CS 322: (Social and Information) Network Analysis Jure Leskovec Stanford University

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1 CS 322: (Social and Inormation) Network Analysis Jure Leskovec Stanord University

2 Initially some nodes S are active Each edge (a,b) has probability (weight) p ab b a g 0.2 Node a becomes active: c e i activates t node b with prob. p ab Activations spread through the network 0.4 d 0.2 h 10/29/2009 Jure Leskovec, Stanord CS322: Network Analysis 2

3 I S is initial active set, let (S) denote expected size o inal active set S is more inluential i (S) is larger 10/29/2009 Jure Leskovec, Stanord CS322: Network Analysis 3

4 Blogs inormation epidemics Which are the inluential/inectious blogs? Viral marketing Who arethe trendsetters? Inluential people? Disease spreading Where to place monitoring stations to detect epidemics? 10/29/2009 Jure Leskovec, Stanord CS322: Network Analysis 4

5 Most inluential set o size k: set S o k nodes producing largest expected cascade size (S) i activated [Domingos Richardson 01] 10/29/2009 Jure Leskovec, Stanord CS322: Network Analysis 5

6 Optimization problem: max S o size k ( S) How hard is this problem? NP HARD! Showthat inding most inluential set is at least as hard as set cover 10/29/2009 Jure Leskovec, Stanord CS322: Network Analysis 6

7 Set cover: Given U={u 1,,u n } and sets S 1,, S m U Are there k sets among S 1,, S m such that their union is U? Goal: Encode set cover as an instance o max S o size k ( S) 10/29/2009 Jure Leskovec, Stanord CS322: Network Analysis 7

8 Given a set cover instance with sets S 1,, S m Build a graph: Create edge (S i, u) S i us i. Directed edge rom all sets to elements. Put weight 1 on each edge There exists a set S o size k with (S)=k+n i there exists a size k set cover 10/29/2009 Jure Leskovec, Stanord CS322: Network Analysis 8

9 Bad news: Inluence maximization is NP hard Good news: There exists an approximation algorithm! Consider: Greedy hill climbing to ind a good set S 10/29/2009 Jure Leskovec, Stanord CS322: Network Analysis 9

10 Start with S 0 ={} For i=1 k Choose node v that max (S i 1 {v}) Let S i = S i 1 {v} What is the runtime? Each step just runs n time steps or each node v b c d a e (S i 1 {v}) a b c d e 10/29/2009 Jure Leskovec, Stanord CS322: Network Analysis 10

11 Hill climbing produces a solution S where (s) (1 1/e) o optimal value (~63%) [Hemhauser, Fisher, Wolsey 78, Kempe, Kleinberg, Tardos 03] Claim holds or unctions with 2 properties: is monotone: i S T then (S) (T) and ({})=0 is submodular: adding element to a set gives less improvement than adding to one o subsets 10/29/2009 Jure Leskovec, Stanord CS322: Network Analysis 11

12 Diminishing returns: size o set (S {u}) (S) (T {u}) (T) Gain o adding a node to a small set Gain o adding a node to a large set 10/29/2009 Jure Leskovec, Stanord CS322: Network Analysis 12

13 Show 2 things: 1) Our (S) () is submodular 2) Hill climbing works well or monotone submodular unctions 10/29/2009 Jure Leskovec, Stanord CS322: Network Analysis 13

14 Keep adding nodes that give the largest gain Start with S 0 ={}, produce sets S 1, S 2,,S k Add elements one by one Marginal gain: i = (S i ) (S i 1 ) Let T={t 1 t k } be the best set o size k We need to show: (S) (1 1/e) (T) 10/29/2009 Jure Leskovec, Stanord CS322: Network Analysis 14

15 Claim: (AB) (A) (A) k j (A {b j }) (A) where: B = {b 1,,b k } and is submodular, Proo: Let B i = {b 1, b i } (AB) (A) = 10/29/2009 Jure Leskovec, Stanord CS322: Network Analysis 15

16 (T) () (S i T) ) T={t 1 t k } = (S i T) (S i ) + (S i ) jk [(S i {t j }) (S i )] + (S i ) jk i+1 + (S i ) = Thus: (T) () (S i ) + k i+1 10/29/2009 Jure Leskovec, Stanord CS322: Network Analysis 16

