1. © 2012 IBM Corporation IBM Research Gelling, and Melting, Large Graphs by Edge Manipulation Joint Work by Hanghang Tong (IBM) B. Aditya Prakash (Virginia Tech.) Tina Eliassi-Rad (Rutgers) Michalis Faloutsos (UCR) Christos Faloutsos (CMU) Presenter: Hanghang Tong

2. An Example: Flu/Virus/Rumor/Idea Propagation HealthySick Contact 2

3. An Example: Flu/Virus Propagation HealthySick Contact 1: Sneeze to neighbors 2: Some neighbors  Sick 3: Try to recover 3

4. An Example: Flu/Virus Propagation HealthySick Contact 1: Sneeze to neighbors 2: Some neighbors  Sick 3: Try to recover Q: How to guild propagation by opt. link structure? 4

5. An Example: Flu/Virus Propagation HealthySick Contact 1: Sneeze to neighbors 2: Some neighbors  Sick 3: Try to recover Q: How to guild propagation by opt. link structure? - Q1: Understand tipping point  existing work - Q2: Minimize the propagation - Q3: Maximize the propagation 5  This paper

6. IBM Research © 2012 IBM CorporationSocial Analytics & Collaboration Technologies Group 6 Roadmap  Motivation: An Illustrative Example  Q1: Understanding the Tipping Point (Background)  Q2: Minimize Propagation  Q3: Maximize Propagation  Conclusion

7. Eigenvalue is the Key! [ICDM2011] (Informal Description) For, – any arbitrary topology (adjacency matrix A) – any virus propagation model (VPM) in standard literature (~25 in total) the epidemic threshold depends only on – the λ (leading eigenvalue of A), – some model constant C vpm (by prop. model itself) Theorem [ Faloutsos 2 + ICDM 2011 ]: No epidemic If λ x (C vpm ) ≤ 1. 7

8. Epidemic Threshold for Alternating Behavior [PKDD 2010, Networking 2011] Theorem [PKDD 2010, Networking 2011]: No epidemic If λ(S) ≤ 1. System matrix S = Π i S i S i = (1-δ)I + β A i day N N night N N AiAi …… Log (Infection Ratio) Time Ticks At Threshold Below Above 8

9. Why is λ So Important? λ  Capacity of a Graph: Larger λ  better connected 9

10. IBM Research © 2012 IBM CorporationSocial Analytics & Collaboration Technologies Group 10 Roadmap  Motivation: An Illustrative Example  Q1: Understanding the Tipping Point (Background)  Q2: Minimize Propagation  Q3: Maximize Propagation  Conclusion

11. Minimizing Propagation: Edge Deletion Given: a graph A, virus prop model and budget k; Find: delete k ‘best’ edges from A to minimize λ Bad 11 Good

12. Q: How to find k best edges to delete efficiently? Left eigen-score of source Right eigen-score of target 12

13. Minimizing Propagation: Evaluations Time Ticks Log (Infected Ratio) (better) Our Method Aa Data set: Oregon Autonomous System Graph (14K node, 61K edges)

14. Discussions: Node Deletion vs. Edge Deletion Observations: Node or Edge Deletion  λ Decrease Nodes on A = Edges on its line graph L(A) Questions? Edge Deletion on A = Node Deletion on L(A)? Which strategy is better (when both feasible)? Original Graph ALine Graph L(A)

15. Discussions: Node Deletion vs. Edge Deletion Q: Is Edge Deletion on A = Node Deletion on L(A)? A: Yes! But, Node Deletion itself is not easy: 15 Theorem: Hardness of Node Deletion. Find Optimal k-node Immunization is NP-Hard Theorem: Line Graph Spectrum. Eigenvalue of A  Eigenvalue of L(A)

16. Discussions: Node Deletion vs. Edge Deletion Q: Which strategy is better (when both feasible)? A: Edge Deletion > Node Deletion 16 (better) Green: Node Deletion (e.g., shutdown a twitter account) Red: Edge Deletion (e.g., un-friend two users)

17. IBM Research © 2012 IBM CorporationSocial Analytics & Collaboration Technologies Group 17 Roadmap  Motivation: An Illustrative Example  Q1: Understanding the Tipping Point (Background)  Q2: Minimize Propagation  Q3: Maximize Propagation  Conclusion

18. Maximizing Propagation: Edge Addition Given: a graph A, virus prop model and budget k; Find: add k ‘best’ new edges into A. By 1 st order perturbation, we have λ s - λ ≈Gv(S)= c ∑ eєS u(i e )v(j e ) So, we are done  need O(n 2 -m) complexity Left eigen-score of source Right eigen-score of target Low Gv High Gv 18

19. λ s - λ ≈Gv(S)= c ∑ eєS u(i e )v(j e ) Q: How to Find k new edges w/ highest Gv(S) ? A: Modified Fagin’s algorithm k k #3: Search space k+d Search space :existing edgeTime Complexity: O(m+nt+kt 2 ), t = max(k,d) #1: Sorting Sources by u #2: Sorting Targets by v Maximizing Propagation: Edge Addition

20. Maximizing Propagation: Evaluation Time Ticks Log (Infected Ratio) (better) 20 Our Method

21. IBM Research © 2012 IBM CorporationSocial Analytics & Collaboration Technologies Group Conclusion  Goal: Guild Influence Prop. by Opt. Link Structure  Our Observation: Opt. Influence Prop = Opt. λ  Our Solutions: –NetMel to Minimize Propagation –NetGel to Maximize Propagation t = 1t = 2t = 3