Maximizing the Spread of Influence through a Social Network

Transcription

1 Maximizing the Spread of Influence through a Social Network Han Wang Department of omputer Science ETH Zürich

2 Problem Example 1: Spread of Rumor 2012 = end! A D E B F

3 Problem Example 2: Viral Marketing ezpad 1 beats ipad 3 A D E B F

4 Problem Definition G: Model: S: σ S : a social network (n nodes) spread process initially active subset (k seeds) #final active nodes (achievement) Task: hoose S Goal: σ S = max σ S NP-Hard Realistic Goal: Approximate the maximum with a guarantee hoose S: σ S r σ S

5 ontents in This Talk G: Model: S: σ S : a social network (n nodes) spread process Two Models initially active subset (k seeds) #final active nodes (achievement) Task: hoose S Prove: Goal: σ S = max σ S NP-Hard Realistic Goal: Prove: Approximate the maximum with a guarantee hoose S: σ S r σ S

6 Model 1: Independent ascade Model

7 Model 1: ascade Model Each active node try to activate his neighbors p,d = 0.2 D p u,v 1 p u,v Only a single chance p,e = 0.8 p,f = 0.6 E F

8 Model 1: ascade Model A 0.2 D E B 0.6 F

9 Model 1: ascade Model S = A,, σ S = 5 A 0.2 D E B 0.6 F

10 Model 2: Linear Threshold Model

11 Model 2: Threshold Model Each inactive node picks a random θ v,0,1- Active condition: b u,v u: active neigbor of v θ v b,d = 0.2 θ D = 0. 3 E D b E,D = 0.7 Iteration 2: 0.2 < 0.3 Iteration 4: E active Iteration 5: > 0.3 D active

12 Model 2: Threshold Model Iteration: A θ = 0. 3 D B θ = F 0.8 θ = 0. 9 E θ = 0. 6

13 Model 2: Threshold Model S = A,, σ S = 4 A 0.2 D E B 0.6 F

14 How to Prove the Guarantee? Given a spread model??? find S, s.t. σ S r σ S find S, s.t. f S (1 1 ) f S e Nemhauser f(s): Non-negative monotone submodular

15 Submodularity U: a finite ground set P U : power set of U f : P U R Submodularity: node v, S T f S v f S f T v f T

16 Example: Submodularity f S : number of vertexes reachable from vertexes in S v v A D A D B B

17 How to Prove the Guarantee? Given a spread model??? find S, s.t. σ S r σ S Prove: σ S is Submodular f(s): Non-negative monotone submodular find S, s.t. f S (1 1 ) f S e Nemhauser

18 We Want to Prove Model Independent ascade Linear Threshold σ S is Submodular NP-hard

19 Prove: Submodularity ascade Model

20 Submodularity (ascade Model) Recall: flip coin A 0.2 D E B 0.6 F

21 Submodularity (ascade Model) Why not flip all the coins in the begining? A D E 0.7 B 0.6 F

22 Submodularity (ascade Model) Live edges live paths blocked edges A 0.2 D E B 0.6 F

23 Simplify ascade Model Node v ends up active A live path: some seed v

24 Achievement(Simplified Model) X: coin flipping outcome R X v e.g. X1, X2 R X1 A = A, B R X1 =, D, E σ X S = R X v v S σ X1 *A, + = A, B,, D, E = 5 A A B B D E F D E F

25 Submodularity (ascade Model) Fix x, σ X S is submodular Linear combination of submodular functions is still submodular σ S = Prob X σ X S X

26 Summary of the proof Active = Has a live path σ X S is submodular σ S is submodular

27 Prove: NP-hard Simplified ascade Model

28 NP-Hard (ascade Model) Set over Problem: k subsets cover all? K=1: No K=2: No K=3: Yes K=4:

29 NP-Hard (ascade Model) Solve Set over Q: 2 subsets cover all? Influence maximization Q: S = 2, σ S 2 + 5? S1 B A S2 E D S3 S1 S2 S3 A B D E

30 NP-Hard (ascade Model) Influence Maximization Problem is at least as difficult as Set over Problem

31 Prove: Submodularity Linear Threshold Model

32 Recall: Threshold Model A 0.2 θ = 0. 3 D B θ = F 0.8 θ = 0. 9 E θ = 0. 6

33 Gamble: Roulette

34 Gamble: Roulette N v N N2 N6 None N1 N N5 N2 N4 N3 θ = 0. 4 N4 N3

35 Submodularity (Threshold Model) A 0.2 θ = 0. 3 D None 0.7 E None 0.4 A B θ = None 0.6 F 0.8 θ = 0. 9 None E θ = 0. 6

36 Submodularity (Threshold Model) Live edges live paths A B θ = F θ = 0. 9 θ = 0. 3 D E 0.7 θ = 0. 6

37 orrectness of Simplification For node v: P active in Iteration t + 1 inactive in Iterations t) = P(active in Iteration t + 1) P(inactive in Iterations t)

38 Simplified Model Active before iteration 5 becomes active in iteration 5 N v N N N2 N6 N5 None N1 N2 N4 N3 N4 N3

39 Simplified Model A t : Nodes becoming active in iteration t u At b u,v 1 b u,v u A 1 A 2 A t 1

40 Original Model N2 N6 N4 N3 N1 N5 None N v N N2 N N N3

41 Original Model A t : Nodes becoming active in iteration t u At b u,v 1 b u,v u A 1 A 2 A t 1

42 Simplify Threshold Model Node v ends up active A live path: some seed v

43 Similarly, we have Active = Has a live path σ X S is submodular σ S is submodular

44 Prove: NP-hard Linear Threshold Model

45 NP-Hard (Threshold Model) Vertex over Problem k vertexes (S) each edge is incident to at least one vertex in S

46 NP-Hard (Threshold Model) Vertex Set over Q: 3 vertexes cover all? Influence maximization Q: S = 3, σ S = 6? A D E A D E B F B F

47 Influence Maximization Q: S = 3, σ S = 6? Q: S = 2, σ S = 6? A D E A D E B F B F

48 NP-Hard (Threshold Model) Influence Maximization Problem is at least as difficult as Vertex over Problem

49 End of Proofs Influence Maximization Problem Model Independent ascade Linear Threshold σ S is Submodular NP-hard

50 Initial Problem Given a spread model find S, s.t. σ S (1 1 e find S, s.t. ε) σ S Prove: σ S is Submodular f(s): Non-negative monotone submodular f S (1 1 ) f S e Greedy Hill limbing MAX v f S v f S (Maximize Marginal Gain)

51 Summary Problem Description Two Models Independent ascade Model Linear Threshold Model Submodular Functions Proof of Approximation Guarantee Proof of NP-Hardness

52 Q&A