leskovec, Faloutsos ‘05] • Kronecker graphs [Leskovec, Chakrabarti, Kleinberg, Faloutsos ‘07] • Optimization-based models [Carlson,Doyle,’00] [Fabrikant et al. ’02]

  7. Resolution Entropy Burstiness • Edge and weight additions are bursty, and self-similar. • Entropy plots [Wang+’02] is a measure of burstiness. Bursty: 0.2 < slope < 0.9 Weights Entropy slope = 5.9 Time Resolution

  8. Outline • Motivation • Related Work - Patterns - Generators • Datasets • Laws and Observations • Proposed graph generator: RTM • Sketch of proofs • Experiments • Conclusion

  9. Datasets 1 Bipartite networks: |N| |E| time 1. AuthorConference 17K, 22K, 25 yr. 2. KeywordConference 10K, 23K, 25 yr. 3. AuthorKeyword 27K, 189K, 25 yr. 4. CampaignOrg 23K, 877K, 28 yr. 9

  10. Datasets 3 Bipartite networks: |N| |E| time 1. AuthorConference 17K, 22K, 25 yr. 2. KeywordConference 10K, 23K, 25 yr. 3. AuthorKeyword 27K, 189K, 25 yr. 4. CampaignOrg 23K, 877K, 28 yr.

  11. Datasets 3 Bipartite networks: |N| |E| time 1. AuthorConference 17K, 22K, 25 yr. 2. KeywordConference 10K, 23K, 25 yr. 3. AuthorKeyword 27K, 189K, 25 yr. 4. CampaignOrg 23K, 877K, 28 yr. Unipartite networks: |N| |E| time 5. BlogNet 60K, 125K, 80 days 6. NetworkTraffic 21K, 2M, 52 months 20MB 11

  12. Datasets 3 Bipartite networks: |N| |E| time 1. AuthorConference 17K, 22K, 25 yr. 2. KeywordConference 10K, 23K, 25 yr. 3. AuthorKeyword 27K, 189K, 25 yr. 4. CampaignOrg 23K, 877K, 28 yr. Unipartite networks: |N| |E| time 5. BlogNet 60K, 125K, 80 days 6. NetworkTraffic 21K, 2M, 52 months 20MB 25MB 5MB 12

  13. Outline • Motivation • Related Work - Patterns - Generators • Datasets • Laws and Observations • Proposed graph generator: RTM • Sketch of proofs • Experiments • Conclusion

  14. Observation 1: λ1Power Law(LPL) Q1: How does the principal eigenvalue λ1 of the adjacency matrixchange over time? Q2: Why should we care? 14

  15. Observation 1: λ1Power Law(LPL) Q1: How does the principal eigenvalue λ1 of the adjacency matrixchange over time? Q2: Why should we care? A2: λ1 is closely linked to density and maximumdegree, also relates to epidemic threshold. A1: λ1(t) ∝ E(t) α, α ≤ 0.5

  16. λ1Power Law (LPL) cont. Theorem: For a connected, undirected graph G with N nodes and E edges, without self-loops and multiple edges; λ1(G) ≤ {2 (1 – 1/N) E}1/2 For large N, 1/N  0 and λ1(G) ≤ cE1/2 DBLP Author-Conference network

  17. Observation 2:λ1,wPower Law (LWPL) Q: How does the weighted principal eigenvalue λ1,wchange over time? A: λ1,w(t) ∝ E(t) β DBLP Author-Conference network Network Traffic

  18. Observation 3:Edge Weights PL(EWPL) Q: How does the weight of an edge relate to “popularity” if its adjacent nodes? A: wi,j ∝ wi * wj Wi,j j i FEC Committee-to- Candidate network Wi Wj

  19. Outline Motivation Related Work - Patterns - Generators Datasets Laws and Observations Proposed graph generator: RTM Sketch of proofs Experiments Conclusion 19

  20. Problem Definition • Generate a sequence of realistic weighted graphs that will obey all the patterns over time. • SUGP: staticun-weighted graph properties • small diameter • power law degree distribution • SWGP: staticweighted graph properties • the edge weight power law (EWPL) • the snapshot power law (SPL)

  21. Problem Definition • DUGP: dynamicun-weighted graph properties • the densification power law (DPL) • shrinking diameter • bursty edge additions • λ1 Power Law (LPL) • DWGP: dynamicweighted graph properties • the weight power law (WPL) • bursty weight additions • λ1,w Power Law (LWPL)

  22. 2D solution: Kronecker Product • Idea: Recursion • Intuition: • Communities within communities • Self-similarity • Power-laws

