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

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Fast Eigen-Functions Tracking on Dynamic Graphs

Chen Chen and Hanghang Tong

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Graphs are Ubiquitous!

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Hospital Network Collaboration Network Autonomous Network Transportation Network

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Key Graph Parameters

§P1: Epidemic Threshold (Propagation network) §P2: Centrality of nodes (All networks) §P3: Clustering Coefficient (Social network) §P4: Graph Robustness (Router/Transportation)

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P1: Epidemic Threshold

§ Questions: How easy is it to spread disease? § Intuition § Solution: Related to the leading eigenvalue

• f the adjacency matrix for ANY cascade model

[ICDM 2011]

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1 1 1 1

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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

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P3: Clustering Coefficient

§ Question: How the nodes in the graph

cluster together?

§ Intuition: § Solution:

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P4: Graph Robustness

§ Question: How robust is a graph under

external attack?

§ Intuition: § Solution:

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Power Grid [wikipedia.com]

Sandy Aftermath [forbes.com] [SDM2014]

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Challenge: Graphs are Dynamic!

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Social Networks Propagation Netoworks Router Netoworks [www.cisco.com] Transportation Netoworks [www.mapofworld.com]

How to track key graph parameters?

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Eigen-Function Tracking

§ Q1. Track key graph parameters § Q2. Estimate the error of tracking algorithms § Q3. Analyze attribution for drastic changes

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Roadmap

§ Motivations § Q1: Efficient tracking algorithms § Q2: Error estimation methods § Q3: Attribution analysis § Conclusion

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Key Graph Parameters

§ Observations: P1-P4 are all eigen-functions

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• P1. Epidemic Threshold
• P2. Eigenvector Centrality
• P3. Clustering Coefficient

(Triangles)

• P4. Robustness Score

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Goal: Tracking Top Eigen-Pairs

§ Method 1.

– Calculate from scratch whenever the

structure changes

– Lanczos algorithm

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Too costly for fast-changing large graphs!

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Key Idea

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+ =

Initialize Update Too Expensive

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Key Idea: Incrementally Update

§ Intuition: § Solution: Matrix Perturbation Theory

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Time stamp omitted for brevity.

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Details: Step 1

Challenge: two equation with four variables

Constraints Assumptions

Solution: Introduce additional constraints and assumptions

First order perturbation terms High order perturbation terms

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Details: Estimate

§ Discard high order term

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Multiply on both side

, , 1 2 3

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Estimate (Option 1)

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Multiply

• n both side

(Trip-Basic)

, ,

Time Complexity: (Discard high order)

Lanczos: 1 2 3 4

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Estimate (Option 2)

§ Keep high order perturbation terms

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Time Complexity:

(Trip-Basic) (Trip)

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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

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Trip-Basic vs. Trip: Effectiveness

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Time Stamp

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Day15 Day30 Day45 Day60 Day75 Day90 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

First Eigenvalue Error Rate First Eigenvalue Trip-Basic Trip Iter Low-Rank Nystrom

Ours

Effectiveness Comparison

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Day15 Day30 Day45 Day60 Day75 Day90

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Effectiveness vs. Efficiency

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5 10 15 20 25 30 0.1 0.2 0.3 0.4 0.5 0.6 0.7 >=0.8

Speedup First Eigenvector Error Rate k=5 Trip-Basic Trip Iter Low-Rank SVD delta Nystrom

Speed-up

Speed-up

Ours

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Roadmap

§ Motivations § Q1: Efficient tracking algorithms § Q2: Error estimation methods § Q3: Attribution analysis § Conclusion

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Q2.Error Estimation

§ Setting:

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10 20 30 40 50 60 70 80 90 100 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Time Stamp Estimated Error (First Eigenvalue) Error Estimate Trip-Basic Trip Option1 Option2

Time Stamps Error Rates

Time Steps True Errors Estimated Errors

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• Q3. Attribution Analysis

§ Precision (Edge Addition)

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Number of Eigen-Pairs Number of Eigen-Pairs Top 10 Added Edges Precision Top 10 Added Edges Precision First Eigenvalue Robustness Score

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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

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