nips nips2012 nips2012-59 knowledge-graph by maker-knowledge-mining

59 nips-2012-Bayesian nonparametric models for bipartite graphs

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Author: Francois Caron

Abstract: We develop a novel Bayesian nonparametric model for random bipartite graphs. The model is based on the theory of completely random measures and is able to handle a potentially inﬁnite number of nodes. We show that the model has appealing properties and in particular it may exhibit a power-law behavior. We derive a posterior characterization, a generative process for network growth, and a simple Gibbs sampler for posterior simulation. Our model is shown to be well ﬁtted to several real-world social networks. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Bayesian nonparametric models for bipartite graphs Francois Caron ¸ INRIA IMB - University of Bordeaux Talence, France Francois. [sent-1, score-0.203]

2 fr Abstract We develop a novel Bayesian nonparametric model for random bipartite graphs. [sent-3, score-0.179]

3 We derive a posterior characterization, a generative process for network growth, and a simple Gibbs sampler for posterior simulation. [sent-6, score-0.221]

4 In this article, we shall focus on bipartite networks, also known as twomode, afﬁliation or collaboration networks [16, 17]. [sent-10, score-0.201]

5 In bipartite networks, items are divided into two different types A and B, and only connections between items of different types are allowed. [sent-11, score-0.219]

6 Following the readers-books example, we will refer to items of type A as readers and items of type B as books. [sent-13, score-0.336]

7 An example of bipartite graph is shown on Figure 1(b). [sent-14, score-0.165]

8 An important summarizing quantity of a bipartite graph is the degree distribution of readers (resp. [sent-15, score-0.619]

9 The degree of a vertex in a network is the number of edges connected to that vertex. [sent-17, score-0.202]

10 A bipartite graph can be represented by a set of binary variables (zij ) where zij = 1 if reader i has read book j, 0 otherwise. [sent-19, score-0.653]

11 In many situations, the number of available books may be very large and potentially unknown. [sent-20, score-0.392]

12 In this case, a Bayesian nonparametric (BNP) approach can be sensible, by assuming that the pool of books is inﬁnite. [sent-21, score-0.426]

13 To formalize this framework, it will then be convenient to represent the bipartite graph by a collection of atomic measures Zi , i = 1, . [sent-22, score-0.196]

14 , n with ∞ Zi = zij δθj (1) j=1 where {θj } is the set of books and typically Zi only has a ﬁnite set of non-zero zij corresponding to books reader i has read. [sent-25, score-1.226]

15 The so-called Indian Buffet Process (IBP) is a simple generative process for the conditional distribution of Zi given Z1 , . [sent-27, score-0.155]

16 Such process can be constructed by considering that the binary measures Zi are i. [sent-31, score-0.093]

17 from some random measure drawn from a beta process [19, 10]. [sent-34, score-0.132]

18 Teh and Gor¨ r [18] proposed a three-parameter extension of the IBP, named stable IBP, that enables to u 1 model a power-law behavior for the degree distribution of books. [sent-36, score-0.274]

19 Although more ﬂexible, the stable IBP still induces a Poissonian distribution for the degree of readers. [sent-37, score-0.25]

20 In this paper, we propose a novel Bayesian nonparametric model for bipartite graphs that addresses some of the limitations of the stable IBP, while retaining computational tractability. [sent-38, score-0.261]

21 We assume that each book j is assigned a positive popularity parameter wj > 0. [sent-39, score-0.422]

22 Similarly, each reader i is assigned a positive parameter γi which represents its ability to read books. [sent-41, score-0.217]

23 The higher γi , the more books the reader i is willing to read. [sent-42, score-0.516]

24 Given the weights wj and γi , reader i reads book j with probability 1 − exp(−γi wj ). [sent-43, score-0.796]

25 We will consider that the weights wj and/or γi are the points of a Poisson process with a given L´ vy measure. [sent-44, score-0.444]

26 We show that depending on the choice of the L´ vy e e measure, a power-law behavior can be obtained for the degree distribution of books and/or readers. [sent-45, score-0.71]

27 Moreover, using a set of suitably chosen latent variables, we can derive a generative process for network growth, and an efﬁcient Gibbs sampler for approximate inference. [sent-46, score-0.238]

28 We provide illustrations of the ﬁt of the proposed model on several real-world bipartite social networks. [sent-47, score-0.206]

