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FastEigen-FunctionsTrackingonDynamicGraphs

Chen Chen and Hanghang Tong

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Graphs are Ubiquitous!

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Hospital NetworkCollaboration 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 of 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 CentralityThe 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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+ =

InitializeUpdate

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 on 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 Day900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Fi

rst E

igen

valu

e Er

ror R

ate

First Eigenvalue

Trip-BasicTripIterLow-RankNystrom

Ours

Effectiveness Comparison

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Day15 Day30 Day45 Day60 Day75 Day90

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Effectiveness vs. Efficiency

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0 5 10 15 20 25 30 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

>=0.8

Speedup

Firs

t Eig

enve

ctor

Err

or R

ate

k=5

Trip-BasicTripIterLow-RankSVD deltaNystrom

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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0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time Stamp

Estim

ated

Err

or (F

irst E

igen

valu

e)Error Estimate

Trip-BasicTripOption1Option2

Time Stamps

Erro

rRat

esTime 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 A

dded

Edg

es P

reci

sion

Top

10 A

dded

Edg

es P

reci

sion

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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