Network Reconstruction under Compressive Sensing

1. Network Reconstruction under Compressive Sensing • By: MotaharehEslamiMehdiabadi • eslami@ce.sharif.edu • Sharif University of Technology • Authors: PayamSiyari, Hamid R. Rabiee • MostafaSalehi, MotaharehEslamiMehdiabadi

2. Outline • Introduction • Related Work • Network Reconstruction • Compressive Sensing • Problem Formulation • Proposed Framework: CS-NetRec • Experimental Evaluation • Conclusion

3. Introduction Large Scale Unknown Structure Many Systems Modeled as Networks Partial Observations

4. Introduction (cont’d) • Network Reconstruction Problem: • Given a network with missing edges • Assumptions: • Certain observable quantities on the network • Can have partial observations • Process-> Node values • Goal: Uncover network structure

5. Introduction (cont’d) • Network Reconstruction Problem: (a) An example network Figure 1: An example of the network reconstruction problem.

6. Introduction (cont’d) • Network Reconstruction Problem: (b) The example network with no Information about the edges Figure 1: An example of the network reconstruction problem.

7. Introduction (cont’d) • Network Reconstruction Problem: (c) A partial observation from the network structure. The process output is f(v1,v2,v3,v5,v6). Figure 1: An example of the network reconstruction problem.

8. Introduction (cont’d) • Network Reconstruction Problem: (c) Another partial observation from the network structure. The process output is f(v2,v3,v4,v5,v6). Figure 1: An example of the network reconstruction problem.

9. Introduction (cont’d) • Network Reconstruction Problem: • Encountered in many real-world applications: • Inaccuracies in uncovering the Protein interaction data [1]. Figure 2: Protein interaction network in yeast image from http://www.bordalierinstitute.com/images/yeastProteinInteractionNetwork.jpg

10. Introduction (cont’d) • Network Reconstruction Problem: • Encountered in many real-world applications: • In the social networks analysis, particularly online social networks (OSNs), there is missing data due to several reasons: • Security • User privacy • Data aggregation overhead, etc. • In recommender systems, especially in OSNs

11. Introduction (cont’d)

12. Related Work

13. Network Reconstruction

14. Compressive Sensing

15. Compressive Sensing

16. Compressive Sensing

17. Compressive Sensing Looking for sparse solutions Combinatorial, NP-Hard

18. Compressive Sensing

19. Compressive Sensing LASSO [22,23]

20. Compressive Sensing Mainly studied in signal & image processing [26-28]

21. Problem Formulation 21

22. Preliminaries • Diffusion of information, e.g. news headlines, virus, rumor, etc.

23. Problem Formulation • Information Diffusion & The Cascading Behavior (a) A news blogs network Figure 3:An example of information diffusion on a news blogs network.

24. Problem Formulation • Information Diffusion & The Cascading Behavior: (b) Example cascade Figure 3:An example of information diffusion on a news blogs network.

25. Problem Formulation • Information Diffusion & The Cascading Behavior: Cascade Hit times = <(A, tA), (B, tB), (D, tD), (G, tG), (E, tE), > (b) Example cascade Figure 3:An example of information diffusion on a news blogs network.

26. Problem Formulation Conditional probability of observing cascade c spreading from u to v [16]

27. Problem Formulation Conditional probability of observing cascade c spreading from u to v [16] The likelihood of a cascade Spreading in a given tree pattern T [16] β: The probability that a cascade will continue

28. Problem Formulation Conditional probability of observing cascade c spreading from u to v [16] The likelihood of a cascade Spreading in a given tree pattern T [16] The probability that a cascade c can occur in the graph G [16] Computationally expensive!

29. Problem Formulation Conditional probability of observing cascade c spreading from u to v [16] The likelihood of a cascade Spreading in a given tree pattern T [16] The probability that a cascade c can occur in the graph G [16] The approximated tree and its corresponding probability

30. Problem Formulation (b) Example cascade Figure 4:An example of information diffusion on a news blogs network.

31. Problem Formulation (c) Most likely cascade tree shown by dotted links Figure 4:An example of information diffusion on a news blogs network.

32. Proposed Framework • Defining cascade probabilities as an inner product: • Where:

33. Proposed Framework(CS-NetRec) Each equation = a cascade.

34. Proposed Framework

35. Experimental Evaluation 35

36. Dataset * Synthetic • Erdos-Reyni (ER) • Small-World • Barabasi-Albert(BA) • Core-Priphery (Kronecker) * Real 36

37. Evaluation 37

38. Cascade Dependency Real Networks Synthetic Networks

39. The Effect of Sparsity Performed in ER network

40. Performance Comparison (with NetInf) BA ER Small World Core Kron.

41. Performance Comparison (with NetInf) C.elegans Football US Top500

42. Conclusion

43. Future Work

44. Q&A

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