© 2012 IBM Corporation

IBM Research

Gelling, and Melting, Large Graphs by Edge Manipulation

Joint Work by

Hanghang Tong(IBM)

B. Aditya Prakash(Virginia Tech.)

Tina Eliassi-Rad (Rutgers)

Michalis Faloutsos (UCR)

Christos Faloutsos (CMU)

Presenter: Hanghang Tong

An Example: Flu/Virus/Rumor/Idea Propagation

HealthySick

Contact

2

An Example: Flu/Virus Propagation

HealthySick

Contact

1: Sneeze to neighbors

2: Some neighbors Sick

3: Try to recover

3

An Example: Flu/Virus Propagation

HealthySick

Contact

1: Sneeze to neighbors

2: Some neighbors Sick

3: Try to recover

Q: How to guild propagation by opt. link structure?

4

An Example: Flu/Virus Propagation

HealthySick

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

5

This paper

IBM Research

© 2012 IBM CorporationSocial Analytics & Collaboration Technologies Group6

Roadmap

Motivation: An Illustrative Example

Q1: Understanding the Tipping Point (Background)

Q2: Minimize Propagation

Q3: Maximize Propagation

Conclusion

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

Epidemic Threshold for Alternating Behavior[PKDD 2010, Networking 2011]

Theorem [PKDD 2010, Networking 2011]: No epidemic Ifλ(S) ≤ 1.

System matrix S = Πi Si

Si = (1-δ)I + β Ai

dayday

N

N nightnight

N

NAi……

Log (Infection Ratio)

Time Ticks

At Threshold

Below

Above

8

Why is λ So Important?

• λ Capacity of a Graph:

Larger λ better connected9

IBM Research

© 2012 IBM CorporationSocial Analytics & Collaboration Technologies Group10

Roadmap

Motivation: An Illustrative Example

Q1: Understanding the Tipping Point (Background)

Q2: Minimize Propagation

Q3: Maximize Propagation

Conclusion

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

11

Good

Q: How to find k best edges to delete efficiently?

Left eigen-score of source

Right eigen-score of target

12

Minimizing Propagation: Evaluations

Time Ticks

Log (Infected Ratio)

(better)

Our Method

Aa Data set: Oregon Autonomous System Graph (14K node, 61K edges)

Discussions: Node Deletion vs. Edge Deletion•Observations:

• Node or Edge Deletion λ Decrease• Nodes on A = Edges on its line graph L(A)

•Questions?• Edge Deletion on A = Node Deletion on L(A)? • Which strategy is better (when both feasible)?

Original Graph A Line Graph L(A)

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:

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Theorem: Hardness of Node Deletion.Find Optimal k-node Immunization is NP-Hard

Theorem: Line Graph Spectrum. Eigenvalue of A Eigenvalue of L(A)

Discussions: Node Deletion vs. Edge Deletion•Q: Which strategy is better (when both feasible)?•A: Edge Deletion > Node Deletion

16

(better)

Green: Node Deletion (e.g., shutdown a twitter account)Red: Edge Deletion (e.g., un-friend two users)

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© 2012 IBM CorporationSocial Analytics & Collaboration Technologies Group17

Roadmap

Motivation: An Illustrative Example

Q1: Understanding the Tipping Point (Background)

Q2: Minimize Propagation

Q3: Maximize Propagation

Conclusion

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

Left eigen-score of source

Right eigen-score of target

Low GvHigh Gv 18

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

k

k

#3:Searchspace k+d

k+d

Searchspace

:existing edgeTime Complexity: O(m+nt+kt2), t = max(k,d)

#1: Sorting Sources by u

#2: Sorting Targets by v

Maximizing Propagation: Edge Addition

Maximizing Propagation: Evaluation

Time Ticks

Log (Infected Ratio)

(better)

20

Our Method

IBM Research

© 2012 IBM CorporationSocial Analytics & Collaboration Technologies Group

Conclusion

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

t = 1 t = 2 t = 3