a bayesian model for recommendation in social rating networks with trust relationships

A Bayesian Model for Recommenda3on in Social Ra3ng Networks with Trust Rela3onships

Gianni Costa, Giuseppe Manco, Riccardo Ortale

Mo3va3ng example

•  Joe is looking for a restaurant –  Likes fish –  Enjoys rock music –  No smoker

Chez Marcel

•  Ra3ng 2 –  “Came there with some

friends. Too loud, and the choice was very limited. I had one steak which wasn’t great”

–  Doesn’t like fish –  Doesn’t like rock music

1

•  Ra3ng 2 –  “Too noisy. But good

assortment of cigars” –  Doesn’t like rock music –  Smoker 2

•  Ra3ng 5 –  “GoSa try the seabass.

Wonderful!” –  Member of “Slow Food” 3

•  Ra3ng 4 –  “Jam night every

Wednesday. Good local groups. A must-‐see place.”

–  Writes on “Rolling Stone” 4

Overall ra3ng:

Mo3va3ng example


Chez Marcel

•  Ra3ng 2 –  “Came there with some

friends. Too loud, and the choice was very limited. I had one steak which wasn’t great”

–  Doesn’t like fish –  Doesn’t like rock music

1

•  Ra3ng 2 –  “Too noisy. But good

assortment of cigars” –  Doesn’t like rock music –  Smoker 2

•  Ra3ng 5 –  “GoSa try the seabass.





Overall ra3ng:

•  Joe’s profile doesn’t match 1 and par3ally matches 2

•  3 and 4 are authorita3ve in their fields

Mo3va3ng example


Chez Marcel •  Ra3ng 5 –  “GoSa try the seabass.





Overall ra3ng:

Recommenda3on with trust (and distrust)

•  We need to only consider compa3ble profiles •  Authorita3veness and suscep3bility play a role •  Recommenda3on is twofold

–  Who should we trust? –  What should we get suggested according to our trustees’ preferences?

Formal Framework

Input: Users, items

Basic assump3on: an underlying social network of trust rela3onships exists among users

Formal Framework Output: (Signed) Network of trust rela3onships + item adop3ons

-‐ +

Related works •  Ra3ng predic3on for item recommenda3on in social networks with –  unilateral rela3onships

•  e.g., trust networks –  coopera-ve and mutual rela3onships

•  e.g., friends, rela3ves, classmates and so forth •  Link predic3on –  temporal vs structural approaches

•  Assume graphs with evolving (resp. fixed) sets of nodes –  unsupervised vs supervised approaches

•  Compute scores for node pairs based on the topology of network graph alone.

•  Cast link predic3on as a binary classifica3on task

Basic Idea: Latent Factor Modeling

•  Three factor matrices: P, Q, F – Pu,k represents the suscep3bility of user u to factor k

– Fu,k represents the exper3se of user u into factor k

– Qi,k represents the characteriza3on of item i within factor k

Modeling item adop3ons

Ru,i | P,Q,F,↵ ⇠ N ((Pu + Fu)0 Qi,↵

�1)

•  Likes fish •  Enjoys rock

music •  No smoker

u i

•  Seafood •  Live music •  Smoking areas

The Bayesian Genera3ve Model

ra ↵�

PF Q

⇤P

µP⇤

F

µF

⇤Q

µQ

µ0,�0W0, ⌫0

N ⇥MN ⇥N

N M

Fig. 1. Graphical representation of the proposed Bayesian hierarchical model.

1. Sample

⇥P ⇠NW(⇥0)

⇥Q ⇠NW(⇥0)

⇥F ⇠NW(⇥0)

2. For each item i 2 I sample

Qi ⇠ N (µQ,⇤�1Q )

3. For each user u 2 N sample

Pu ⇠N (µP,⇤�1P )

Fu ⇠N (µF,⇤�1F )

4. For each pair hu, vi 2 N ⇥N sample

Au,v ⇠ N (�P0

uFv

�,��1)

5. For each pair hu, ii 2 N ⇥ I sample

Ru,i ⇠ N ((Pu + Fu)Q0j ,↵

�1)

Fig. 2. Generative process for the proposed Bayesian hierarchical model.

