Learning Personal+Social Latent Factor Modelfor Social Recommendation

Yelong ShenDepartment of Computer Science

Kent State Universityyshen@cs.kent.edu

Ruoming JinDepartment of Computer Science

Kent State Universityjin@cs.kent.edu

ABSTRACTSocial recommendation, which aims to systematically leveragethe social relationships between users as well as their past be-haviors for automatic recommendation, attract much attentionrecently. The belief is that users linked with each other in socialnetworks tend to share certain common interests or have similartastes (homophily principle); such similarity is expected to helpimprove the recommendation accuracy and quality. There havebeen a few studies on social recommendations; however, theyalmost completely ignored the heterogeneity and diversity of thesocial relationship.In this paper, we develop a joint personal and social latent

factor (PSLF) model for social recommendation. Specifically,it combines the state-of-the-art collaborative filtering and the so-cial network modeling approaches for social recommendation.Especially, the PSLF extracts the social factor vectors for eachuser based on the state-of-the-art mixture membership stochas-tic blockmodel, which can explicitly express the varieties of thesocial relationship. To optimize the PSLF model, we develop ascalable expectation-maximization (EM) algorithm for fast ex-pectation computation. We compare our approach with the latestsocial recommendation approaches on two real datasets, Flixterand Douban. With similar training cost, our approach has showna significant improvement in terms of prediction accuracy criteriaover the existing approaches.

Categories and Subject DescriptorsH.3.3 [Information Search and Retrieval]: :Information Filter-ing; J.4 [Computer Applications]: :Social and Behavioral Sci-ences

KeywordsSocial Recommender Systems, Personal + social Factor

1. INTRODUCTIONRecommendation systems are playing increasingly important

role for Etailers and service providers: providing each individualwith the “best” options matching with their tastes and needs. By

Permission to make digital or hard copies of all or part of this work forpersonal or classroom use is granted without fee provided that copies arenot made or distributed for profit or commercial advantage and that copiesbear this notice and the full citation on the first page. To copy otherwise, torepublish, to post on servers or to redistribute to lists, requires prior specificpermission and/or a fee.KDD’12, August 12–16, 2012, Beijing, China.Copyright 2012 ACM 978-1-4503-1462-6 /12/08 ...$15.00.

doing so, companies can foster customers’ satisfaction and loy-alty, and even boost the sale and attract more potential customers.In the meantime, users may view recommendation itself as anindispensable service as it can help them save time and effortin discovering what they like or need. In the past, recommenda-tion systems mainly focus on utilizing users’ historical behaviors,such as their preferences on the purchased products/services (of-ten represented in terms a user-item rating matrix R, illustratedin Figure 1(a)) for collaborative filtering [1, 10].Thanks to the booming online social networking websites, users

are explicitly linked and often (un)knowingly recommend prod-ucts/services to friends. Numerous websites [16] directly utilizethe social networks to enable users to share opinions and recom-mends stuffs, such as books, movies, and/or musics, etc. to oneanother. More recently, there have been lots of interest in socialrecommendation, which aims to systematically leverage the so-cial relationships between users (Figure 1(b)) as well as their pastbehaviors for automatic recommendation. The belief here is thatusers linked with each other in social networks tend to share cer-tain common interests or have similar tastes (homophily princi-ple [12]), which can help increase the recommendation accuracy.However, how to discover and utilize the common denominatorshidden in the complex and massive social network is not a trivialproblem.There have been a few recent attempts at social recommenda-

tions [5, 11, 18], which are all based on the assumption that anypair of friends in the social network shall have similar interests.The recent studies [11, 18] incorporate such a network-basedsimilarity property between users into the state-of-the-art matrixfactorization recommendation approaches [10]. The matrix fac-torization approaches map both users and products to a commonlatent factor space R

k (each dimension encodes a latent factor).In movie recommendation, for instance, one latent factor (dimen-sion) may represent the degree of actions and/or violence andanother one may encode the movie cost. Each user ui is charac-terized by a latent factor vector Ui ∈ R

k, which represents her orhis personal interests/tastes; and each item vj is also representedby a vector Vj ∈ R

k, describing its key features and composi-tions. Given this, the inner product (UT

i Vj ) of these two latentfactor vectors (Ui and Vj ) is used to predict the likely rating ofuser ui on item vj . The higher is the value (UT

i Vj ), the morelikely is a match (between user ui and item vj ). Mao et al. [11]propose a social-network based regulation on top of the latentfactor model, which targets to constrain the difference betweenthe latent vectors of user ui’s friends and user ui, i.e., ||Ui−Uf ||shall be small where uf is ui’s friend. Yang et al. [18] take onestep further and they require the inner product (UT

i Uf ) betweentwo users ui and uf can directly help to recover whether ui anduj is linked in the social network. Basically, the higher is the

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(a) User-Item Rating MatrixR (b) User-Social Network GFigure 1: Social Recommendation Data Sources

value UTi Uf , the more similar they are and the more likely they

are friends. Both approaches have shown reasonable improve-ment over the state-of-art recommendation approaches which donot utilize the social factors.However, the basic underlying assumption about social rela-

tionships used in the existing social recommendation approachesis too strict to be realistic: they almost completely ignored theheterogeneity and diversity of the social relationship. Users in anonline social network, such as Facebook, are linked or become“friends” because of different reasons, including family, friends,classmates, or even business relationships, among many others.For different types of interactions, the common denominators(shared interests or preferences) might be quite different. Froma user-centered viewpoint, each user ui may be characterized bya set of social tags (factors), such as “car dealer”, “skiing enthu-siast”, “food-lover”, etc.; one of his neighbors uf in the socialnetwork might only share the common attribute “car dealer”, andanother one u′

f with may be his ski-buddies who teaches in thenearby college. Thus, an interest on an adventure movie, maybe propagated between ui and his “skiing enthusiast" friend, butmay not necessarily be shared with his “car dealer” colleague.On the other hand, two users at different states with many similarinterests may not know one another and never become friends.Thus, the (dis)similarity measure of the latent factors, such as

the difference ||Ui − Uf || or the inner product UTi Uf may not

be an appropriate factor to determine the likelihood of existingsocial relationship. In the meantime, however, the social factorsof an individual do directly affect her or his preferences towardsan item. Given this, the key challenge is how to discover those la-tent social factors of individual user and how to effectively utilizethem to boost the recommendation accuracy.