17 We know: i+1 1/k ((T) (S (S i )) What is (S i+1 )= What is (S k )? 10/29/2009 Jure Leskovec, Stanord CS322: Network Analysis 17

18 Claim: ( S ) 1 1 ( T ) Proo by induction: i=0: i 1 k i 10/29/2009 Jure Leskovec, Stanord CS322: Network Analysis 18

19 Claim: ) ( ) ( T S i Claim: Induction: At i+1: ) ( 1 1 ) ( T k S i At i+1: 10/29/2009 Jure Leskovec, Stanord CS322: Network Analysis 19

20 Thus: ) ( ) ( ) ( T S S k Thus: Then: ) ( 1 1 ) ( ) ( T k S S k Then: (S ) = (S k ) = 10/29/2009 Jure Leskovec, Stanord CS322: Network Analysis 20

21 Next, we must show is submodular: S T F(S {u}) (S) (T {u}) (T) Gain o adding a node to a small set Gain o adding a node to a large set Basic act: I 1,, K are submodular, and c 1,,c k 0 then is also submodular 10/29/2009 Jure Leskovec, Stanord CS322: Network Analysis 21

22 (S {u}) (S) (T {u}) (T) A simple submodular unction: Sets S 1,,S k (S) = i S i 10/29/2009 Jure Leskovec, Stanord CS322: Network Analysis 22

23 Principle o deerred decision: Generate randomness ahead o time a b e g c 0.4 d 0.2 h i Flip a coin or each edge to decide whether it will succeed when (i ever) it attempts to transmit Edges on which h activation will succeed are live (S i ) = size o the set reachable by live edge paths 10/29/2009 Jure Leskovec, Stanord CS322: Network Analysis 23

24 Fix outcome i o coin lips Let i (S) be size o cascade rom S given these coin lips a b e g c 0.4 d 0.2 h i Let i (v) = set o nodes reachable rom v on live edge paths i (S) () = size o union i (v) () i is submodular = i is submodular 10/29/2009 Jure Leskovec, Stanord CS322: Network Analysis 24

25 Hill climbing l reward a b b d a What do we know about optimizing submodular unctions? A hill climbing (i.e., greedy) is near optimal (1-1/e (~63%) o optimal) But c d e c e Hill climbing algorithm is slow At each iteration we need to re evaluate marginal gains It scales as O(n k) Add sensor with highest marginal gain 10/29/2009 Jure Leskovec, Stanord CS322: Network Analysis 25

26 [Leskovec et al., KDD 07] Observation: Submodularity guarantees that marginal beneits decrease with the solution size Marginal gain (S i {x}) x Idea: exploit submodularity, doing lazy evaluations! 10/29/2009 Jure Leskovec, Stanord CS322: Network Analysis Part 2 26

27 [Leskovec et al., KDD 07] Lazy hill climbing: Keep an ordered list o marginal beneits b i romprevious iteration Re evaluate evaluate b i only or top node Re sortand prune Marginal gain a b c d e b c d a e 10/29/2009 Jure Leskovec, Stanord CS322: Network Analysis Part 2 27

28 [Leskovec et al., KDD 07] Lazy hill climbing: Keep an ordered list o marginal beneits b i romprevious iteration Re evaluate evaluate b i only or top node Re sortand prune Marginal gain a b c d e b c d a e 10/29/2009 Jure Leskovec, Stanord CS322: Network Analysis Part 2 28

29 [Leskovec et al., KDD 07] Lazy hill climbing: Keep an ordered list o marginal beneits b i romprevious iteration Re evaluate evaluate b i only or top node Re sortand prune Marginal gain a d b e c b c d a e 10/29/2009 Jure Leskovec, Stanord CS322: Network Analysis Part 2 29

30 [Leskovec et al., KDD 07] Given a real city water distribution network And ddt data on how contaminants spread in the network Problem posed by US Environmental Protection Agency S 10/29/2009 Jure Leskovec, Stanord CS322: Network Analysis 30