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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

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

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  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

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