29 1 Statistical Model Completely Random Measures We ﬁrst provide a brief overview of completely random measures (CRM) [12, 13] before describing the BNP model for bipartite graphs in Section 2. [sent-50, score-0.221]

30 CRM can be decomposed into a sum of three independent parts: a non-random measure, a countable collection of atoms with ﬁxed locations, and a countable collection of atoms with randoms masses at random locations. [sent-60, score-0.183]

31 The mean measure Λ of this Poisson process is known as the L´ vy measure. [sent-68, score-0.185]

32 We will assume in the following that the L´ vy measure decomposes as a product e e of two non-atomic densities, i. [sent-69, score-0.123]

33 from h, while the masses are distributed according to a Poisson process ∞ over R+ with mean intensity λ. [sent-75, score-0.173]

34 We will further assume that the total mass G(Θ) = j=1 wj is positive and ﬁnite with probability one, which is guaranteed if the following conditions are satisﬁed ∞ ∞ λ(w)dw = ∞ (1 − exp(−w))λ(w)dw < ∞ and 0 (2) 0 and note g(x) its probability density function evaluated at x. [sent-76, score-0.28]

35 We will refer to λ as the L´ vy intensity e in the following, and to h as the base density of G, and write G ∼ CRM(λ, h). [sent-77, score-0.144]

36 5) and the stable process (τ = 0) as special cases. [sent-79, score-0.12]

37 2 A Bayesian nonparametric model for bipartite graphs Let G ∼ CRM(λ, h) where λ satisﬁes conditions (2). [sent-84, score-0.203]

38 A draw G takes the form ∞ G= wj δθj (6) j=1 where {θj } is the set of books and {wj } the set of popularity parameters of books. [sent-85, score-0.702]

39 , n, let consider the latent exponential process ∞ Vi = vij δθj (7) j=1 deﬁned for j = 1, . [sent-89, score-0.195]

40 , ∞ by vij |wj ∼ Exp(wj γi ) where Exp(a) denotes the exponential distribution of rate a. [sent-92, score-0.103]

41 We then deﬁne the binary process Zi conditionally on Vi by ∞ Zi = zij δθj zij = 1 if vij < 1 zij = 0 otherwise with j=1 (8) By integrating out the latent variables vij we clearly have p(zij = 1|wj , γi ) = 1 − exp(−γi wj ). [sent-94, score-1.059]

42 Hence, the total mass Zi (Θ) = j=1 zij , which corresponds to the total number of books read by reader i is ﬁnite with probability one and admits a Poisson(ψλ (γi )) distribution, where ψλ (z) is deﬁned in Equation (3), while the locations ∗ θj are i. [sent-99, score-0.792]

43 As an example, for the gamma process we have Zi (Θ) ∼ Poisson α log 1 + γi . [sent-104, score-0.212]

44 τ It will be useful in the following to introduce a censored version of the latent process Vi , deﬁned by ∞ Ui = uij δθj (9) j=1 where uij = min(vij , 1), for i = 1, . [sent-105, score-0.705]

45 5 a Gibbs sampler for our model, that both rely on the introduction of these latent variables. [sent-131, score-0.094]

46 We write Ki = Zi (Θ) = j=1 zij the degree n n of reader i (number of books read by reader i) and mj = i=1 Zi ({θj }) = i=1 zij the degree of book j (number of people having read book j). [sent-136, score-1.849]

47 Un |G) = γi ij wj ij exp (−γi wj uij ) exp (−γi G(Θ\{θ1 , . [sent-143, score-0.955]

48 , θK }))   i=1 j=1   n K K γi i = i=1 n m  j=1 wj j exp −wj n γi (uij − 1)  exp − γi G(Θ) (10) i=1 i=1 1 In the case where γi = γ, it is possible to derive P (Z1 , . [sent-146, score-0.38]

49 Un ) = K exp −ψλ γi i=1 n h(θj )κ mj , i=1 j=1 γi uij (11) i=1 where ψλ and κ are resp. [sent-160, score-0.467]

50 Proposition 3 The conditional distribution of G given the latent processes U1 , . [sent-165, score-0.123]

51 Un can be expressed as K G = G∗ + wj δθj (12) j=1 where G∗ and (wj ) are mutually independent with n G∗ ∼ CRM(λ∗ , h) λ∗ (w) = λ(w) exp −w γi (13) i=1 and the masses are m P (wj |rest) = n λ(wj )wj j exp (−wj i=1 γi Uij ) n κ(mj , i=1 γi uij ) (14) Proof. [sent-168, score-0.744]