1.1 Our ContributionIn this paper, we propose a joint personal and social latent fac-

tor (PSLF) model for social recommendation, which utilize bothusers’ past behaviors and the social relationships. Similar to thematrix factorization approaches, each user ui (and each item vj)is represented by a personal (item) latent factor vector, Ui (Vj)in a unified user-item factor space R

k . However, instead of usingthe social relationship to constraint those personal factors, thenew model explicitly represent each user ui with an additionalsocial factor vector Πi in another space S

d, which can also belinearly transformed and mapped to the user-item factor spaceR

k, denoted as MT Πi, where M is the linear transformationmatrix. Given this, the model utilizes the joint predictions fromboth personal and social latent factors, UiVj + ΠT

i MVj , for thelikely rating from ui on item vj . Clearly, the major challengehere is how to discover the right social latent factors which cancomplement and also reinforce the personal latent factors, whichhave been extensively studied [10].In this work, we extract the social factor vectors for each user

based on the mixture membership stochastic blockmodel [2], whichcan simultaneously factorize the social network and infer the la-

tent social structure. Specifically, in this blockmodel, the net-work is assumed to base on a number of social factors (attributes),and these social factors may interact with one another in certainprobability (represented as a social interaction/connection matrixB). Furthermore, each user ui is represented as a mixture (socialfactor vector Πi) of social factors; and ΠT

i BΠf represents thelikelihood whether user ui and uf are linked.Even though both personal latent factor model and mixture

membership blockmodel for social factors have been studied in-dependently before, to the best of our knowledge, this is the firststudy which bridges them. Here, we combine the latent factormodel and mixture membership blockmodel in a unified proba-bilistic model for social recommendation; and we study how tooptimize the joint model and scale it on large social networks.Especially, we note that in the joint personal and social model-ing, the social latent factors should not only reflect the underlyingsocial structures, but also directly contribute to improvise the ac-curacy of the ratings. In other words, users’ past behaviors arealso served as the feedback for factorizing the social network.To sum, we make the following contributions in this work.

We first propose a novel joint personal and social latent factormodel which can effectively utilize both users’ past behaviorsand the social network. Our approach combines the collaborativefiltering and the social network modeling approaches for socialrecommendation. We develop a fast expectation-maximization(EM) algorithm to optimize the joint factor model. We compareour approach with the latest social recommendation approacheson two real datasets, Flixster and Douban. In the experiments,our approach has shown a significant improvement in terms ofprediction accuracy criteria over the existing approaches.Paper Organization: In Section 2, we formally describe the so-cial recommendation problem and discuss the basic latent factormodeling and its adoption for the problem. In Section 3, wepresent the joint personal and social latent factor (PSLF) mod-eling. In Section 4, we introduce the fast EM based parameterinference algorithm for the PSLF model. In Section 5, we reportthe detailed experiments results. Finally, we provide a short re-view of the related work in Section 6 and conclude the paper inSection 7.

2. PROBLEM DEFINITION ANDLATENT FACTORMODELING

The social recommendation problem considers two major datasources: 1) the user-item rating matrix R (recording users’ pastbehaviors), where Rij indicates the user ui’s preference to itemvj ; 2) the social network G = (V, E), where (i, f) ∈ E indicateusers ui and uf are linked. Here, we consider the social rela-tionship is mutual and thusG is an undirected graph. Figure 1(a)illustrates the rating matrix R and Figure 1(b) shows the socialnetwork R. Given this, the social recommendation problem canbe defined as learning a model which can effectively predict themissing values in the user-item rating matrix R by employingbothR and G.In the following, we will introduce the basics of latent fac-

tor modeling for collaborative filtering (Subsection 2.1). Then,in Subsection 2.2, we will given an overview on the latest stud-ies [11, 18] which incorporates the social networks with the latentfactor modeling for social recommendation.

2.1 Latent Factor ModelingThe latent factor modeling approach maps both users and items

to a common latent factor space Rk (each dimension encodes a

latent factor). Each user ui is characterized by a latent factor vec-

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torUi ∈ Rk, which represents her or his personal interests/tastes;

and each item vj also represented by a vector Vj ∈ Rk, describ-

ing its key features and compositions. Given this, the inner prod-uct (UT

i Vj) of these two latent factor vectors (Ui and Vj) is usedto predict Rij , the rating of user ui on item vj .Now, we follow the probabilistic matrix factorization (PMF)

notation to formally describe the latent factor modeling approach.The ratingRij is modeled using the normal distributionN (x|μ, σ2)with mean μ and variance σ2:

Rij ∼ p(Rij |UTi Vj , σ

2) = N (Rij |UTi Vj , σ

2) (1)

Furthermore, the user (item) latent factor vectors are also mod-eled using the normal distribution with zero mean:

Ui ∼ p(Ui|σ2u) = N (Ui|0, σ2

uI) (2)Vj ∼ p(Vj |σ2

v) = N (Vj |0, σ2vI) (3)

Parameters U = (Ui),V = (Vj) are estimated through maxi-mizing the log-posterior over user rating behaviors (R):

ln p(U,V |R, σ2, σ2u, σ2

v) = ln p(R|U,V, σ2) +

ln p(U |σ2u) + ln p(V |σ2

v) + C, (4)

where C is a parameter-independent constant. Specifically,

p(R|U, V, σ2) =Yi,j

N (Rij |UTi Vj , σ

2)Iij

p(U |σ2u) =

Yi

N (Ui|0, σ2uI), p(V |σ2

v) =Y

j

N (Vj |0, σ2vI) (5)

where Iij is the indicator: Iij = 1 if user ui rates item vj , andIij = 0, otherwise. Maximizing the log-posterior with respectto U and V is equivalent to minimize the sum-of-squared-errorsobjective function with quadratic regulation terms:

minU,V

Xi,j

Iij(Rij − UTi Vj)

2 + λU

Xi

||Ui||2 + λV

Xj

||Vj ||2 (6)

where λU = σ2/σ2U and λV = σ2/σ2

V . Usually, an iterativemethod, such as gradient descent method, can be employed tofind the local optimum of U and V .Recently, Koren in [9] make amajor improvement over the ex-

isting latent factor model by taking the similarity between items(users) into consideration. The neighborhood-based latent factormodel naturally combines the neighborhood model with the la-tent factor model, yielding better performances over pure neigh-borhood and pure latent factor models. Specially, the basic neigh-borhood based latent factor model utilizes the sum:

UTi Vj + (Bij + |R(i)|−1/2 P

j′∈R(i)(Rij′ − Bij′)Wjj′) topredict Rij , where R(i) is the set of items user ui rated, Bij isthe baseline estimator ofRij

1, and (Wjj′) are the user-independentparameters for describing the similarity between item j and j′ . In[9], both the baseline estimator and similarity parameters (Wjj′)can be computed to minimize the estimation accuracy, and dif-ferent variants such as using Rk(j; i), which is the set of top ksimilar items user i has rated, to replace R(i) are also discussed.

