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

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. Burstiness • Edge and weight additions are bursty, and self-similar. • Entropy plots [Wang+’02] is a measure of burstiness. D Weights Entropy Time Resolution

8. Entropy plots • From time series data, begin with resolution T/2. • Record entropy HR. D Weights Entropy Time Resolution

9. Entropy plots • From time series data, begin with resolution T/2. • Record entropy HR. • Recursively take finer resolutions. D Weights Entropy Resolution Time

10. Resolution Entropy Entropy Plots • Self-similarity  Linear plot slope = 5.9

11. Resolution Entropy Entropy Plots • Self-similarity  Linear plot Uniform: slope=1 time slope = 5.9

12. Resolution Entropy Entropy Plots • Self-similarity  Linear plot Uniform: slope=1 Point mass: slope=0 time time slope = 5.9

13. Resolution Entropy Entropy Plots • Self-similarity  Linear plot Uniform: slope=1 Point mass: slope=0 Bursty: 0.2 < slope < 0.9 time time slope = 5.9 McGlohon, Akoglu, Faloutsos KDD08

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

15. Datasets

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

17. Observation 1:λ1Power Law (LPL) Q: How does the principal eigenvalue λ1 change over time A: λ1 (t) and the number of edges E(t) over time follow a power law with exponent less than 0.5, especially after the ‘gelling point’. λ1(t) ∝ E(t) α, α ≤ 0.5

18. λ1Power Law (LPL) cont. Theorem: For a connected, undirected graph G with N nodes and E edges, without self-loops and multiple edges; DBLP Author-Conference network

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

20. Observation 3: Edge Weights PL Q: How does the weight of an edge relate to “popularity” if its adjacent nodes A: Weight of the link wi,j between two given nodes i and j in a given graph G has a power law relation with the weights wiand wj of the nodes; FEC Committee-to- Candidate network

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

22. 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)

23. Problem Definition cont. • 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)

24. One solution: Kronecker Product Intuition: Self-similarity! • Communities within communities • Recursion yields modular network behavior

25. One solution: Kronecker Product

26. Recursive Tensor Product(RTM) • Use of tensors: • 3rd mode is time • Initial tensor I is a realistic graph itself RTM of a (3x3x3) tensor by itself

27. RTM cont.

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

29. Experimental Results (I) BLOG NETWORK RTM MODEL

30. Experimental Results (II) BLOG NETWORK RTM MODEL

31. Conclusion • Largest (un)weighted principal eigenvalues are power-law related to the number of edges in real graphs. • Weight of an edge is related to the total weights of its incident nodes. • Recursive Tensor Multiplication is a recursive method to generate weighted, time-evolving, self-similar, modular networks.

32. Future Directions • Largest eigenvalues of the Laplacian matrices • Second largest eigenvalue – related to global connectivity – conductance – mixing rate of random walk on graph • Probabilistic version of RTM • Fitting graphs