Pr(A⇤uv|R,A,⌅) relative to the prior ⌅ = {⇥0,�,↵}. Exact inference consists

in computing these predictive distributions as reported at Eq. 3.3 and Eq. 3.4,where we set ⇥ = {P,⇥

P

,F,⇥F

,Q,⇥Q

} for readability sake.

ra ↵�

PF Q

⇤P

µP⇤

F

µF

⇤Q

µQ

µ0,�0W0, ⌫0

N ⇥MN ⇥N

N M

Fig. 1. Graphical representation of the proposed Bayesian hierarchical model.

1. Sample

⇥P ⇠NW(⇥0)

⇥Q ⇠NW(⇥0)

⇥F ⇠NW(⇥0)

2. For each item i 2 I sample

Qi ⇠ N (µQ,⇤�1Q )

3. For each user u 2 N sample

Pu ⇠N (µP,⇤�1P )

Fu ⇠N (µF,⇤�1F )

4. For each pair hu, vi 2 N ⇥N sample

Au,v ⇠ N (�P0

uFv

�,��1)

5. For each pair hu, ii 2 N ⇥ I sample

Ru,i ⇠ N ((Pu + Fu)Q0j ,↵

�1)

Fig. 2. Generative process for the proposed Bayesian hierarchical model.

Pr(A⇤uv|R,A,⌅) relative to the prior ⌅ = {⇥0,�,↵}. Exact inference consists

in computing these predictive distributions as reported at Eq. 3.3 and Eq. 3.4,where we set ⇥ = {P,⇥

P

,F,⇥F

,Q,⇥Q

} for readability sake.

Inference and Predic3on •  Given observed trust rela3onships (A) and item adop3ons (R)

we want to infer

•  Problem: trust bias –  Observed rela3onships in a social network are rarely nega3ve: people

only make posi3ve connec3ons explicit

Ru,i | P,Q,F,↵ ⇠ N ((Pu + Fu)0 Qi,↵

�1)

Au,v | P,F,� ⇠ N (P0uFu,�

�1)


Inference and Predic3on

•  Solu3on: latent variable modeling

•  Yu,v represents a (bernoulli) latent variable sta3ng

whether a nega3ve trust rela3onship exists between u and v

v

u

Evalua3on •  Two datasets

–  Product evalua3on, trust rela3onships

–  5-‐star ra3ng system

Gibbs sampling(N , ⇥0 = {µ0, �0, ⌫0,W0}, �, ↵, �)1: Sample a subset U ✓ N ⇥ N such that u ! v 62 A;

2: Initialize P

(0), F(0), Q(0), Y(0);3: for h = 1 to H do

4: Sample ⇥(h)P ⇠ NW(⇥n) where ⇥n is computed by updating ⇥0 with P, SP;

5: Sample ⇥(h)F ⇠ NW(⇥n) where ⇥n is computed by updating ⇥0 with F, SF;

6: Sample ⇥(h)F ⇠ NW(⇥n) where ⇥n is computed by updating ⇥0 with Q, SQ

7: for each (u, v) 2 U do

8: Sample ✏(h)u,v according to Eq. 4.4;

9: end for

10: for each (u, v) 2 U do

11: Sample Y (h)uv according to Eq. 4.3;

12: end for

13: for each u 2 N do

14: Sample Pu ⇠ N✓µ⇤(u)P ,

h⇤

⇤(u)P

i�1◆

;

15: Sample Fu ⇠ N✓µ⇤(u)F ,

h⇤

⇤(u)F

i�1◆;

16: end for

17: for each i 2 I do

18: Sample Qi ⇠ N✓µ⇤(i)Q ,

h⇤

⇤(i)Q

i�1◆;

19: end for

20: end for

Fig. 4. The scheme of Gibbs sampling algorithm in pseudo code

Ciao Epinions

Users 7,375 49,289Trust Relationships 111,781 487,181

Items 106,797 139,738Ratings 282,618 664,823

InDegree (Avg/Median/Min/Max) 15.16/6/1/100 9.8/2/1/2589OutDegree (Avg/Median/Min/Max) 16.46/4/1/804 14.35/3/1/1760

Ratings on items (Avg/Median/Min/Max) 2.68/1/1/915 4.75/1/1/2026Ratings by Users (Avg/Median/Min/Max) 38.32/18/4/1543 16.55/6/1/1023

Table 1. Summary of the chosen social rating networks.