2.2 Social Recommendation ModelsAs we mentioned before, there have been a few studies [11,

18] on social recommendations recently. They all based on theassumption that any pair of friends in the social network shallhave similar interests and try to incorporate such a network-based1Bij = μ + Bi + Bj , where μ is the overall average rating andBi and Bj are the observed deviations of user ui and item vj ,respectively.

similarity property between users into the latent factor model-ing. Mao et al. [11] introduces the social regulation, i.e., usingthe social relationship explicitly regulate the latent factor model.Specifically, the individual-based regulation (the best approachin [11]) is modeled as follows:

minU,V

Xi,j

Iij(Rij − UTi Vj)

2 + λU

Xi

||Ui||2 + λV

Xj

||Vj ||2

+λE

X(i,f)∈E

Sim(i, f)||Ui − Uf ||2, (7)

where the last one is the social regulation term: it targets to min-imize the difference between the latent vectors of user ui and heror his friend uf . Sim(i, f) is the similarity between users ui anduf and can be defined in terms of either Vector Space Similarity(VSS) or Person Correction Coefficient (PCC) [11].In [18], Yang et al. proposes the Friendship-Interest Propaga-

tion (FIP) model, which utilizes the latent space vectors not onlyfor predicting the rating (UT

i Vj), but also for link prediction. Ba-sically, the inner product (UT

i Uf ) between two users ui and uf

can directly help to recover whether ui and uj is linked in thesocial network. Note that Yang et al. [18] study slightly differ-ent contexts where both users and items have additional features(contents) and the rating is only binary (item recommendation).To facilitate the discussion, we adopt their approach to the typ-ical social recommendation. Thus, the basic FIP model can bedescribed as follows:

minU,V

Xi,j

Iij(Rij − UTi Vj)

2 + λE

X(i,f)∈E

L(1, UTi Uf )

+λU

Xi

||Ui||2 + λV

Xj

||Vj ||2, (8)

where the second term is the error induced by using the user’slatent factor similarity to predict their social relationship. Thefunction L(x, y) is a loss function for measuring the differencebetween a binary variable x and a continuous variable y [18]. Forinstance, the logistic regression log(x, y) = log[1 + exp(−xy)]can be applied here.As we argued earlier, even though linked users often share cer-

tain common denominators (interests, tastes, etc.), (homophilyprinciple), this is not equivalent to say that they shall be “simi-lar”. In fact, the similarity on certain interests and tastes betweentwo friends can often be outweighed by their difference on mostother aspects. Clearly, the existing approaches have largely ig-nored the heterogeneity and diversity of the social relationship.Using the (dis)similarity measure of the latent factors, such as thedifference ||Ui −Uf || or the inner product UT

i Uf , to regulate orpredict social relationship is clearly not fully reflect or exploitthe underlying dimensionality of social networks. In this work,we utilize the social network modeling to help discover the latentsocial factors Si of an individual user and combine it with the(personal) latent factors (Ui) to solve the social recommendationproblems.

3. PERSONAL+SOCIALLATENT FACTORMODEL

In the section, we present a joint personal and social latentfactor (PSLF) model, which utilize both users’ past behaviorsand the social relationships for social recommendation. For thepersonal latent factor, it is similar to the matrix factorization ap-proaches, where each user ui is represented by a personal la-tent factor vector, Ui in a unified user-item factor space R

k . For

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the social latent factor, it is based on the state-of-the-art mix-ture membership stochastic blockmodel [2], which can simulta-neously factorize the social network and infer the latent socialstructure. The social factor vector of each user ui is representedin terms of his or her mixture membership Πi in the social net-work. Given this, the model utilizes the joint predictions fromboth personal and social latent factors, UiVj + ΠT

i MVj , for thelikely rating from ui on item vj .

Figure 2: Social-interest Factor model for recommendationFigure 2 shows the graphical model of PSLF. The generative

process of PSLF model is as follows:

1. For each user ui,

(1.a) Draw a personal latent factor vector Ui ∈ Rk ∼

p(Ui|σ2u) = N (Ui|0, σ2

uI).

(1.b) Draw a social latent factor vector (mixture member-ship) Πi ∈ S

d ∼ p(Πi|β) = Dirichlet(β).

2. For each item vj , draw an item latent vector Vj ∈ Rk ∼p(Vj |σ2

v) = N (Vj |0, σ2vI).

3. For each pair of users (ui,uf ),

(3.a) Draw membership indicator zi→f ∼ p(zi→f |Πi) =Multi(zi→f |Πi).

(3.b) Drawmembership indicator zf→i ∼ p(zf→i|Πf ) =Multi(zf→i|Πf ).

(3.c) Sample the value of their social relationship, Eif ∼p(Eif |zT

i→fBzf→i) = Bernoulli(zTi→fBzf→i).

4. For each user-item pair (ui, vj ), draw the response Rij ∼p(Rij |Pij + Sij , σ

2) = N (Rij |UTi Vj + ΠT

i MVj , σ2).

HerePij and Sij can be considered as the predictions draw-ing from the personal latent factors (UT

i Vj) and social la-tent factors (ΠT

i MVj ), respectively. Furthermore, eachcell Mij in the matrix M is also considered to be gener-ated from a normal distribution: Mij ∼ p(Mij |0, σ2

M ) =N (Mij |0, σ2

M ).

5. For the item pair similarity measure S = V T LT LV , drawits value S ∼ p(S|0, σ2

s) = N(S|0, σ2s). Here, L is the

Laplace matrix defined as L = D − W , where matrixW is the pearson coefficient matrix, and D is the diag-onal matrix with Djj =

Pj′∈Ωj

Wjj′ . Also, Ωi is theneighborhoods of item vi that the two items are neighborsindicates that they at least rated by one common user.

