Maximizing the Spread of Influence through a Social Network Han - - PowerPoint PPT Presentation

Maximizing the Spread of Influence through a Social Network

Han Wang

Department of Computer Science ETH Zürich

Problem Example 1: Spread of Rumor

 2012 = end!

C F E B D A

Problem Example 2: Viral Marketing

 ezPad 1 beats iPad 3

C F E B D A

Problem Definition

 G:

a social network (n nodes)

 Model: spread process  S:

initially active subset (k seeds)

 𝝉 𝑻 :

#final active nodes (achievement)

 Task: Choose 𝑇∗  Goal: 𝜏 𝑇∗ = max 𝜏 𝑇 NP-Hard

Realistic Goal: Approximate the maximum with a guarantee Choose S: 𝜏 𝑇 ≥ 𝑠 ∙ 𝜏 𝑇∗

Contents in This Talk

 G:

a social network (n nodes)

 Model: spread process  S:

initially active subset (k seeds)

 𝝉 𝑻 :

#final active nodes (achievement)

 Task: Choose 𝑇∗  Goal: 𝜏 𝑇∗ = max 𝜏 𝑇 NP-Hard

Realistic Goal: Approximate the maximum with a guarantee Choose S: 𝜏 𝑇 ≥ 𝑠 ∙ 𝜏 𝑇∗

Two Models Prove: Prove:

Model 1: Independent Cascade Model

Model 1: Cascade Model

 Each active node try to activate his neighbors

 𝑞𝑣,𝑤 1 − 𝑞𝑣,𝑤  Only a single chance

C F E D

𝑞𝐷,𝐸 = 0.2 𝑞𝐷,𝐹 = 0.8 𝑞𝐷,𝐺 = 0.6

Model 1: Cascade Model

C F E B D A

0.2 0.8 0.7 0.6 0.3 0.4

 𝑇 = 𝐵, 𝐷 , 𝜏 𝑇 = 5

C F E B D A

0.2 0.8 0.7 0.6 0.3 0.4

Model 1: Cascade Model

Model 2: Linear Threshold Model

Model 2: Threshold Model

 Each inactive node picks a random 𝜄𝑤 ∈ ,0,1-

 Active condition:

𝑐𝑣,𝑤

𝑣: 𝑏𝑑𝑢𝑗𝑤𝑓 𝑜𝑓𝑗𝑕𝑖𝑐𝑝𝑠 𝑝𝑔 𝑤

≥ 𝜄𝑤

C E D

𝜾𝑬 = 𝟏. 𝟒 Iteration 2: 0.2 < 0.3 𝑐𝐷,𝐸 = 0.2 𝑐𝐹,𝐸 = 0.7 Iteration 4: E  active Iteration 5: 0.2+0.7 > 0.3 D  active

Iteration:

Model 2: Threshold Model

C F E B D A

0.2 0.8 0.7 0.6 0.3 0.4 𝜾 = 𝟏. 𝟔 𝜾 = 𝟏. 𝟕 𝜾 = 𝟏. 𝟒 𝜾 = 𝟏. 𝟘

1 2

Model 2: Threshold Model

 𝑇 = 𝐵, 𝐷 , 𝜏 𝑇 = 4

C F E B D A

0.2 0.8 0.7 0.6 0.3 0.4

How to Prove the Guarantee?

Given a spread model find 𝑇, s.t. 𝜏 𝑇 ≥ 𝑠 ∙ 𝜏 𝑇∗

???

f(S): Non-negative monotone submodular find 𝑇, s.t. f 𝑇 ≥ (1 − 1 𝑓) ∙ 𝑔 𝑇∗

Nemhauser

Submodularity

 𝑉: a finite ground set  𝑄 𝑉 : power set of 𝑉  𝑔 ∙ : 𝑄 𝑉 → 𝑆∗  Submodularity: ∀ 𝑜𝑝𝑒𝑓 𝑤, ∀𝑇 ⊆ 𝑈

𝒈 𝐓 ∪ 𝒘 − 𝒈 𝑻 ≥ 𝒈 𝑼 ∪ 𝒘 − 𝒈 𝑼

Example: Submodularity

 𝒈 𝑻 :

number of vertexes reachable from vertexes in S

C D B v A C D B v A

How to Prove the Guarantee?