52 In the case of the GGP, G∗ is still a GGP of parameters (α∗ = α, σ ∗ = σ, τ ∗ = τ + the wj ’s are conditionally gamma distributed, i. [sent-170, score-0.457]

53 n i=1 γi ), while n wj |rest ∼ Gamma mj − σ, τ + γi uij i=1 Corollary 4 The predictive distribution of Zn+1 given the latent processes U1 , . [sent-172, score-0.799]

54 , Un is given by K ∗ Zn+1 = Zn+1 + zn+1,j δθj j=1 ∗ where the zn+1,j are independent of Zn+1 with n κ(mj , τ + γn+1 + i=1 γi uij ) n κ(mj , τ + i=1 γi uij ) ∗ where Ber is the Bernoulli distribution and Zn+1 is a homogeneous Poisson process over Θ of intensity measure ψλ∗ (γn+1 ) h(θ). [sent-175, score-0.756]

55 γi uij Finally, we consider the distribution of un+1,j |zn+1,j = 1, u1:n,j . [sent-177, score-0.318]

56 This is given by n p(un+1,j |zn+1,j = 1, u1:n,j ) ∝ κ(mj + 1, un+1,j γn+1 + γi uij )1un+1,j ∈[0,1] (15) i=1 In supplementary material, we show how to sample from this distribution by the inverse cdf method for the GGP. [sent-178, score-0.347]

57 B1 B2 (a) B6 B7 (b) Figure 1: Illustration of the generative process described in Section 2. [sent-188, score-0.111]

58 4 A generative process In this section we describe the generative process for Zi given (U1 , . [sent-191, score-0.222]

59 This reinforcement process, where popular books will be more likely to be picked, is reminiscent of the generative process for the beta-Bernoulli process, popularized under the name of the Indian buffet process [8]. [sent-195, score-0.632]

60 Let xij = − log(uij ) ≥ 0 be latent positive scores. [sent-196, score-0.08]

61 Consider a set of n readers who successively enter into a library with an inﬁnite number of books. [sent-197, score-0.284]

62 The ﬁrst reader picks a number K1 ∼ Poisson(ψλ (γ1 )) books. [sent-202, score-0.124]

63 Now consider that reader i enters into the library, and knows about the books read by previous readers and their scores. [sent-204, score-0.871]

64 Let K be the total number of books chosen by the previous i − 1 readers, and mj the number of times each of the K books has been read. [sent-205, score-0.906]

65 This generative process is illustrated in Figure 1 together with the underlying bipartite graph . [sent-211, score-0.276]

66 In Figure 2 are represented draws from this generative process with a GGP with parameters γi = 2 for all i, τ = 1, and different values for α and σ. [sent-212, score-0.111]

67 5 Gibbs sampling From the results derived in Proposition 3, a Gibbs sampler can be easily derived to approximate the posterior distribution P (G, U |Z). [sent-214, score-0.082]

68 , K, set uij = 1 if zij = 0, otherwise sample uij |zij , wj , γi ∼ rExp(γi wj , 1) where rExp(λ, a) is the right-truncated exponential distribution of pdf λ exp(−λx)/(1 − exp(−λa))1x∈[0,a] from which we can sample exactly. [sent-223, score-1.332]

69 , K, sample n wj |U, γi ∼ Gamma mj − σ, τ + γi uij i=1 n and the total mass G∗ (Θ) follows a distribution g ∗ (w) ∝ g(w) exp (−w i=1 γi ) where g(w) is the distribution of G(Θ). [sent-227, score-0.793]

70 In the case of the GGP, g ∗ (w) is an exponentially tilted stable distribution for which exact samplers exist [4]. [sent-228, score-0.113]

71 In the particular case of the gamma process, we have the simple n update G∗ (Θ) ∼ Gamma (α, τ + i=1 γi ) . [sent-229, score-0.15]

72 9 Figure 2: Realisations from the generative process of Section 2. [sent-233, score-0.111]

73 We may also go a step further and consider that there is an inﬁnite number of readers with weights γi associated to a given CRM Γ ∼ CRM(λγ , hγ ) and a measurable space of readers Θ. [sent-237, score-0.547]