We make the following notes of the joint personal and social la-tent factor (PSLF) model:1) The generative steps (Steps 1.b and 3) forms the core of themixture membership stochastic blockmodeling as described in

[2]. Recall that in the typical block modeling, each user is con-sidered to be associated with only one group and a block struc-ture can represent at high level how likely one group may interactwith another group. The stochastic blockmodel assumes there isan underlying block structure (describe in the matrixBd×d), andeach user may belong to several groups (latent social factors),which are specified by Πi. Thus, the social factor vector Πi canhelp characterize each user’s unique underlying social interestsand preferences.2) Because the social latent factor vector Πi is located in thedifferent space S

d as the unified user-item latent space Rk, an

additional matrixM (Step 4) is utilized to linearly transform andmap each social vector Πi to the user-item space MT Πi. Thenew vector MT Πi can be viewed as the social latent factor inthe user-item space R

k which complements the personal latentvector Ui. We note that the generative representation of matrixM is for penalizing the model complexity.3) The last step (Step 5) is to incorporate the neighborhood itemsimilarity constraint into the latent factor model, which was ini-tially proposed by Koren in [9]. However, since our approachis based on generative modeling, for the simplicity (and unifor-mity) of technical treatment, we utilize the neighborhood pre-serving regulation (V T LT LV ) used in matrix factorization [6]and consider it as a response being generated through a normaldistribution. We note that also for simplicity, Figure 2 does notexplicitly reflect this constraint.Now, let Θ = (U, V, Π, M, B) be the model parameter set in

the PSLF model; let Φ = (σ, β, σs, σu, σv, σM ) be the hyper-parameters. Given this, the posterior probability given the user-item rating matrix R and the social work G (p(Θ,Φ|R,G)) isproportional to:

∝Y

(ui,vj)

p(Rij |UTi Vj + πT

i MVj , σ2)IijYi,j

p(Mij |σ2M )

Y(ui,uf )

Zzi→f

Zzf→i

p(zi→f |Πi)p(zf→i|Πj)p(Eif |zTi→fBzf→i)

Yui

p(Πi|β)Yui

p(Ui|σ2u)

Yvi

p(Vi|σ2v)

Yvi,vj

p(V Ti LT LVj |σ2

s) (9)

Thus, the social recommendation problem needs to find theoptimal model parameters Θ which can maximize the posteriorprobability given the user-item rating matrix R and the socialwork G:

maxΘ

ln p(Θ,Φ|R,G) (10)

In the next section, we will discuss how to efficiently infer themodel parameters Θ.

4. PARAMETERS INFERENCE ON PSLFMODEL

In this section, we present an efficient algorithm for parameterinference , which aims to maximize the posteriori distribution(MAP) of the PSLF model.Since there are hidden variables Z (in the integral form) being

included in the PSLFmodel, naturally the expectation-maximization(EM) method can be utilized: computing Z first in the E-stepgiven parameters Θ, and then update model parameters in theM-step. In general, the M-step can be solved by the gradientbased optimization to update model parameters Θ when the hid-den variables Z is explicit estimated; for the E-step, the hidden

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parameters Z can be estimated by Gibbs sampling (or variationalbayesian method), such as:

p(zi→f = m, zf→i = n|Eif , Πi, Πf , B)

∝ πim ∗ πfn ∗ Bernoulli(Eif , Bmn) (11)

However, both Gibbs sampling and variational bayesian method[2] can lead to O(N2K2) 2 time complexity in each E-step,which can be too computationally expensive for large networks.To address this problem, in this work, we propose a novel ap-proximate mean-field approximation method for E-step. The newmethod can reduce the time complexity of E-step to O(EK2) 3.In the following two subsections, we elaborate M-step and scal-able E-step for model parameter inference.

4.1 Parameters Inference : M-stepThe posterior distribution over social latent factors Πi for user

ui can be written asp(Πi|R, E, Θ/ Π, Z; Φ) ∝ L[Πi]

=

Dir(Πi; Zi,• + β) ∗Y

vj∈Rij

p(Rij ; UTi Vj + ΠT

i MVj , σ), (12)

where the dimension of latent social factors denotesK, and letZ(i,•) be a K dimensional vector, i.e.,Zi,• = (Zi,1, Zi,2, ..Zi,K)T .Here each element Zi,k indicates the count of the kth social at-tribute (factor) sampled by user ui, Z(i,k) =

Puj

I(zi→j = k),I is the indicator function. The posterior distribution over Πi isthe joint probability of modeling social networks and user ratingbehaviors.Since the normal distribution and Dirichlet distribution are non-

conjugate, we update Π using the gradient-based approach tomaximize posterior distribution over Π.Update Π:

∂L[Πi]

∂Πi= (Z(i,•) + β − 1) ◦ (

1

Πi) +

Xvj∈Rij

1

σ∗ (Rij − UT

i Vj − ΠTi MVj)MVj (13)

where ◦ is the pairwise product operator. SinceΠi is specifiedas the Dirichlet distribution, which places linear constraints onΠi (Eqn 14). Thus, the active set method can be utilized toproject gradient into feasible region [3]. Here we omit the detailsdue to space limitation.

KXk=1

Πi,k = 1 0 ≤ Πi,k ≤ 1 (14)

Similarly, by isolating U , V and M terms in posterior distri-bution of model, the parameters can be updated by moving in thedirection of the gradient, yielding:Update U , V ,M : (eij = Rij − UT

i Vj − ΠTi MVj)

Ui = Ui − λσuUi + λσXvj

eijVj) (15)

Vj = Vj − λσvVj + λσXui

eij(Ui + ΠiM) +

λσs

Xj′∈Ωj

LT LVj′ (16)

M = M − λσM M + λσX

(ui,vj)

eij(ΠiVTj ) (17)

2N is the user number in social networks,K is the dimension oflatent social factors3E is the number of connections in social networks

4.2 Parameters Inference : Scalable E-stepIn the earlier discussion, we point out that the classical sta-

tistical methods, such as Gibbs sampling and variational infer-ence, can be too costly to estimate Z (O(N2K2) time com-plexity). Here, we introduced an scale EM method to reducethe time complexity to O(EK2). The basic idea is to utilizean clustering based social network modeling method. Let C =(C1, C2, ...CT ) be a set of clusters forΠ, eachΠi has an mixturemembership to the clusters C,

PTt=1 p(Ct|Πi) = 1. Figure 3

illustrates the clustering based approximation. Given this, thedistribution over zi→f and zf→i can be rewritten in Eqn. 18.