  23. 2D solution: Kronecker Product

  24. 3D solution: Recursive Tensor Multiplication(RTM) I 2 3 4 X I1,1,1 24

  25. 3D solution: Recursive Tensor Multiplication(RTM) I 2 3 4 X I1,2,1 25

  26. 3D solution: Recursive Tensor Multiplication(RTM) I 2 3 4 X I1,3,1 26

  27. 3D solution: Recursive Tensor Multiplication(RTM) I 2 3 4 X I1,4,1 27

  28. 3D solution: Recursive Tensor Multiplication(RTM) I 2 3 4 X I2,1,1 28

  29. 3D solution: Recursive Tensor Multiplication(RTM) I 2 3 4 X I3,1,1 29

  30. 3D solution: Recursive Tensor Multiplication(RTM) I 2 3 4 30

  31. 3D solution: Recursive Tensor Multiplication(RTM) I 2 3 4 X I1,1,2 31

  32. 3D solution: Recursive Tensor Multiplication(RTM) I 2 3 4 X I1,2,2 32

  33. 3D solution: Recursive Tensor Multiplication(RTM) 22 I 2 32 3 4 42 33

  34. 3D solution: Recursive Tensor Multiplication(RTM) t-slices time senders recipients 34

  35. 3D solution: Recursive Tensor Multiplication(RTM) t1 t2 t3 35

  36. 3D solution: Recursive Tensor Multiplication(RTM) 2 3 4 2 3 4 2 3 4 1 1 1 1 1 1 2 3 2 1 1 2 5 3 3 2 3 2 4 4 4 t2 t3 t1 3 4 4 4 3 2 3 2 2 2 3 3 2 2 2 1 1 5 2 1 4 1 36

  37. Outline Motivation Related Work - Patterns - Generators Datasets Laws and Observations Proposed graph generator: RTM (Sketch of proofs) Experiments Conclusion 37

  38. Experimental Results • SUGP: • small diameter • PL Degree Distribution • SWGP: • Edge Weights PL • Snaphot PL • DUGP: • Densification PL • shrinking diameter • bursty edge additions • λ1 PL • DWGP: • Weight PL • bursty weight additions • λ1,w PL diameter Time

  39. Experimental Results • SUGP: • small diameter • PL Degree Distribution • SWGP: • Edge Weights PL • Snaphot PL • DUGP: • Densification PL • shrinking diameter • bursty edge additions • λ1 PL • DWGP: • Weight PL • bursty weight additions • λ1,w PL count degree 39

  40. Experimental Results • SUGP: • small diameter • PL Degree Distribution • SWGP: • Edge Weights PL • Snaphot PL • DUGP: • Densification PL • shrinking diameter • bursty edge additions • λ1 PL • DWGP: • Weight PL • bursty weight additions • λ1,w PL |E| |N| 40

  41. Experimental Results • SUGP: • small diameter • PL Degree Distribution • SWGP: • Edge Weights PL • Snaphot PL • DUGP: • Densification PL • shrinking diameter • bursty edge additions • λ1 PL • DWGP: • Weight PL • bursty weight additions • λ1,w PL |W| |E| 41

  42. Experimental Results • SUGP: • small diameter • PL Degree Distribution • SWGP: • Edge Weights PL • Snaphot PL • DUGP: • Densification PL • shrinking diameter • bursty edge additions • λ1 PL • DWGP: • Weight PL • bursty weight additions • λ1,w PL 42

  43. Experimental Results • SUGP: • small diameter • PL Degree Distribution • SWGP: • Edge Weights PL • Snaphot PL • DUGP: • Densification PL • shrinking diameter • bursty edge additions • λ1 PL • DWGP: • Weight PL • bursty weight additions • λ1,w PL In-weight In-degree Out-weight Out-degree 43

  44. Experimental Results • SUGP: • small diameter • PL Degree Distribution • SWGP: • Edge Weights PL • Snaphot PL • DUGP: • Densification PL • shrinking diameter • bursty edge additions • λ1 PL • DWGP: • Weight PL • bursty weight additions • λ1,w PL 44

  45. Experimental Results • SUGP: • small diameter • PL Degree Distribution • SWGP: • Edge Weights PL • Snaphot PL • DUGP: • Densification PL • shrinking diameter • bursty edge additions • λ1 PL • DWGP: • Weight PL • bursty weight additions • λ1,w PL λ1 |E| λ1,w |E| 45

  46. Conclusion Wi,j Wj Wi In real graphs, (un)weighted largest eigenvalues are power-law related to number of edges. Weight of an edge is related to the total weights and of its incident nodes. Recursive Tensor Multiplication is a recursive method to generate (1)weighted, (2)time-evolving, (3)self-similar, (4)power-law networks. Future directions: • Probabilistic version of RTM • Fitting the initial tensor I 46

  47. Contact us Mary McGlohon www.cs.cmu.edu/~mmcgloho mmcgloho@cs.cmu.edu Christos Faloutsos www.cs.cmu.edu/~christos christos@cs.cmu.edu Leman Akoglu www.andrew.cmu.edu/~lakoglu lakoglu@cs.cmu.edu 47

More Related