On the Submodularity of Influence in Social Networks

Size: px

Start display at page:

Download "On the Submodularity of Influence in Social Networks"

Phyllis Hudson
6 years ago
Views:

1 On he ubmodulariy o Inluence in ocial Neworks Elchanan Mossel & ebasien Roch OC07 peaker: Xinran He Xinranhe1990@gmail.com

2 ocial Nework ocial nework as a graph Nodes represen indiiduals. Edges are social relaions wih dieren srenghs: Neighbors, Coworkers relaion in real lie Virual Friship in Facebook Follower-Followee relaions in wier

3 iusion In ocial Nework he adopion o new producs can propagae in he social nework iusion in he social nework Inormaion, rumors, innoaion,...

4 Inluence Maximizaion Inluence maximizaion: Find k people ha generaes he larges inluence spread i.e. expeced number o aciaed nodes [KK 2003]

5 Linear hreshold Model Gien a social nework wih edge weigh w u and a se o Iniially acie indiiduals as seed. Eery indiidual indepenly chooses a hreshold Θ uniormly in [0,1]. any sep laer, sill inacie nodes become aciaed i u N wu where N is he se o aciaed direc neighbors o. he diusion s when no more nodes are aciaed. he inluence spread σ=e[ P ], is he expeced number o acie nodes when he diusion process s. θ

6 Linear hreshold Example Inacie Node cie Node hreshold 0.4 X 0.1 U cie neighbors op! w 0.5 ep 01 23

7 Inluence Maximizaion Find a seed se, k, σ is maximized. Inluence Maximizaion Problem is NP-hard under linear hreshold model[kempe e.al 2003]. We hae o sole i approximaely. Main ool or analysis heorem: he greedy algorihm is a 1-1/e approximaion or maximizing monoone and submodular se uncions[nemhauser/wolsey 1978].

8 ubmodular & Monoone se uncion : 2 V R is monoone i se uncion : 2 V R is submodular i or all, V V + +, all or

9 ubmodulariy uncion se is submodular i Or equialenly ubmodulariy can be considered as diminishing reurn propery. V + +, all or, V all or, } { } {

10 ubmodulariy: Examples Maximum coerage problem: Gien a collecion o ses ={ 1,, m } and a number k, ind ', ' k, maximize σ = σ is submodular. ' i i. he inluence spread σ under he linear hreshold model is submodular[kempe e.al 2003]. Inluence Maximizaion Problem under linear hreshold model can be soled approximaely.

11 General hreshold Model wu θ Linear hreshold Model: General hreshold Model: : aciaion uncion o node oer. is he se o already aciaed nodes. General hreshold model is generalizaion o many diusion models: = u N u wu u N Linear hreshold Model [KK 2003] 1 1 p u Indepen Cascade Model r i= 1 N 1-1- p ω i,i-1 θ [KK 2003] ecreasing Cascade Model [KK 2005]

12 General hreshold Model2 For Linear hreshold model, he inluence spread σ is submodular [KK 2003]. Conjecure: Under he general hreshold model wih monoone and submodular, σ is monoone and submodular [KK 2003].

13 Main Resul heorem: Under he general hreshold model wih monoone and submodular, σ is monoone and submodular [Mossel/Roch 2007]. Corollary: he greedy algorihm is a 1-1/e approximaion o sole he inluence maximizaion problem under general hreshold model.

14 Proo: General Idea1 y coupling our diusion process: ={ 0 =, 1, 2,, } ={ 0 =, 1, 2,, } C={C 0 =,C 1,C 2,,C } ={ 0 =, 1, 2,, } uch ha C and

15 Proo: General Idea2 we hae hen aking expecaion, hen and I C C σ σ σ σ

16 C Couple he our processes wih he same hresholds θ. how C, C ase Case: C = = ssume C. by inducion. 0 0 For a node sill inacie a sep, we hae C. hereore i is aciaed in sep +1 in C, i mus also be aciaed in. C C

:Firs emp Le s ry he same coupling mehod or. 1 2 0.3 0.

17 :Firs emp Le s ry he same coupling mehod or Θ 3 =0.5 Θ 3 =0.5 Θ 3 =

18 nisense Coupling hen how could we keep? Inuiiely, using ϴ or aciaion o and 1- ϴ or aciaion o will maximize heir union.

19 Piecemeal Growh eine P = P 1 process, where,..., 1 k,..., k as he he piecemeal growh diusion is a pariion o seed se. Grow 1 Unil i s Grow 2 Unil i s Grow k Unil i s dd 1 dd 2 dd k Lemma: he disribuion oer he aciaed node se a he o original process wih seed se and he piecemeal growh process P 1,, k is idenical.

20 Piecemeal Growh: Proo y coupling hree piecemeal growh processes,, and original process wih same θ. Grow Grow nohing dd a sage 1 dd nohing a sage 2 Grow 1 Grow 2 dd 1 a sage 1 dd 2 a sage 2 Grow nohing Grow dd nohing a sage 1 dd a sage 2 '' s so ha s ' = s and ' = '' =

21 Need-o-know Represenaion1 Consider he diusion in a dieren way: Need-o-know Represenaion. Principle o eerred ecisions: We don decide all hresholds a he beginning; insead we reeal he alue o hresholds wheneer needed. For example: i node is inacie a sep -1, we only wan o know wheher i is aciaed a sep. Θ Θ -2-1

22 Need-o-know Represenaion2 Lemma: he ollowing process is equialen o he original one: 1.Iniialize 0 2. sep1 = - Wih probabiliy and we pick θ n 1, we iniialize 1 1 uniormly in [ 2 2 =, becomes aciaed 2 1, and or each sill inacie node - Oherwise we do nohing ]

23 nisense Coupling1 eine he anisense diusion P = P 1,..., k ; where 1,..., k is a pariion o seed se. Grow 1 Unil i s Grow k Unil i s Grow Unil i s K sage piecemeal growh dd a he beginning o sage k+1 ny sep in he inal sage, aciae nodes under he condiion. P P + 1 θ

24 nisense Coupling2 Grow 1 Grow k Grow Grow 1 Grow k Grow P θ θ Θ = P +1- Θ P P Θ θ ' P Q Q

25 nisense Coupling3 Grow 1 Grow k Grow Grow 1 Grow k Grow Lemma: he disribuions oer he aciaed node se a he o he piecemeal growh process P 1,, k ; and he anisense diusion process Q 1,, k ; are idenical.

