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Gelling, and Melting, Large Graphs by Edge Manipulation

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

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Gelling, and Melting, Large Graphs by Edge Manipulation

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

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

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

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

  5. Sick Healthy An Example: Flu/Virus Propagation 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  This paper 5

  6. Motivation: An Illustrative Example Q1: Understanding the Tipping Point (Background) Q2: Minimize Propagation Q3: Maximize Propagation Conclusion Roadmap

  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 Cvpm (by prop. model itself) Theorem [Faloutsos2 + ICDM 2011]: No epidemic Ifλ x (Cvpm) ≤ 1. 7

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

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

  10. Motivation: An Illustrative Example Q1: Understanding the Tipping Point (Background) Q2: Minimize Propagation Q3: Maximize Propagation Conclusion Roadmap

  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 Good 11

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

  13. Minimizing Propagation: Evaluations Log (Infected Ratio) (better) Our Method Time Ticks 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) Original Graph A Line Graph L(A) • Questions? • Edge Deletion on A = Node Deletion on L(A)? • Which strategy is better (when both feasible)?

  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: Theorem: Line Graph Spectrum. Eigenvalue of A Eigenvalue of L(A) Theorem: Hardness of Node Deletion. Find Optimal k-node Immunization is NP-Hard 15

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

  17. Motivation: An Illustrative Example Q1: Understanding the Tipping Point (Background) Q2: Minimize Propagation Q3: Maximize Propagation Conclusion Roadmap

  18. Maximizing Propagation: Edge Addition • Given: a graph A, virus prop model and budget k; • Find: add k ‘best’ new edges into A. • By 1st order perturbation, we have λs - λ ≈Gv(S)= c ∑eєS u(ie)v(je) • So, we are done  need O(n2-m) complexity Right eigen-score of target Left eigen-score of source Low Gv High Gv 18

  19. Maximizing Propagation: Edge Addition λs - λ ≈Gv(S)= c ∑eєS u(ie)v(je) • Q: How to Find k new edges w/ highest Gv(S) ? • A: Modified Fagin’s algorithm #2: Sorting Targets by v k k+d #3: Search space Search space k k+d #1: Sorting Sources by u Time Complexity: O(m+nt+kt2), t = max(k,d) :existing edge

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

  21. 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 Conclusion t = 1 t = 2 t = 3

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