Given a spread model find 𝑇, s.t. 𝜏 𝑇 ≥ 𝑠 ∙ 𝜏 𝑇∗

???

f(S): Non-negative monotone submodular find 𝑇, s.t. f 𝑇 ≥ (1 − 1 𝑓) ∙ 𝑔 𝑇∗

Nemhauser

Prove: 𝛕 𝑻 is Submodular

We Want to Prove…

Model 𝛕 𝑇 is Submodular NP-hard Independent Cascade Linear Threshold

Prove: Submodularity

Cascade Model

Submodularity (Cascade Model)

C F E B D A

0.2 0.8 0.7 0.6 0.3 0.4

 Recall: flip coin

Submodularity (Cascade Model)

C F E B D A

0.2 0.8 0.7 0.6 0.3 0.4

 Why not flip all the coins in the begining?

Submodularity (Cascade Model)

 Live edges  live paths  blocked edges

C F E B D A

0.2 0.8 0.7 0.6 0.3 0.4

Simplify Cascade Model

Node v ends up active A live path: some seed  v

 X: coin flipping outcome

 e.g. X1, X2

 𝑆𝑌 𝑤

 𝑆𝑌1 𝐵 = 𝐵, 𝐶  𝑆𝑌1 𝐷 = 𝐷, 𝐸, 𝐹

 𝜏𝑌 𝑇 = |

𝑆𝑌 𝑤 |

𝑤∈𝑇

 𝜏𝑌1 *𝐵, 𝐷+ =

𝐵, 𝐶, 𝐷, 𝐸, 𝐹 = 5

C F E B D A C F E B D A

Achievement(Simplified Model)

 Fix x, 𝜏𝑌 𝑇 is submodular  Linear combination of submodular functions

is still submodular

𝜏 𝑇 = 𝑄𝑠𝑝𝑐 𝑌 ∙ 𝜏𝑌 𝑇

𝑌

Submodularity (Cascade Model)

Summary of the proof

Active = Has a live path 𝜏𝑌 𝑇 is submodular

𝜏 𝑇 is submodular

Prove: NP-hard

Simplified Cascade Model

NP-Hard (Cascade Model)

 Set Cover Problem: k subsets cover all?  K=1: No  K=2: No  K=3: Yes  K=4: …

NP-Hard (Cascade Model)

 Solve Set Cover

Q: 2 subsets cover all ?

 Influence maximization

Q: 𝑇 = 2, 𝜏 𝑇 ≥ 2 + 5?

A B C D E S1 S3 A B C D E S1 S2 S3 S2

NP-Hard (Cascade Model)

Influence Maximization Problem

is at least as difficult as

Set Cover Problem

Prove: Submodularity

Linear Threshold Model

Recall: Threshold Model

C F E B D A

0.2 0.8 0.7 0.6 0.3 0.4 𝜾 = 𝟏. 𝟔 𝜾 = 𝟏. 𝟕 𝜾 = 𝟏. 𝟒 𝜾 = 𝟏. 𝟘

Gamble: Roulette

Gamble: Roulette

N1 N2 N3 N4 N5 N6 None N5

v

N4 N3 N2 N1 N6 0.2 0.15 0.07 0.23 0.1 0.14

𝜾 = 𝟏. 𝟓

A None C E None C None

𝜾 = 𝟏. 𝟔

C None

Submodularity (Threshold Model)

C F E B D A

0.2 0.8 0.7 0.6 0.3 0.4 𝜾 = 𝟏. 𝟕 𝜾 = 𝟏. 𝟒 𝜾 = 𝟏. 𝟘

Submodularity (Threshold Model)