Figure 3: clustering social factors to approximate social net-work modeling

p(zi→f = m, zf→i = n|Eif , Πi, Πf , B, C) ∝TX

t=1

ΠimCtnBernoulli(Eif ; Bmn)p(Ct|Πf ) (18)

Furthermore, a key observation is as follows: Based on Eqn13, it is not necessary to infer hidden variables Z for any pair ofusers, i.e. z(i→f), z(f→i). Instead, only the count of social at-tributes (factors), i.e. Zi,• = (Zi,1, Zi,2, ..Zi,K)T is needed forfurther parameters updating process. Given this, let us introduceZ+

i,m,n and Z−i,m,n:

Z+i,m,n : the count of mth and nth social attributes (factors)

sampled by user ui with her/his connected users. According toEqn. 18, the expectation of Z+

i,m,n can be rewritten as:

E(Z+i,m,n) =

Xuf |Eif=1

TXt=1

ΠimCtnBmnPm′,n′ Πim′Ctn′Bm′n′

p(Ct|Πf ) =

TXt=1

ΠimCtnBmnPm′,n′ Πim′Ctn′Bm′n′

Xuf |Eif=1

p(Ct|Πf ) (19)

Z−i,m,n : the count of mth and nth social attributes (factors)

sampled by user ui with her/his non-connected users.

E(Z−i,m,n) =

Xuf |Eif =0

TXt=1

p−(Ctn, Πim, Bmn)p(Ct|Πf ) =

TXt=1

p−(Ctn, Πim, Bmn)(T (Ct) − p(Ct|Πi) −X

uf |Eif=1

p(Ct|Πf )) (20)

p−(Ctn, Πim, Bmn) =ΠimCtn(1 − Bmn)P

m′,n′ Πim′Ctn′ (1 − Bm′n′ )(21)

where T (Ct) =Xui

p(Ct|Πi)

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Given this, the expectation of Zi,m can be calculated based onZ+

i,m,n and Z−i,m,n:

E(Zi,m) =KX

n=1

E(Z+i,m,n) + E(Z−

i,m,n) (22)

Note that in this approach, the time complexity for estimatingZi,• is O(TEK2) (at each E-step), where T is the number ofthe clusters.Update B : In addition, we can utilize this method to infer B

in the M-step:

B(m, n) =

Pui

E(Z+i,m,n)P

ui(1 − ρ)(E(Z+

i,m,n) + E(Z−i,m,n))

(23)

where ρ is the sparsity parameter to capture non-interaction ofthe social network [2]. And ρ can be estimated by:

ρ =

Pui

Pm,n E(Z−

i,m,n)Pui

Pm,n(E(Z+

i,m,n) + E(Z−i,m,n))

(24)

4.3 Overall EM Algorithm

Algorithm 1 EM-PSFL-Inference(R, G, Φ)Parameter: R:user-item rating matrix; G: user social network; Φ:

model hyperparameters;1: Random Initialize parameters Θ = (U, V,Π, M, B).2: repeat3: Scalable E-Step:4: Online K-Means Clustering on Θ.5: EstimateE(Z+

i,k,l) ,E(Z−i,k,l) andE(Zi,k) using Eqn. 19, 20

and 226: M-Step:7: Update parameters U, V, M using Eqn. 15, 16 and 178: Update Π according Eqn. 13 and 14.9: Update B using Eqn. 2310: until Repeat the E-M steps until Θ convergence.

The parameters inference method for PSFL is outlined in Al-gorithm 1. The time complexity for the E-step and M-step areO(TEK2) and O(RKN) 4, respectively. In the Algorithm 1,we use the online K-Means clustering algorithm to obtain C,which leads to O(KTN) time complexity. In practice, the num-ber of clusters T has a relative small affect to our proposed PSFLmodel (we use 300 in this work). For the hyper-parameters Φ inPSFL model, they are determined by cross-validation. We willfurther elaborate it in the experiment section.

5. EXPERIMENTSIn this section, we use the real world user rating data and their

corresponding social network to empirically validate the effec-tiveness of the proposed joint personal and social latent factor(PSLF) model for social recommendation. We compare it withthe existing social recommendation methods and the state-of-the-art collaborative filtering approaches without considering the so-cial relationships. Specifically, the following five recommenda-tion models are compared:SIM item-oriented neighborhood model(SIM) is the classical

collaborative filtering method based on item similarity. SIM pre-dicts the rating score of the item according to the user historicalratings on the similar items. We implement the SIM as describedin [15].4N is the dimension of latent interest factors

MMMFmaximum-margin matrix factorization model(MMMF)places a latent factor vector on each user and item. In the trainingstep, user-item latent factors are computed by user-rating matrixdecomposition. For prediction, rating scores is calculated as theinner product between user latent factors and item latent factors.We implement MMMF as described in [14]. In addition, we alsoincorporate the item-neighborhood information based on [9]. InMMMF, there are three hyper-parameters, regularization termsσu,σv on factors U and V respectively and σs item similarityconstraint terms. The three hyper-parameters are determined bycross-validation and the results reported in all experimental re-sults are based on the parameter configurations which producethe best results.FIP Friend-Interest Propagation(FIP) model [18] utilizes the

latent factors of users not only for predicting the ratings, but alsofor link prediction. The inner product between two user latentfactors can directly help whether they are linked in social net-work. Note that the original FIP model mainly targets for theitem recommendation (rating is only binary). For comparisonpurpose, we extend their approach to the rating prediction, as inEqn. 8. Four hyper-parameters in the model, λs, λu, λv (thefirst three are the same as the MMMF model), and λE are alldetermined by cross-validation.SR-LFM Social regularization based latent factor model (SR-

LFM) [11] utilizes the social relationships explicitly to regu-late the latent facts of users. Specifically, the individual-basedregulation is modeled as in Eqn 7. For comparison purpose,we also add neighborhood item similarity constraint to the SR-LFMmodel. There are four parameters, λs,λu, λv (the first threeare the same as the MMMF model), and λE , are determined bycross-validation.PSLF Joint Personal and Social latent Factor model(PSLF)

proposed in the paper, utilizes both users’ past behaviors and so-cial relationships for social recommendation. In the PSLFmodel,there are six hyper-parameters: λs,λu and λv are the same asMMMF model; β is a prior Dirichlet parameter (assigned as afixed value i.e. 1.0); and σM and σ. All these parameters exceptβ are determined by cross-validation.In the experimental study, we use the two popular metrics,

Mean Absolute Error (MAE) and RootMean Square Error (RMSE)to measure the prediction quality of our proposed PSLF model incomparison with other collaborative filtering and social recom-mendation methods.MAE is defined as:

MAE =1

T

X(ui,vj)

|Rij − R′ij | (25)

RMSE is defined as:

RMSE =

vuut 1

T

X(ui,vj)

(Rij − R′ij)

2 (26)

where Rij denotes the rating score user ui gave to movie vj ,R′

ij denotes the predicted rating score user ui gave to movie vj ,and T denotes the number of tested ratings. The smaller MAE orRMSE value means a better performance.

