CSCI 3210: Computational Game Theory. Cascading Behavior in Networks Ref: [AGT] Ch 24

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1 CSCI 3210: Computational Game Theory Cascading Behavior in Networks Ref: [AGT] Ch 24 Mohammad T. Irfan Web: Course Website: Cascading behavior u Analogy: epidemic u Starts with a few sick people u Propagates Alice 2 1

2 Diffusion of innovations u Studied in sociology since 1940s u Examples u Indirect/informational effects u Adoption of new medical or agricultural innovations u Photos/video going viral u Sudden success of new products u Rise of celebrities u Direct-benefit effects u Technology adoption Xbox/PS4, phone, fax, , online social networking apps Viral Facebook posts (2013) 2

3 #TheDress (February 2015) 3

4 Twitter vs Facebook Questions u What is the process by which these happen? u Models u Depart from one-shot game setting u Who are the most influential entities? u Algorithms 8 4

5 Modeling cascades Threshold models Morris' contagion model (2000) u "Networked coordination game" u Game u 2 actions: A (old: Yahoo messenger) and B (new: Google hangout) u Payoffs of two friends v & w in network q, q 1-q, 1-q u Any node v's total payoff is the sum of all payoffs from v's friends 5

6 Contagion model (cont...) u What action will v adopt (A or B)? Degree of v = d d A neighbors choose A d B neighbors choose B # d A # d B v will adopt B if: d B (1-q) >= d A q d B >= q d v adopts B if at least q fraction of her neighbors do so q = threshold of adoption Dynamic process u Initial adopters: A set of nodes adopting B at the start u Time steps 1, 2, 3,... u At each time step u Every node simultaneously updates their behavior in response to the current behaviors of neighbors u 2 versions u Nonpregressive: Nodes may switch between A & B u Progressive: Once switch to B, stay at B 6

7 Complete cascade u Complete cascade: everyone gets "converted" u Initial adopters are called contagious set u Depends on u Initial adopters u Network structure u Threshold q u Contagious threshold: Max q s.t. there is a finite contagious set Examples u Infinite line graph with q = 1/2 u A A A B A A A Vs u A A B B B A A Oscillations Complete cascade with contagious threshold = 1/2 7

8 Complete contagion with q = 1/2 2 main results u Setting: graph has countably infinite nodes, with finite maximum degree 1. Progressive and nonprogressive versions are equivalent w.r.t. a finite contagious set 2. Contagious threshold q cannot be > ½ for any (infinite) graph 8

9 More General Models Linear threshold model General threshold model Cascade model Linear threshold model u Actions of node i, x i is 0 for old behavior A and 1 for new behavior B u Threshold of each node i, b i u Assume: b i is chosen randomly between 0 and 1 j w ji b i i u Influence level from j to i, w ji u Assume: w ji >= 0 and j N (i) w ji 1 u Decision rule u Adopt B iff w ji x j b i j N (i) 18 9

10 General threshold model u The influence on a node is any general function of its neighbors u Assume: this function is (weakly) monotonous u Influence can't decrease as more people adopt B g i (x N (i)) b i? j b i i 19 Dynamics u Starts with the initial adopters of B u Progressive: no switching back to A u Time steps: 1, 2, 3,... (same as before) 10

11 Cascade model u Probabilistic model u Dynamics u For any edge u à v s.t. x u = 1, x v = 0 u u is given one chance to convert v u u's success probability is a function of u, v, and F v u u F v is set of neighbors of v that have already tried and failed Equivalent to the general threshold model! (Kempe et al., 2005) 21 Independent Cascade Model (ICM) u u's success probability is a function of u and v only u Independent of other nodes 11

12 Finding most influential nodes Most influential nodes problem u Posed by Domingos and Richardson [2001] u Kleinberg et al s formulation [2003]: u Input u Graph instance and cascade model u Integer (budget) k >= 1 u Function f(s), giving the expected number of nodes converted by initial adopters S. u We want to u Find a set S of k nodes such that f(s) is maximized 24 12

13 Kleinberg et al s formulation u NP-hard to find the optimal set S u NP-hard to approximate 25 Special case u The function f is submodular u Shows the property of diminishing marginal return u Greedy hill-climbing search gives 0.63 approximation (Nemhauser+, 1978) u Greedily selecting k nodes will lead to 0.63 times the maximum spread u Greedy algorithm: u For iteration 1 to k: u Pick the node that increases f the most f(s) Smalle r set Larger set S 26 13

14 Wish list beyond linear threshold model (LTM) u Non-probabilistic model, instantiated using data u Something is more general u Allows switching back and forth u Allows negative influences (Why is negative influence troublesome in LTM?) u Threshold values not required to be in [0, 1] u A model focused on outcome (LTM focuses on process) u Most influential nodes problem should be w.r.t. a desirable outcome u Must ensure stable outcomes (LTM allows unstable initial adopters) Linear Influence Game (LIG) Model Next 14