C F E B D A

0.2 0.8 0.7 0.6 0.3 0.4 𝜾 = 𝟏. 𝟔 𝜾 = 𝟏. 𝟕 𝜾 = 𝟏. 𝟒 𝜾 = 𝟏. 𝟘

 Live edges  live paths

Correctness of Simplification

𝐺𝑝𝑠 𝑜𝑝𝑒𝑓 𝑤: 𝑄 𝑏𝑑𝑢𝑗𝑤𝑓 𝑗𝑜 𝐽𝑢𝑓𝑠𝑏𝑢𝑗𝑝𝑜 𝑢 + 1 𝑗𝑜𝑏𝑑𝑢𝑗𝑤𝑓 𝑗𝑜 𝐽𝑢𝑓𝑠𝑏𝑢𝑗𝑝𝑜𝑡 ≤ 𝑢) = 𝑄(𝑏𝑑𝑢𝑗𝑤𝑓 𝑗𝑜 𝐽𝑢𝑓𝑠𝑏𝑢𝑗𝑝𝑜 𝑢 + 1) 𝑄(𝑗𝑜𝑏𝑑𝑢𝑗𝑤𝑓 𝑗𝑜 𝐽𝑢𝑓𝑠𝑏𝑢𝑗𝑝𝑜𝑡 ≤ 𝑢)

Simplified Model

N1 N2 N3 N4 N5 N6 None N5

v

N4 N3 N2 N1 N6 0.2 0.15 0.07 0.23 0.1 0.14

Active before iteration 5 becomes active in iteration 5

𝐵𝑢: Nodes becoming active in iteration t

Simplified Model

𝑐𝑣,𝑤

𝑣∈𝐵𝑢

1 − 𝑐𝑣,𝑤

𝑣∈ 𝐵1∪𝐵2∪⋯∪𝐵𝑢−1

Original Model

N5

v

N4 N3 N2 N1 N6 0.2 0.15 0.07 0.23 0.1 0.14 N2 N6 N4 N3 N1

N5 None

𝐵𝑢: Nodes becoming active in iteration t

Original Model

𝑐𝑣,𝑤

𝑣∈𝐵𝑢

1 − 𝑐𝑣,𝑤

𝑣∈ 𝐵1∪𝐵2∪⋯∪𝐵𝑢−1

Simplify Threshold Model

Node v ends up active A live path: some seed  v

Similarly, we have…

Active = Has a live path 𝜏𝑌 𝑇 is submodular

𝜏 𝑇 is submodular

Prove: NP-hard

Linear Threshold Model

NP-Hard (Threshold Model)

 Vertex Cover Problem

 k vertexes (S)

each edge is incident to at least one vertex in S

NP-Hard (Threshold Model)

 Vertex Set Cover

Q: 3 vertexes cover all ?

 Influence maximization

Q: 𝑇 = 3, 𝜏 𝑇 = 6?

C F E B D A C F E B D A

Influence Maximization

Q: 𝑇 = 3, 𝜏 𝑇 = 6?

C F E B D A C F E B D A

Q: 𝑇 = 2, 𝜏 𝑇 = 6?

NP-Hard (Threshold Model)

Influence Maximization Problem

is at least as difficult as

Vertex Cover Problem

End of Proofs

 Influence Maximization Problem

Model 𝛕 𝑇 is Submodular NP-hard Independent Cascade Linear Threshold

Initial Problem

Given a spread model find 𝑇, s.t. 𝜏 𝑇 ≥ (1 − 1 𝑓 − 𝝑) ∙ 𝜏 𝑇∗ f(S): Non-negative monotone submodular find 𝑇, s.t. f 𝑇 ≥ (1 − 1 𝑓) ∙ 𝑔 𝑇∗

Greedy Hill Climbing 𝑵𝑩𝒀𝒘 𝒈 𝐓 ∪ 𝒘 − 𝒈 𝑻 (Maximize Marginal Gain)

Prove: 𝛕 𝑻 is Submodular

Summary

 Problem Description  Two Models

 Independent Cascade Model  Linear Threshold Model



Submodular Functions



Proof of Approximation Guarantee



Proof of NP-Hardness

Q&A