– Thirdly, we analyze the structure of the model and investigate the propertiesthat can be derived, such as relationships among factors and propensities ofusers within given factors.

Datasets. We conducted experiments on two datasets representing social ratingnetworks from the popular product review sites Epinions and Ciao, describedin [29]. Users in these sites can share their reviews about products. Also they canestablish their trust networks from which they may seek advice to make decisions.Both sites employ a 5-star rating system. Some statistics of the datasets areshown in Table 1 and in Fig. 5. We can notice that both the trust relationshipsand the rating distributions are heavy-tailed. Epinions exhibits a larger numberof users, as well as a larger sparsity coe�cient on A.

Evaluation setting. We chose some state-of-the-art baselines from the current lit-erature. For rating prediction, we compared our approach against SocialMF [14].The metric used here is the standard RMSE. We exploited the implementationof SocialMF made available at http://mymedialite.net. For trust prediction, we

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InDegree − Epinions

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ItemRatings − Epinions

ItemRatings

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Fig. 5. Distributions of trust relationships and ratings in Epinions and Ciao.

adapted the framework described in [20]. For each user, we considered the rat-ings as user features and we trained the factorization model which minimizes theAUC loss. We exploited the implementation made available by the authors athttp://cseweb.ucsd.edu/ akmenon/code. We refer to this method as AUC-MFin the following. In addition, we considered a further comparison in terms ofboth RMSE and AUC against a basic matrix factorization approach based onSVD named Joint SVD (JSVD) [11]. We computed a low-rank factorization ofthe joint adjacency/feature matrix X = [A R] as X ⇡ U · diag(�1, . . . ,�K) ·VT ,where K is the rank of the decomposition and �1, . . . ,�K are the square roots ofthe K greatest eigenvalues of XT

X. The matrices U and V resemble the rolesof P, F and Q: The term Uu,k can be interpreted as the tendency of u to trustusers, relative to factor k. Analogously, Vu,k represents the tendency of u to betrusted, and Vi,k represents the rating tendency of item i in k. The score can be

hence computed as [26] score(u, x) =PK

k=1 Uu,k�kVx,k, where x denotes eithera user v or an item i.

In all the experiments, we performed a Monte-Carlo Cross Validation, byperforming 5 training/test splits. Within the partitions, 70% of the data wereretained as training, and the remaining 30% as test. The splitting was accom-plished for the sole data upon which to measure the performance (i.e., ratingsfor the RMSE and links for the AUC).

Concerning the AUC, it is worth noticing that Epinions and Ciao only con-tain positive trust relationships, and the computation of the AUC relies onthe presence of negative values. Negative values are indeed crucial in the ap-proach [20], since the latter relies on a loss function which penalizes situationswhere the score of negative links is higher than the score of positive links. Inprinciple, we can consider all links in the test-set as positive examples, and allnon-existing links as negative example. However, the sparsity of the networksposes a major tractability issue, as it would make the computation of the AUCinfeasible. A better estimation strategy in [2, 26] consists in narrowing the nega-

Evalua3on •  RMSE on Ra3ng Predic3on •  AUC on Link Predic3on •  Compe3tors

–  RMSE: SocialMF, JSVD (SVD on the combined matrices) –  AUC: Matrix Factoriza3on tuned on AUC loss (AUC-‐MF), JSVD

•  Experiments –  5-‐Fold Monte-‐Carlo Cross Valida3on (70/30 split on each trial, for the

matrix to predict)

RMSE

tive examples to all the 2-hops non-existing links, i.e., all triplets (u, v, w) whereboth (u, v) and (v, w) exhibit a trust relationship in A, but (u,w) does not.

Results. Fig. 6 reports the averaged results of the evaluation. We ran the exper-iments on a variable number of latent factors, ranging from 4 to 128. We cannotice that the proposed hierarchical model, denoted as HBPMF, achieves theminimum RMSE on both datasets. There is a tendency of the RMSE to pro-gressively decrease. However, this tendency is more evident on SocialMF, whilethe other two methods exhibit negligible di↵erences.

4 8 16 32 64 128

0.0

0.2

0.4

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0.8

1.0

1.2

1.4

4 8 16 32 64 128

Epinions

N. of factors

RM

SE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4 HBPMF

JSVDSocialMF

4 8 16 32 64 128

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Ciao

N. of factors

RM

SE

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HBPMFJSVDSocialMF

4 8 16 32 64 128

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Epinions

N. of factors

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HBPMFJSVDAUC−MF

4 8 16 32 64 128

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Ciao

N. of factors

AUC

0.0

0.2

0.4

0.6

0.8

1.0

HBPMFJSVDAUC−MF

Fig. 6. Prediction results.

