Probability Models of Information Exchange on Networks Lecture 6

Transcription

1 Probability Models of Information Exchange on Networks Lecture 6 UC Berkeley

2 Many Other Models There are many models of information exchange on networks. Q: Which model to chose? My answer good features of models include: Canonical models. Amenable to analysis. Studied intensively before. Ok to invent your own models. Models are always just models

3 Ariel Rubinstein on theoretical economics When talking about economics: Everything I say is personal, based upon the entire range of my life experience which also includes the fact that professionally I engage in economics theory. However, to the best of my understanding, economic theory has nothing to do say about the heart of the issue under discussion here. I am not sure I know what an opinion is. I am not attempting to predict the rate of inflation tomorrow

4 Some other natural models Growth models: percolation models, DLA etc. Competition models: Competing growth. Infection models: Contact process, SIR, SIS Aspects of modeling: Dynamic networks Random networks Today: two examples of percolation based processes.

5 Example1: models of collective behavior examples: joining a riot adopting a product going to a movie model features: binary decision cascade effect network structure

6 viral marketing referrals, word-of-mouth can be very effective ex.: google+ viral marketing goal: mining the network value of potential customers how: target a small set of trendsetters, seeds example [Domingos-Richardson 02] collaborative filtering system use MRF to compute influence of each customer

7 independent cascade model when a node is activated it gets one chance to activate each neighbour probability of success from u to v is p u,v

8 generalized models graph G=(V,E); initial activated set S 0 generalized threshold model [Kempe-Kleinberg-Tardos 03, 05] activation functions: f u (S) where S is set of activated nodes threshold value: θ u uniform in [0,1] dynamics: at time t,set S t to S t-1 and add all nodes with f u (S t-1 ) θ u (note the process stops after (at most) n-1 steps) generalized cascade model [KKT 03, 05] when node u is activated: gets one chance to activate each neighbours probability of success from u to v: p u (v,s) where S is set of nodes who have already tried (and failed) to activate u assumption: the p u (v,.) s are order-independent theorem [KKT 03] - the two models are equivalent

9 influence maximization definition - the influence σ(s) given the initial seed S is the expected size of the infected set at termination σ(s) = E S [ S n 1 ] definition - in the influence maximization problem (IMP), we want to find the seed S of fixed size k that maximizes the influence theorem [KKT 03] - the IMP is NP-hard { } S* = argmax σ(s) : S V, S = k reduction from Set Cover: ground set U = {u 1,,u n } and collection of cover subsets S 1,,S m independent cascade model u 1 u 2 u 3 (u i,s j ) E S 1 S 2 S 3 S, S = k, σ(s) n + k? u n u i S j S m

10 submodularity definition - a set function f : V -> R is submodular if for all A, B in V f (A) + f (B) f (A B) + f (A B) example: f(s) = g( S ) where g is concave interpretation: discrete concavity or diminishing returns, indeed submodularity equivalent to S T, v V, f (T {v}) f (T) f (S {v}) f (S) threshold models: it is natural to assume that the activation functions have diminishing returns supported by observations of [Leskovec-Adamic-Huberman 06] in the context of viral marketing

11 main result theorem [M-Roch 06; first conjectured in KKT 03] - in the generalized threshold model, if all activation functions are monotone and submodular, then the influence is also submodular corollary [M-Roch 06] - IMP admits a (1 - e -1 - ε)-approximation algorithm (for all ε > 0) this follows from a general result on the approximation of submodular functions [Nemhauser-Wolsey-Fisher 78] known special cases [KKT 03, 05]: linear threshold model, independent cascade model decreasing cascade model, normalized submodular threshold model S T, p u (v,s) p u (v,t) or equiv. f u (S {v}) f u (S) 1 f u (S) f u (T {v}) f u (T) 1 f u (T)

12 related work sociology threshold models: [Granovetter 78], [Morris 00] cascades: [Watts 02] data mining viral marketing: [KKT 03, 05], [Domingos-Richardson 02] recommendation networks: [Leskovec-Singh-Kleinberg 05], [Leskovec- Adamic-Huberman 06] economics game-theoretic point of view: [Ellison 93], [Young 02] probability theory Markov random fields, Glauber dynamics percolation interacting particle systems: voter model, contact process

