Fast Eigen-Functions Tracking on Dynamic Graphs Chen Chen and - PowerPoint PPT Presentation

Fast Eigen-Functions Tracking on Dynamic Graphs Chen Chen and Hanghang Tong - 1 - Arizona State University

Graphs are Ubiquitous! Collaboration Network Hospital Network Autonomous Network Transportation Network - 2 - Arizona State University

Key Graph Parameters § P1: Epidemic Threshold (Propagation network) § P2: Centrality of nodes (All networks) § P3: Clustering Coefficient (Social network) § P4: Graph Robustness (Router/Transportation) - 3 - Arizona State University

P1: Epidemic Threshold § Questions : How easy is it to spread disease? § Intuition 1 1 1 1 § Solution : Related to the leading eigenvalue of the adjacency matrix for ANY cascade model [ICDM 2011] - 4 - Arizona State University

P2: Node Centrality § Question : How important is a node? § Intuition : Having more important friends are considered influential § Commonly used : Eigenvector Centrality The eigenvector corresponding to the leading eigenvalue - 5 - Arizona State University

P3: Clustering Coefficient § Question : How the nodes in the graph cluster together? § Intuition : § Solution : - 6 - Arizona State University

P4: Graph Robustness § Question : How robust is a graph under external attack? § Intuition : Power Grid Sandy Aftermath [wikipedia.com ] [forbes.com] § Solution : [SDM2014] - 7 - Arizona State University

Challenge: Graphs are Dynamic! Social Networks Propagation Netoworks Router Netoworks Transportation Netoworks [www.cisco.com] [www.mapofworld.com] How to track key graph parameters? - 8 - Arizona State University

Eigen-Function Tracking § Q1. Track key graph parameters § Q2. Estimate the error of tracking algorithms § Q3. Analyze attribution for drastic changes - 9 - Arizona State University

Roadmap § Motivations § Q1: Efficient tracking algorithms § Q2: Error estimation methods § Q3: Attribution analysis § Conclusion - 10 - Arizona State University

Key Graph Parameters § Observations: P1-P4 are all eigen-functions P1. Epidemic Threshold P2. Eigenvector Centrality P3. Clustering Coefficient (Triangles) P4. Robustness Score - 11 - Arizona State University

Goal: Tracking Top Eigen-Pairs § Method 1. – Calculate from scratch whenever the structure changes – Lanczos algorithm Too costly for fast-changing large graphs! - 12 - Arizona State University

Key Idea + = Too Expensive Initialize Update - 13 - Arizona State University

Key Idea: Incrementally Update § Intuition: § Solution: Matrix Perturbation Theory Time stamp omitted for brevity. - 14 - Arizona State University

Details: Step 1 First order perturbation terms High order perturbation terms Challenge: two equation with four variables Solution: Introduce additional constraints and assumptions Constraints Assumptions - 15 - Arizona State University

Details: Estimate § Discard high order term 1 , 2 Multiply on both side 3 , - 16 - Arizona State University

Estimate (Option 1) (Discard high order) 1 Multiply on both side 2 , 3 (Trip-Basic) 4 , Time Complexity: Lanczos: - 17 - Arizona State University

Estimate (Option 2) § Keep high order perturbation terms (Trip) (Trip-Basic) Time Complexity: - 18 - Arizona State University

Evaluation § Data set: – Autonomous systems AS-733 (https://snap.stanford.edu/data/as.html) – 100 days time spans • (11/08/1997-02/16/1998) • (03/15/1998-06/26/1998) – Maximum #nodes = 4,013 – Maximum #edges = 14,399 - 19 - Arizona State University

Trip-Basic vs. Trip: Effectiveness Time Stamp - 20 - Arizona State University

Effectiveness Comparison First Eigenvalue 1 Trip-Basic 0.9 Trip Iter 0.8 First Eigenvalue Error Rate Low-Rank 0.7 Nystrom 0.6 0.5 0.4 0.3 0.2 Ours 0.1 0 Day15 Day30 Day45 Day60 Day75 Day90 Day15 Day30 Day45 Day60 Day75 Day90 - 21 - Arizona State University

Effectiveness vs. Efficiency Speed-up k=5 >=0.8 Trip-Basic 0.7 Trip First Eigenvector Error Rate Iter 0.6 Low-Rank SVD delta 0.5 Nystrom 0.4 0.3 0.2 Ours 0.1 0 0 5 10 15 20 25 30 Speedup Speed-up - 22 - Arizona State University

Roadmap § Motivations § Q1: Efficient tracking algorithms § Q2: Error estimation methods § Q3: Attribution analysis § Conclusion - 23 - Arizona State University

Q2.Error Estimation § Setting: Time Steps Error Estimate 0.7 Estimated Errors 0.6 Estimated Error (First Eigenvalue) 0.5 Error Rates 0.4 True Errors 0.3 0.2 Trip-Basic Trip Option1 0.1 Option2 0 0 10 20 30 40 50 60 70 80 90 100 Time Stamp Time Stamps - 24 - Arizona State University

Q3. Attribution Analysis § Precision (Edge Addition) First Eigenvalue Robustness Score Top 10 Added Edges Precision Top 10 Added Edges Precision Number of Eigen-Pairs Number of Eigen-Pairs - 25 - Arizona State University

Conclusion § Goal: Tracking key graph parameters § Solutions: – Key idea: • Fixed eigen-space, Matrix perturbation theory – Algorithms: Trip-Basic, Trip § More Details: – Error Estimation – Attribution Analysis - 26 - Arizona State University