The opposite trend is observed in trust prediction. Here, all methods tend toprefer a low number of factors, as the best results are achieved with K = 4. Thedevised HBPMF model achieves the maximum AUC on the Epinions dataset,and results comparable to JSVD on Ciao. The detailed results are shown inFig. 7, where the ROC curves are reported. In general, the predictive accuracyof the Bayesian hierarchical model is stable with regards to the number of factors.This is a direct result of the Bayesian modeling, which makes the model robustto the growth of the model complexity. Fig. 8 also shows how the accuracyvaries according to the distributions which characterize the data. We can noticea correlation between accuracy and node degrees, as well as the number of ratingsprovided by a user or received by an item.

Epinions

False positive rate

True

pos

itive

rate

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

HBPMFJSVDAUC−MF

Ciao

False positive rate

True

pos

itive

rate

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

HBPMFJSVDAUC−MF

Fig. 7. ROC curves on trust prediction for K = 4.

To evaluate the e↵ects of the joint modeling of both the trust relationshipsand the ratings, we conducted some further experiments withK = 4. In a first ex-periment, we performed the sampling without considering the trust relationships.

AUC



4 8 16 32 64 128

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N. of factors

RM

SE

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Fig. 8. Data distribution vs. AUC and rating prediction.

More precisely, we performed a simple BPMF (as described in [25]). Dually, wediscarded the rating matrix and performed the sampling by only considering thetrust relationships. The first graph of Fig. 9 shows the comparison between theresults of these partial models against those achieved through the full HBPMFmodel. The e↵ects of the joint modeling can be appreciated on the RMSE: inpractice, the additional information provided by the trust relationships refinesthe modeling of the data, thus lowering the RMSE. By contrast, the e↵ects ofthe joint modeling on the AUC do not highlight substantial improvements.

Finally, the last two graphs of Fig. 9 report the running times relative to themethods. For the HBPMF, we achieved stable results for the RMSE after 100iterations, whereas the AUC result was stable after 20 iterations. Both SocialMFand AUC-MF exhibited stable results with 20 iterations. The computationaloverhead of the Gibbs Sampling procedure plays a crucial role here. Therein,it would be interesting to investigate alternative inference strategies based onvariational approximation, which are known to guarantee fast convergence.

6 Conclusions and Future Research

We presented the first unified approach to the recommendation of interestingitems and trustworthy users in social rating networks with trust relationships.The key intuition is that the interactions from users to users as well as betweenusers and items are explained by the same latent factors, which ultimately allowsto combine user and item recommendation into a simple and intuitive Bayesian

Joint modeling

•  Significant on RMSE

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Fig. 9. (a) E↵ects of the joint modeling. (1 denotes Epinions, and 2 denotes Ciao). (b)Average running time for iteration (JSVD reports the total time).

generative model. A comparative experimentation over real-world social ratingnetworks confirmed such an intuition: the devised model was shown to deliver asuperior predictive performance in terms of both RMSE and AUC.

Future research will focus on two major directions. We planned to study anextension of our model in which the Indian Bu↵et Process [12] is exploited toautomatically infer the most appropriate number of latent factors from the inputsocial rating network. In addition, variational approximate inference and relatedlearning algorithms will be studied to improve the computational e�ciency. Fi-nally, a further line of research is relative to how the proposed models can beadapted to support recommendation tasks behind rating prediction [24, 5, 4].

References

1. E.M. Airoldi, D.M. Blei, S.E. Fienberg, and E.P. Xing. Mixed membership stochas-tic blockmodels. The Journal of Machine Learning Research, 9:1981 – 2014, 2008.

2. L. Backstrom and J. Leskovec. Supervised random walks: Predicting and recom-mending links in social networks. In Proc. ACM WSDM Conf., pages 635 – 644,2011.

3. N. Barbieri, F. Bonchi, and G. Manco. Cascade-based community detection. InProc. of ACM WSDM Conf., pages 33–42, 2013.