5.1 Experimental DatasetsFlixster Dataset. Flixster is a social networking service in

which users can rate movies 5. Users can add other users totheir friend list and create a social network. The social relations5The flixster dataset is public, and it can be download fromhttp://www.sfu.ca/ sja25/datasets/

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Table 1: General Statistic of Flixster and DouBanStatistics Flixster DouBanUsers 786,936 129,490

Social Relations 7,058,819 1,692,950Ratings 8,196,077 8,415,420Items 48,794 58,541

Table 2: Paired t-Test(2-tail) resultsdataset t-Test MMMF FIP SR-LFM

Douban PSLF(MAE) 3.38e − 8 3.16e − 7 2.42e − 4PSLF(RMSE) 5.73e − 9 2.22e − 6 6.52e − 7

flixster PSLF(MAE) 7.47e − 4 1.06e − 3 3.09e − 4PSLF(RMSE) 1.22e − 4 1.08e − 4 1.12e − 3

in Flixster are undirected. It also contains ratings expressed byusers in the period from November 2005 to November 2009.Possible rating values in Flixster dataset are 10 discrete num-

bers in the range [0.5, 5] with step size 0.5. There are totally786, 936 users, and 48, 794 movies in this dataset. The over-all number of social relations and rating historical records are7, 058, 819 and 8, 196, 077, respectively.Douban Dataset. Douban is a Chinese social website provid-

ing user rating, review and recommendation services for movie,books and music 6. It provides Facebook-like social network-ing services that users can make friends with each other throughthe email communication. Douban dataset is crawled and sharedwith us by Ma Hao [11]. In the dataset, users can assign 5 in-teger ratings (1 to 5) to movies, books and musics. There areoverall 129, 490 unique users and 58, 541 unique items. Thenumber of social relations and rating records are 1, 692, 950 and8, 415, 420 respectively. The basic statistics of the Flixster andDouban dataset are shown in Table 5.1.

5.2 Experiment ResultsIn this subsection, we report the performance of different (so-

cial) recommendation approaches. The users’ rating dataset israndomly divided into 2 folds. In the experiments, we use 70percent of the rating data for training and the other 30 percentfor testing. The random selection was carried out 5 times inde-pendently, and we report the average results. Note that exceptSIM, the other four models MMMF, FIP, SR-LFM and PSLFneed select the dimension of the latent (personal) factors whichis denoted byN . Thus, we show the performance of MMMF, FIPSR-LFM and PSLF models with different N , which are (4, 6, 8,10, 20) respectively. For PSLF model, additional social factordimension K is set to beK = 50. Later, we will further analyzethe performance of PSLF when K varies.Rating Prediction Varying N (Number of Latent Factors):First, we evaluate the five different methods by varying the num-ber of latent factors N . Both MAE and RMSE metrics are usedto measure the prediction quality. As shown in Figure 4, theproposed PSLF model consistently outperforms the other fourmethods. The performance of SIM is generally the worst amongall these methods in both Douban and flixster dataset. This is asexpected because it does not utilize any latent factors (a straightline in these figures). The MMMF is better than SIM as it uti-lizes the latent factors and item neighborhood information. Weobserve the three recommendation approaches FIP, SR-LFM, andour PSLF are all consistently better than MMMF and SIM. This

6http://www.douban.com

Table 3: Performance Comparison with MAE and RMSE onflixster dataset

Model MAE RMSESIM 0.1594 0.1963MMMF 0.1429 ± (0.0010) 0.1886 ± (0.0007)FIP 0.1400 ± (0.0006) 0.1863 ± (0.0009)

SR-LFM 0.1389 ± (0.0018) 0.1864 ± (0.0017)PSLF 0.1339 ± (0.0007) 0.1831 ± (0.0013)

Table 4: Performance Comparison with MAE and RMSE ondouban dataset

Model MAE RMSESIM 0.1443 0.1828MMMF 0.1427 ± (0.0011) 0.1785 ± (0.0010)FIP 0.1397 ± (0.0005) 0.1726 ± (0.0010)

SR-LFM 0.1367 ± (0.0008) 0.1718 ± (0.0012)PSLF 0.1326 ± (0.0018) 0.1694 ± (0.0013)

provides a strong evidence that the social recommendation in-deed is useful and can be used to improve the recommendationaccuracy. In these social recommendation approaches, SR-LFMis slightly better than FIP in Douban dataset, and comparablewith FIP in flixster dataset; and our PSLF are always better thanboth of them.From the Figure 4, we observe that a convergence effect ofN :

when we increase the latent factor dimensionN to be around 10,there seem to be little improvement for any large N . This sug-gests that a small number of latent factors (such as 10) is enoughfor all the models, including MMMF, FIP, SR-LFM, and PSLF.Detailed Performance Analysis for N = 10 In Table 5.2 and5.2, we provide a detailed comparison of these five approaches(for the four latent factor based approaches, the number of latentfactorN is set to be 10.) We observe in terms of MAE in flixsterdataset, PSLF model can improve the performance as high as3.60% in contrast to SR-LFM model, 4.36% and 6.30% in con-trast to FIP and MMMF models, respectively. In terms of RMSEin flixster dataset, PSLFmodel improves the performance as highas 1.39%, 1.71% and 2.91% in contrast to SR-LFM,FIP, andMMMF, respectively. In Douban dataset, PSLF model improvesthe performance as high as 2.99%, 5.06%, 7.07% in terms ofMAE, and 1.39%, 1.85%, 2.91% in terms of RMSE in contrastto the other three latent factor based approaches, such as SR-LFM, FIP, and MMMF, respectively.Finally, to validate the statistical significance of our experi-

ments, we perform the paired t-test (2-tail) over the MAE andRMSE of the experiential result. As shown in Table 5.1, all thet-test results are less than 0.01, which means the improvementsof PSLF over other methods are statistically significant.Parameter Sensitivity Analysis: Here, we analyze the perfor-mance of PSLF model when varying the dimension of socialfactors K. In the experiments, we compare the performance onMAEwith different social factor dimensions, which are (20, 50, 80).As shown in the Figure 5, we observe that the performance withK = 50 and the performance withK = 80 are comparable; theyare both slighter better thanK = 20. We note that in most of thecases, the performance withK = 50 is even slightly better thanthe performance withK = 80. This indicates a large number ofsocial factors (beyond 50) may not help much for performanceimprovement.