74 This provides a lot of ﬂexibility in the modelling of the distribution of the degree of readers, allowing in particular to obtain a power-law behavior, as shown in Section 5. [sent-239, score-0.192]

75 We focus here on the case where Γ is drawn from a generalized gamma process of parameters (αγ , σγ , τγ ) for n simplicity. [sent-240, score-0.212]

76 , n,   K γi |G, U ∼ Gamma  K wj uij + G∗ (Θ) zij − σγ , τ + j=1 j=1 K j=1 and Γ∗ ∼ CRM(λ∗ , hγ ) with λ∗ (γ) = λγ (γ) exp −γ γ γ wj + G∗ (Θ) . [sent-244, score-1.064]

77 , K n γi uij + Γ∗ (Θ) wj |U, Γ ∼ Gamma mj − σ, τ + i=1 ∗ ∗ n ∗ ∗ and G ∼ CRM(λ , h) with λ (w) = λ(w) exp −w . [sent-248, score-0.747]

78 For the scale parameter α of the GGP, we can assign a gamma prior α ∼ Gamma(aα , bα ) and update it with α|γ ∼ n ∗ Gamma aα + K, bα + ψλ i=1 γi + Γ (Θ) using a Metropolis-Hastings step. [sent-250, score-0.15]

79 The total number of books read by n readers is O(nσ ). [sent-254, score-0.747]

80 Moreover, for σ > 0, the degree distribution follows a power-law distribution: asymptotically, the proportion of books read by m readers is O(m−1−σ ) (details in supplementary material). [sent-255, score-0.968]

81 However, in our case, a similar behavior can be obtained for the degree distribution of readers when assigning a GGP to it, while it will always be Poisson for the stable IBP. [sent-257, score-0.536]

82 The stable beta process [18] is a particular case of our construction, obtained by setting weights γi = γ and L´ vy measure e λ(w) = α Γ(1 + c) γ(1 − e−γw )−σ−1 e−γw(c+σ) Γ(1 − σ)Γ(c + σ) (16) The proof is obtained by a change of variable from the L´ vy measure of the stable beta process. [sent-259, score-0.522]

83 In a more general setting, we may want γi to depend on a set of metadata associated to reader i. [sent-265, score-0.124]

84 Inference for latent factor models is described in supplementary material. [sent-266, score-0.082]

85 5 Illustrations on real-world social networks We now consider estimating the parameters of our model and evaluating its predictive performance on six bipartite social networks of various sizes. [sent-267, score-0.271]

86 The dataset ‘Boards’ contains information about members of the boards of Norwegian companies sitting at the same board in August 20112 . [sent-269, score-0.188]

87 ‘Forum’ is a forum network about web users contributing to the same forums3 . [sent-270, score-0.127]

88 ‘Books’ concerns data collected from the Book-Crossing community about users providing ratings on books4 where we extracted the bipartite network from the ratings. [sent-271, score-0.208]

89 ‘Movielens100k’ contains information about users rating particular movies5 from which we extracted the bipartite network. [sent-273, score-0.175]

90 php Data for the forum and citation datasets can be downloaded from http://toreopsahl. [sent-312, score-0.099]

91 set containing 3/4 of the readers and a test set with the remaining. [sent-326, score-0.262]

92 Poissonian degree distribution for readers and power-law degree distribution for books. [sent-334, score-0.646]

93 Both methods perform better solely on the Board dataset, where the Poisson assumption on the number of people sitting on the same board makes sense. [sent-335, score-0.127]

94 give the empirical degree distributions of the test network and a draw from the estimated models, for the IMDB dataset and the Books dataset. [sent-339, score-0.224]

95 It is clearly seen that the four models are able to capture the power-law behavior of the degree distribution of actors (Figure 3(e-h)) or books (Figure 4(e-h)). [sent-340, score-0.664]

96 However, only IG and GGP are able to capture the power-law behavior of the degree distribution of movies (Figure 3(a-d)) or readers (Figure 4(a-d)). [sent-341, score-0.499]

97 Random variate generation for exponentially and polynomially tilted stable distributions. [sent-360, score-0.09]

98 Inﬁnite latent feature models and the Indian buffet process. [sent-393, score-0.12]

99 Nonparametric bayes estimators based on beta processes in models for life history data. [sent-403, score-0.075]

100 Random graphs with arbitrary degree distributions and their applications. [sent-438, score-0.193]

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