4. N. Barbieri and G. Manco. An analysis of probabilistic methods for top-n recom-mendation in collaborative filtering. In Proc. ECML/PKDD Conf., pages 172–187,2011.

5. N. Barbieri, G. Manco, R. Ortale, and E. Ritacco. Balancing prediction and rec-ommendation accuracy: Hierarchical latent factors for preference data. In Proc. ofSIAM Int. Conf. on Data Mining, pages 1035 – 1046, 2012.

6. G. Costa and R. Ortale. A bayesian hierarchical approach for exploratory analysisof communities and roles in social networks. In Proc. of the IEEE/ACM ASONAMConf., pages 194 – 201, 2012.

7. G. Costa and R. Ortale. A Unified Generative Bayesian Model for CommunityDiscovery and Role Assignment based upon Latent Interaction Factors. In Proc.of the IEEE/ACM ASONAM Conf., To appear, 2014.

8. G. Costa and R. Ortale. Probabilistic analysis of communities and inner roles innetworks: Bayesian generative models and approximate inference. Social NetworkAnalysis and Mining, 3(4):1015 – 1038, 2013.

9. M. DeGroot. Optimal Statistical Decisions. McGraw-Hill, 1970.

Computa3onal cost


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Fig. 9. (a) E↵ects of the joint modeling. (1 denotes Epinions, and 2 denotes Ciao). (b)Average running time for iteration (JSVD reports the total time).

generative model. A comparative experimentation over real-world social ratingnetworks confirmed such an intuition: the devised model was shown to deliver asuperior predictive performance in terms of both RMSE and AUC.

Future research will focus on two major directions. We planned to study anextension of our model in which the Indian Bu↵et Process [12] is exploited toautomatically infer the most appropriate number of latent factors from the inputsocial rating network. In addition, variational approximate inference and relatedlearning algorithms will be studied to improve the computational e�ciency. Fi-nally, a further line of research is relative to how the proposed models can beadapted to support recommendation tasks behind rating prediction [24, 5, 4].

References

1. E.M. Airoldi, D.M. Blei, S.E. Fienberg, and E.P. Xing. Mixed membership stochas-tic blockmodels. The Journal of Machine Learning Research, 9:1981 – 2014, 2008.

2. L. Backstrom and J. Leskovec. Supervised random walks: Predicting and recom-mending links in social networks. In Proc. ACM WSDM Conf., pages 635 – 644,2011.

3. N. Barbieri, F. Bonchi, and G. Manco. Cascade-based community detection. InProc. of ACM WSDM Conf., pages 33–42, 2013.

4. N. Barbieri and G. Manco. An analysis of probabilistic methods for top-n recom-mendation in collaborative filtering. In Proc. ECML/PKDD Conf., pages 172–187,2011.

5. N. Barbieri, G. Manco, R. Ortale, and E. Ritacco. Balancing prediction and rec-ommendation accuracy: Hierarchical latent factors for preference data. In Proc. ofSIAM Int. Conf. on Data Mining, pages 1035 – 1046, 2012.

6. G. Costa and R. Ortale. A bayesian hierarchical approach for exploratory analysisof communities and roles in social networks. In Proc. of the IEEE/ACM ASONAMConf., pages 194 – 201, 2012.

7. G. Costa and R. Ortale. A Unified Generative Bayesian Model for CommunityDiscovery and Role Assignment based upon Latent Interaction Factors. In Proc.of the IEEE/ACM ASONAM Conf., To appear, 2014.

8. G. Costa and R. Ortale. Probabilistic analysis of communities and inner roles innetworks: Bayesian generative models and approximate inference. Social NetworkAnalysis and Mining, 3(4):1015 – 1038, 2013.

9. M. DeGroot. Optimal Statistical Decisions. McGraw-Hill, 1970.

Conclusions •  Unified approach item recommenda3on and trust

rela3onships –  Mi3gates the effect of not matching profiles –  Simple, intui3ve, robust mathema3cal formula3on –  Good predic3ve performance

•  Issues –  Inferring the number of factors

•  Indian Buffet Process easy to plug –  Modeling alterna3ves

•  Logis3c, probit –  Computa3onal cost

•  Paralleliza3on •  Reformula3on as tensor decomposi3on?

Thank you

Ques3ons [email protected]

@beman

a bayesian model for recommendation in social rating networks with trust relationships

Data & Analytics