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Gelling, and Melting, Large Graphs by Edge Manipulation

This presentation by Hanghang Tong explores the manipulation of graph edges in order to understand and control the propagation of viruses, rumors, and ideas in large networks. The goal is to minimize or maximize the spread of the propagation through targeted edge deletion or addition. The presentation also discusses the importance of eigenvalues in determining the tipping point for propagation.

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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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