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5 10 15 200.13

0.135

0.14

0.145

0.15

MAE

Latent Factor Number(N)

SIMMMMFFIPSR−LFPSLF

(a) MAE (Douban)

5 10 15 200.165

0.17

0.175

0.18

0.185

RMSE

Latent Factor Number(N)

SIMMMMFFIPSR−LFPSLF

(b) RMSE (Douban)

5 10 15 200.13

0.135

0.14

0.145

0.15

0.155

0.16

0.165

0.17

MAE

Latent Factor Number(N)

SIMMMMFFIPSR−LFPSLF

(c) MAE (flixster)

5 10 15 200.18

0.185

0.19

0.195

0.2

RM

SE

Latent Factor Number(N)

SIMMMMFFIPSR−LFPSLF

(d) RMSE (flixster)

Figure 4: Performance Comparisons with MAE and RMSE on flixster and Douban

0 4 6 8 10 20 400.13

0.132

0.134

0.136

0.138

0.14

MA

E

Latent Factor Number(N)

PSLF K=20PSLF K=50PSLF K=80

(a) flixster

0 4 6 8 10 20 400.13

0.132

0.134

0.136

0.138

0.14

MA

E

Latent Factor Number(N)

PSLF K=20PSLF K=50PSLF K=80

(b) doubanFigure 5: Sensitivity Analysis onK andN

100 101 102 103 104100

101

102

103

104

User Friends Number

User

Rati

ng N

umbe

r

(a) flixster

100 101 102 103 104100

101

102

103

104

User Friends Number

User

Rati

ng N

umbe

r

(b) DoubanFigure 6: Correlation between user’s rating behav-iors and social activities

Table 5: Comparison of Pearson coefficient with differentsampling strategiesSampling Method flixster DoubanRandomly Sample 0.0504 ± 0.0025 0.2554 ± 0.0027Social-aware Sample 0.0561 ± 0.0026 0.2654 ± 0.0016

5.3 Statistical Data Analysis for Social Rec-ommendation

In this subsection, we perform a detailed statistical analysis tohelp analyze some underlying assumptions which form the basisof the existing social recommendation methods. Such an anal-ysis also provides an insights into their recommendation perfor-mance. First, we analyze the correlations between users’ ratingbehaviors and their social activities. As in the Figure 6, thecorrelation coefficient between users’ rating number and theirfriends number is relative small, i.e 0.045 in flixster dataset. Thisindicates that users who have a large number of friends may rateonly a small number of items. In other words, the behaviorsof user’s rating and making friends in social network seem tobe independent. This also provides a statistical evidence for theproposed PSLF model as it treats the personal and social latentfactors as being independent. Note that both FIP and SR-LFMmodels all try to use the same latent factors to represent bothusers’ rating behaviors and their social relationships.Secondly, we further analyze whether friends share similar

opinions (with similar rating to the same items). To achieve this,we utilize two sampling strategies: the first one is to randomlysample 100,000 user pairs, from the overall 786, 936 users inflixster and 129, 490 users in Douban; the second sampling ap-proach is to randomly sample 100,000 user pairs, who are di-rectly linked in the sample, from 7,058,819 social relations inflixster and 1,692,950 social relations in Douban. Then, for eachsample set, we compute the interest coefficient 7 between anytwo user in each sample. The higher interest coefficient is, themore similar interests users have.In the Table 5.3, we report the comparison with average in-

terest coefficient on the two sampling approaches. In flixsterdataset, the average interest coefficient is 0.0524 based on ran-7PCC between the two users, ranges between [-1, 1]. We onlyreport the PCC between two users with no less than 5 commonrated items

dom sampling method (random pair), and 0.0561 based on sec-ond sampling method (random friends) with nearly 7 percent im-provement. In Douban dataset, the average interest coefficientsare 0.2554 and 0.2654 based on random sampling method andfriends sampling method, respectively. There is 4 percent im-provement for social-aware sampling method. Therefore, simplyassuming users have similar interest with their friends in socialnetworks may still be helpful when incorporating into recom-mender system, though the improvement maybe limited.Finally, we compare the distribution of Pearson correlation co-

efficient(PCC) between the two sample sets generate by the twosampling approaches. First, PCC is sorted in ascending and de-scending order respectively. For each order, we get Top N per-centage of samples, and calculate the average PCC. In Figure5.3, we show the comparison with Top(N) average PCC in as-cending order on random sampling set and friend sampling set.From the Figure 5.3, we can observe that the friend samplingmethod’s PCC curve is above random sampling method’s in bothflixster and douban dataset. This indicates that friends in socialnetwork have less probability to have opposite interests. In Fig-ure 5.3, we report the comparison with Top(N) average PCC indescending order between the two sampling approaches. Here,we found that the friend sampling method’s PCC curve is be-low random sampling method’s. It indicates that there are manyusers who have similar interests may not be friends with eachother in social network. This again suggests that using the (per-sonal) latent factor to measure the likelihood of being friends isnot accurate.

0 10 20 30 40 50 60−1

−0.8

−0.6

−0.4

−0.2

0

PC

C

(Percentage % N) User Paris

Random Sampling MethodSocial−aware Sampling Method

(a) flixster

0 10 20 30 40 50 60

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

PC

C

(Percentage % N) User Paris

Random Sampling MethodSocial−aware Sampling Method

(b) DoubanFigure 7: Comparison with Top(N) PCC in descending order

6. RELATED WORKRecently studies show users’ trust relationship can be employed

to enhance traditional recommender systems[13]. A few trust-aware recommendation methods have been proposed [7][4]. How-