13 proof sketch

14 coupling we use the generalized threshold model arbitrary sets A, B; consider 4 processes: (A t ) started at A (B t ) started at B (C t ) started at A B (D t ) started at A B it suffices to couple the 4 processes in such a way that for all t indeed, at termination C t A t B t D t A t B t A n 1 + B n 1 A n 1 B n 1 + A n 1 B n 1 C n 1 + D n 1 (note this works with. replaced with any w monotone, submodular)

15 proof ideas our goal: C t A t B t (1) D t A t B t (2) antisense coupling obvious way to couple: use same θ u s for all 4 processes satisfies (1) but not (2) antisense : using θ u for (A t ) and (1-θ u ) for (B t ) maximizes union we combine both couplings piecemeal growth seed sets can be introduced in stages we add A B then A\B and finally B\A need-to-know not necessary to pick all θ u s at beginning can unveil only what we need to know: θ v [ f v ( S t 2 ), f v ( S t 1 )]?

16 piecemeal growth process started at S: (S t ) partition of S: S (1),,S (K) consider the process (T t ): pick θ u s run the process with seed S (1) until termination add S (2) and continue until termination add S (3) and so on lemma - the sets S n-1 and T Kn-1 have the same distribution

17 antisense coupling disjoint sets: S, T partition of S: S (1),,S (K) piecemeal process with seeds S (1),,S (K),T: (S t ) consider the process (T t ): pick θ u s run piecemeal process with seeds S (1),,S (K) until termination add T and continue with threshold values θ v '=1 θ v + f v ( ) T Kn 1 lemma - the sets S (K+1)n-1 and T (K+1)n-1 have the same distribution

18 need-to-know proof of lemma run the first K stages identically in both processes note that for all v not in S Kn-1 = T Kn-1, θ v is uniformly distributed in [f v (T Kn-1 ),1] but θ v = 1 - θ v + f v (T Kn-1 ) has the same distribution θ v [ f v ( S t 2 ), f v ( S t 1 )]? simulation 1 simulation 2

19 proof I ANTI 19/31

20 proof II ANTI 20/31

21 proof III new processes have correct final distribution up to time 2n-1, B t = C t and A t = D t so that C t A B t t D t A t B t for time 2n, note that so by monotonicity and submodularity then proceed by induction B 2n 1 D 2n 1 B 2n = B 2n 1 (T \ S) D 2n = D 2n 1 (T \ S) f v (B 2n ) f v (B 2n 1 ) f v (D 2n ) f v (D 2n 1 ) 21/31

22 general result we have proved: theorem [Mossel-R 06] - in the generalized threshold model, if all activation functions are submodular, then for any monotone, submodular function w, the generalized influence is submodular σ w (S) = E S [w(s n 1 )] Note: A closure property for sub-modular functions! 22/31

23 Competing first passage percolation on random regular graphs University of California, Berkeley July 24, 2013 Based on a joint work with Tonći Antunović Yael Dekel, and Yuval Peres

24 First passage percolation First passage percolation: Fix a graph G = (V, E), consider iid edge lengths (l e ) e E. Define the random metric on V d(x, y) = inf Γ l(γ), where the infimum is taken over all paths Γ connecting x and y and l(γ) is the sum of lengths of the edges on Γ. An important case: l e exp(λ). Process r B(0, r) evolves as a Markov process, new vertices are added at the rate λ the number of neighbors in B(0, r).

25 First passage percolation

26 First passage percolation

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52 First passage percolation

53 First passage percolation Theorem (Cox-Durrett shape theorem) There exists a compact convex set A such that for any δ > 0 lim P( (1 δ)ra B(0, r) (1 + δ)ra ) = 1. r

54 Competing first passage percolation Competing first passage percolation (also called Two-type Richardson Model by Häggström, Pemantle): Start with one red vertex and one blue vertex, other uncolored. Uncolored vertices become red at the rate (λ R the number of red neighbors) and blue at the rate (λ B the number of blue neighbors). Once colored, vertices never change the color.

55 Competing first passage percolation

56 Competing first passage percolation

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80 Competing first passage percolation

81 Competing first passage percolation Theorem (Häggström, Pemantle) On 2D lattice, for λ R = λ B for at most countable set S P(both red and blue ) > 0; λ R λ B / S P(both red and blue ) = 0.