26 nisense Coupling: Proo1 Grow 1 Grow k Grow Grow 1 Grow k Grow From Need-o-know Represenaion poin o iew: For any node sill inacie a ime =, we hae θ uniormly disribued in [ P,1] = [ Q,1]

27 nisense Coupling: Proo2 hen or any sill inacie node, we pick is Θ uniormly in [ P,1]. We deine Θ = Q +1- Θ. ince Θ and Θ hae he same disribuion, he inal sage in growing in P and Q is idenical. hereore P and Q hae he same disribuion.

$Coupling: Oeriew Grow Unil i s Grow \ Unil$ $Grow \ Unil i s Grow Unil i s Grow \ Unil i$

28 Coupling: Oeriew Grow Unil i s Grow \ Unil i s Grow nohing Grow Unil i s Grow Nohing Grow \ Unil i s Grow Unil i s Grow \ Unil i s Grow \ Unil i s or any sep in all hree sages

$or all seps in he irs or any sep in inal Grow Grow \ Grow nohing Grow Grow Nohing Grow \ Grow Grow \ Grow \$

29 Coupling: Firs wo sages = or all in he irs wo sages. hereore wo sages. We will show sage. or all seps in he irs or any sep in inal Grow Grow \ Grow nohing Grow Grow Nohing Grow \ Grow Grow \ Grow \ Firs wo sages Las sage

30 Coupling: nisense Coupling We irs proe \ \ or any sep in he inal sage by inducion on. ase case: \ \ ecause: = = \ \

31 Coupling: nisense Coupling ssume. We need o show ha. \ \ \ \ \ \ θ θ \ \ \, \ ', ', = = = = Lemma: For any and and submodular, we hae. ' ' ' ' '

$Coupling: Wrapup hereore we hae: \ \ = or all in he inal sage, C Preiously proed$

32 Coupling: Wrapup hereore we hae: \ \ = or all in he inal sage, C Preiously proed Grow Grow \ Grow nohing Grow Grow Nohing Grow \ Grow Grow \ Grow \ Firs wo sages Las sage

33 Furher Generalizaion We hae deined σ=e[ P ]. We can inroduce a se uncion ω on P and deine he inluence spread as σ ω =E[ωP ] insead. heorem: Under he general hreshold model wih monoone and submodular and ω, σ ω is monoone and submodular. [Mossel/Roch 2007]

34 Furher Generalizaion: Proo. we hae aking expecaion,. hen. and ssume C C ω ω ω ω σ σ σ σ ω ω ω ω ω ω

35 Conclusion General hreshold Model generalizes many popular diusion models. heorem: Under he general hreshold model wih monoone and submodular and ω, σ ω is monoone and submodular. [Mossel/Roch 2007] Proo mehodology: Coupling piecemeal growh & anisense coupling

36 lgorihm or Inluence Maximizaion Corollary: he greedy algorihm is a 1-1/e approximaion o sole he inluence maximizaion problem under general hreshold model. lgorihm1: Greedyk 1: iniialize o empy se 2 :or i = 1o k do 3: 4 : selec u = = { u} arg max V \ σ { } σ 5 : or 6 : reurn

37 lgorihm or Inluence Maximizaion lgorihm1: Greedyk 1: 2 :or i = 1o k do 3: 4 : iniialize o empy se 5 : or 6 : reurn selec u = { u} = arg max V \ σ { } σ ime complexiy: OknCm Where n= V, m= E, C he imes o Mone- Carlo simulaion.

38 lgorihm or Inluence Maximizaion Name Main Idea Model Guaranee Reerence CELF Lazy Forward opimizaion ll 1-1/e CELF++ Furher opimizaion o CELF ll 1-1/e PMI Use direced ree srucure IC No LG Use G srucure L No IRIE Use PageRank o iniialize and updae locally CG Use communiy srucure IC IC No 1 + dθ 1 e 1 M imulaed nnealing ll No Leskoec e al Goyal e al Chen e al Chen e al Chen e al Wang e al Jiang e al. 2011

39 Open Quesions ieren classes o aciaion uncion. Local subaddiie se uncion Global subaddiie inluence spread σ? Find approximaion algorihm or soling he inluence maximizaion problem under diusion models wih non-submodular inluence spread σ.

T L. t=1. Proof of Lemma 1. Using the marginal cost accounting in Equation(4) and standard arguments. t )+Π RB. t )+K 1(Q RB

T L. t=1. Proof of Lemma 1. Using the marginal cost accounting in Equation(4) and standard arguments. t )+Π RB. t )+K 1(Q RB Elecronic Companion EC.1. Proofs of Technical Lemmas and Theorems LEMMA 1. Le C(RB) be he oal cos incurred by he RB policy. Then we have, T L E[C(RB)] 3 E[Z RB ]. (EC.1) Proof of Lemma 1. Using he marginal