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0 10 20 30 40 50 600.2

0.3

0.4

0.5

0.6

0.7

0.8

PC

C

(Percentage% N) of User Pairs

Random Sampling MethodSocial−aware Sampling Method

(a) flixster

0 10 20 30 40 50 600.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

PC

C

(Percentage% N) of User Pairs

Random Sampling MethodSocial−aware Sampling Method

(b) DoubanFigure 8: Comparison with Top(N) PCC in ascending order

ever, trust-based recommendation approaches are different fromsocial recommendation in many aspects. Typically, on the web-site, when a user A likes a review by another user B, user Aprobably will add user B to his/her trust list [11]. This processof trust generation is a unilateral action that does not require userA to confirm the relationship. However, the social relationshiprefers to the cooperative and mutual relationships that surroundus, such as classmates, colleagues, or relatives, etc. In addition,trust-based recommender systems are based on the assumptionthat users have similar tastes with other users they trust. Thishypothesis may not always be true in social recommendation sys-tems since the tastes among the user and her/his connected usersmay vary significantly. A more detailed analysis on the differ-ence between the trust-based recommendation approach and thesocial recommendation approach can be found in [11].There have been a few attempts on social recommendations

lately [5, 11, 18], which are all based on the assumption that anypair of friends in the social network shall have similar interests.The recent studies [11, 18] incorporate such a network-basedsimilarity property between users into the state-of-the-art matrixfactorization recommendation approaches [10]. However, theyalmost completely ignored the heterogeneity and diversity of thesocial relationship.Social attribute (factor) extraction was first proposed by Hol-

land et al. [8] and Harrison et al.[17]. They are also referredto as the blockmodel, which can be used to describe the role orposition of actors in the social network. In [8] and [17], usersin social network have only one factor(position) being extracted,and the interaction of the positions is captured in the latent so-cial structures. Recently, more complicated blockmodel, mixed-membership blockmodel [2], proposed to relax the assumptionof single position bloackmodel, allow users to have a more nat-ural, mixed-membership of social attributes. However, the exit-ing parameter inference method has scalability bottleneck, whichrequires O(n2) time complexity in each iteration, where n isthe user number of social networks. In this work, we develop ascale EM approach with time complexityO(E) in each iteration,where E is the number of edges in the social network.

7. CONCLUSIONIn this paper, we develop a joint personal and social latent fac-

tor (PSLF) model which combines the collaborative filtering andsocial network modeling approaches for social recommendation.Our approach has shown to have better prediction accuracy overthe existing social network approaches. In the future work, weplan to study the following aspects of social recommendations:1) can we scale the social recommendation used the map-reduce(Hadoop) computation framework to handle tens or hundreds ofmillions of users? 2) how to deal with privacy issue? Can socialcommendation be done with minimal (or coarse) network infor-mation?

8. ACKNOWLEDGMENTSThe authors appreciate the anonymous reviewers for their ex-

tensive and informative comments to help improve the paper.We also thank Dr Hao Ma for sharing Douban Dataset. Ye-long Shen and Ruoming Jin’s work in this paper is partially sup-ported by National Science Foundation under CAREER AwardIIS-0953950.

9. REFERENCES[1] Gediminas Adomavicius and Alexander Tuzhilin. Toward the next generation

of recommender systems: A survey of the state-of-the-art and possibleextensions. IEEE Trans. on Knowl. and Data Eng., 17:734–749, June 2005.

[2] Edoardo M. Airoldi, David M. Blei, Stephen E. Fienberg, and Eirc P. Xing.Mixed membership stochastic blockmodels. In JMLR’ 08. The Journal ofMachine Learning Research, pages 1981–2014. ACM, 2008.

[3] Paul H. Calamai and Jorge J. More:9A. Projected gradient methods forlinearly constrained problems. Math. Program., 39:93–116, October 1987.

[4] J.O’ Donovan and B. Smyth. Trust in recommender systems. In IUI ’05.Proceedings of the 10th international conference on Intelligent UserInterfaces, pages 167–174. ACM, 2005.

[5] H. J. Lee F. Liu. Use of social network information to enhace collaborativefiltering performance. In ESA’ 10. Expert Systems with Applications, pages4772–4778. Elsevier Ltd, 2010.

[6] Quanquan Gu and Jie Zhou. neighborhood preserving nonnegative matrixfactorization. In BMVC, pages 1–11, 2009.

[7] Michael R. Lyu Hao Ma, Irwin King. Learning to recommend with socialtrust ensemble. In SIGIR ’09. Proceedings of the 32th InternationalConference on information retrieval, pages 203–210. ACM, 2009.

[8] P W Holland, K B Laskey, and S Leinhardt. Stochastic blockmodels: Firststeps. Social Networks, 5(2):109–137, 1983.

[9] Yehuda Koren. Factorization meets the neighborhood: a multifacetedcollaborative filtering model. In KDD’ 08. Proceedings of the 14th ACMSIGKDD international conference on Knowledge discovery and data mining.ACM, 2008.

[10] Yehuda Koren, Robert Bell, and Chris Volinsky. Matrix factorizationtechniques for recommender systems. Computer, 42:30–37, August 2009.

[11] Hao Ma, Dengyong Zhou, Chao Liu, Michael R. Lyu, and Irwin King.Recommender systems with social regularization. In WSDM ’11.Proceedings of the fourth ACM international conference on Web search anddata mining, pages 287–196. ACM, 2011.

[12] Miller McPherson, Lynn S. Lovin, and James M. Cook. Birds of a feather:Homophily in social networks. Annual Review of Sociology, 27(1):415–444,2001.

[13] Sudeep Marwaha Punam Bedi, Harmeet Kaur. Trust based recommendersystem for semantic web. In IJCAI ’07. Proceedings of the 2007International Joint Conference on Artificial Intelligence, pages 2677–2682.ACM, 2007.

[14] Jason D. M. Rennie and Nathan Srebro. Fast maximum margin matrixfactorization for collaborative prediction. In ICML ’05. Proceedings of the22th International Conference on Machine Learning, pages 713–719. ACM,2005.

[15] Badrul Sarwar, George Karypis, Joseph Konstan, and John Reidl. Item-basedcollaborative filtering recommendation algorithms. In Proceedings of the10th international conference on World Wide Web, WWW ’01, pages285–295, New York, NY, USA, 2001. ACM.

[16] http://en.wikipedia.org/wiki/List_of_social_

networking_websites.[17] Harrison C. White, Scott A. Boorman, and Ronald L. Breiger. Social

structure from multiple networks. i. blockmodels of roles and positions.American Journal of Sociology, 81(4):730–780, 1976.

[18] Shuang Hong Yang, Alex Smola H. Bo Long, Narayanan Sadagopan,Zhaohui Zheng, and Hongyuan Zha. Like like alike - joint friendship andinterest propagation in social networks. In WWW’ 11. Proceedings of the20th international conference on World Wide Web. ACM, 2011.

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