82 Random regular graphs Random regular graphs Have only bounded number of short cycles. Neighborhoods or typical vertices are trees. Expander properties. Configuration model

83 Competing process on random graphs Let G n be random d-regular graph on n vertices. Uniformly choose r(n) vertices of G n and color it red and b(n) vertices and color it blue (r(n) and b(n) are given functions). Run the same dynamics as in the competing first passage percolation model with rates λ R and λ B. Consider the number of red and blue vertices Rn final when the graphs is exhausted. Question: Can we estimate R final n and B final n? and B final n

84 Compare with the Torus

85 Compare with the Torus

86 Compare with the Torus

87 Compare with the Torus

88 Compare with the Torus

89 Compare with the Torus Both processes occupy Θ(n) = Θ(k 2 ) vertices.

90 Results - asymptotics Theorem (Antunović, Dekel, M, Peres) Up to a constant factor with high probability R total n ( n ) λr /λ B r(n) n. b(n) In particular if r(n) = n ρ and b(n) = n β then R total n { n ρ+(1 β)λ R /λ B, for ρ < 1 (1 β)λ R /λ B, n, for ρ 1 (1 β)λ R /λ B. Balance occurs at (1 ρ)λ B = (1 β)λ R.

91 Configuration model

92 Configuration model

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105 Configuration model P(Bad configuration) is bounded away from 1.

106 Configuration model P(Bad configuration) is bounded away from 1. Conditioned that there are no Bad configuration the algorithm generates a random regular graph.

107 Couple CFPP and the configuration model

108 Couple CFPP and the configuration model

109 Couple CFPP and the configuration model

110 Couple CFPP and the configuration model

111 Couple CFPP and the configuration model

112 Couple CFPP and the configuration model

113 Couple CFPP and the configuration model

114 Couple CFPP and the configuration model

115 Couple CFPP and the configuration model

116 Couple CFPP and the configuration model

117 Couple CFPP and the configuration model

118 Couple CFPP and the configuration model

119 Couple CFPP and the configuration model

120 Couple CFPP and the configuration model

121 Couple CFPP and the configuration model We will keep track of: R t and B t... number of red and blue half-edges at step t.

122 Couple CFPP and the configuration model We will keep track of: R t and B t... number of red and blue half-edges at step t. Transition probabilities: (R t + d 2, B t ), w.p. λ R R t dn 2t R t B t λ R R t+λ B B t dn 2t 1 (R t+1, B t+1 ) = (R t, B t + d 2), w.p. (R t 2, B t ), w.p. λ B B t dn 2t R t B t λ R R t+λ B B t dn 2t 1 λ R R t R t 1 λ R R t+λ B B t dn 2t 1 (R t, B t 2), w.p. λ B B t B t 1 λ R R t+λ B B t dn 2t 1 (R t 1, B t 1), w.p. (λ R +λ B )B tr t (λ R R t+λ B B t)(dn 2t 1)

123 Couple CFPP and the configuration model To control R t and B t we use martingale techniques to control X t = R t + B t and Y t = R t B λ R/λ B t

124 Couple CFPP and the configuration model To control R t and B t we use martingale techniques to control X t = R t + B t and Y t = R t B λ R/λ B t X t... number of active half-edges in the configuration model

125 Couple CFPP and the configuration model To control R t and B t we use martingale techniques to control X t = R t + B t and Y t = R t B λ R/λ B t X t... number of active half-edges in the configuration model Y t... is the continuous time martingale when both processes evolve without any interactions (and self-interactions)

126 Couple CFPP and the configuration model X t = R t + B t Process (X t, dn 2t X t ) evolves as an urn model ( 2 0 ) d 2 d

127 Couple CFPP and the configuration model X t = R t + B t Process (X t, dn 2t X t ) evolves as an urn model ( 2 0 ) d 2 d As n X t = (1 ± o(1)) for all 0 t < dn/2. ( ( dn 2t (dn X 0 ) 1 2t ) ) d/2, dn

128 Couple CFPP and the configuration model X t = R t + B t Process (X t, dn 2t X t ) evolves as an urn model ( 2 0 ) d 2 d As n X t = (1 ± o(1)) for all 0 t < dn/2. ( ( dn 2t (dn X 0 ) 1 2t ) ) d/2, dn

129 Couple CFPP and the configuration model As long as X t is large Y t = R t B λ R/λ B t ( Y t = (1 ± o(1))y 0 1 2t ) λ B λ R 2λ B. dn

130 Couple CFPP and the configuration model As long as X t is large Y t = R t B λ R/λ B t ( Y t = (1 ± o(1))y 0 1 2t ) λ B λ R 2λ B. dn Observe the extra advantage of the faster process.

131 THANKS!!!

132 Speakers: Organizers: Laurent, Lionel, Tasia Heather Peterson You! Thank you!