A Bayesian Model for Recommendation in Social Rating Networks with Trust Relationships
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This document presents a Bayesian model for generating recommendations in social rating networks that incorporates trust relationships among users. It outlines a framework for understanding the influence of trust and authority in recommendations, addressing the modeling of user-item interactions and trustworthiness. The authors propose a hierarchical model and utilize Gibbs sampling for inference on trust and rating relationships within the network.
![✓ ⇥ ! v62 2: P ⇠ NW(⇥n) where ⇥n is computed by updating ⇥0 with P, SP;
Evalua3on
Initialize P(0), F(0), Q(0), Y(0);
3: for h = 1 to H do
4: Sample ⇥(h)
• Two
datasets
– Product
evalua3on,
trust
rela3onships
– 5-‐star
ra3ng
system
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
Frequency
◆
10 100 1000 10000
;
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
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InDegree − Epinions
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InDegree
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Fig. 4. The scheme of Gibbs sampling algorithm in pseudo code
Ciao Epinions
Users 7,375 49,289
Trust Relationships 111,781 487,181
Items 106,797 139,738
Ratings 282,618 664,823
InDegree (Avg/Median/Min/Max) 15.16/6/1/100 9.8/2/1/2589
OutDegree (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 4.75/1/1/2026
Ratings 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.
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ItemRatings − Epinions
ItemRatings
Frequency
1 10 100 1000
10 100 1000 10000 1e+05
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UserRatings − Epinions
UserRatings
Frequency
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10 100 1000 10000
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InDegree − Ciao
InDegree
Frequency
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10 100 1000
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OutDegree − Ciao
OutDegree
Frequency
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10 100 1000
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ItemRatings − Ciao
ItemRatings
Frequency
1 10 100
10 100 1000 10000 1e+05
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UserRatings − Ciao
UserRatings
Frequency
10 100 1000
10 100
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 the
AUC loss. We exploited the implementation made available by the authors http://cseweb.ucsd.edu/ akmenon/code. We refer to this method as AUC-MF
in the following. In addition, we considered a further comparison in terms both RMSE and AUC against a basic matrix factorization approach based on
SVD named Joint SVD (JSVD) [11]. We computed a low-rank factorization the 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 the K greatest eigenvalues of XTX. The matrices U and V resemble the roles](https://image.slidesharecdn.com/presentazioneshort-140922022203-phpapp01/85/A-Bayesian-Model-for-Recommendation-in-Social-Rating-Networks-with-Trust-Relationships-16-320.jpg)
![Cold/Warm
start
effects
Epinions
False positive rate
True positive 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
InDegree 10
10 InDegree 100
InDegree 100
Epinions
False positive rate
True positive 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
OutDegree 10
10 OutDegree 100
OutDegree 100
Ciao
False positive rate
True positive 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
InDegree 10
10 InDegree 100
Ciao
False positive rate
True positive 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
OutDegree 10
10 OutDegree 100
OutDegree 100
Epinions
False positive rate
True positive 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
UserRatings(src) 10
10 UserRatings(src) 100
UserRatings(src) 100
Epinions
False positive rate
True positive 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
UserRatings(dst) 10
10 UserRatings(dst) 100
UserRatings(dst) 100
Ciao
False positive rate
True positive 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
UserRatings(src) 10
10 UserRatings(src) 100
UserRatings(src) 100
Ciao
False positive rate
True positive 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
UserRatings(dst) 10
10 UserRatings(dst) 100
UserRatings(dst) 100
0.0 0.2 0.4 0.6 0.8 1.0
Epinions
RMSE
ItemRatings 10
10ItemRatings 100
100 ItemRatings 1000
ItemRatings 1000
UserRatings 10
10UserRatings 100
UserRatings 100
0.0 0.2 0.4 0.6 0.8 1.0
epinions
RMSE
InDegree 10
10InDegree 100
OutDegree 10
10OutDegree 100
OutDegree 100
0.0 0.2 0.4 0.6 0.8 1.0
Ciao
RMSE
ItemRatings 10
10ItemRatings 100
ItemRatings 100
UserRatings 10
10UserRatings 100
UserRatings 100
0.0 0.2 0.4 0.6 0.8
Ciao
RMSE
InDegree 10
10InDegree 100
OutDegree 10
10OutDegree 100
OutDegree 100
Fig. 8. Data distribution vs. AUC and rating prediction.
More precisely, we performed a simple BPMF (as described in [25]). Dually, we](https://image.slidesharecdn.com/presentazioneshort-140922022203-phpapp01/85/A-Bayesian-Model-for-Recommendation-in-Social-Rating-Networks-with-Trust-Relationships-20-320.jpg)
A Bayesian Model for Recommendation in Social Rating Networks with Trust Relationships
- 1. A Bayesian Model for Recommenda3on in Social Ra3ng Networks with Trust Rela3onships Gianni Costa, Giuseppe Manco, Riccardo Ortale
- 2. 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.” – 4 Writes on “Rolling Stone” Overall ra3ng:
- 3. 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.” – 4 Writes on “Rolling Stone” Overall ra3ng: • Joe’s profile doesn’t match 1 and par3ally matches 2 • 3 and 4 are authorita3ve in their fields
- 4. Mo3va3ng example • Joe is looking for a restaurant – Likes fish – Enjoys rock music – No smoker Chez Marcel • 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.” – 4 Writes on “Rolling Stone” Overall ra3ng:
- 5. 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?
- 6. Formal Framework Input: Users, items Basic assump3on: an underlying social network of trust rela3onships exists among users
- 7. Formal Framework Output: (Signed) Network of trust rela3onships + item adop3ons -‐ +
- 8. 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
- 9. 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
- 10. 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
- 11. Modeling trust rela3onships Ru,i | P,Q,F, ↵ ⇠ N((Pu + Fu)0 Qi, ↵−1) Pr(Au,v|P,Q,F)Pr(Y,P,Q,F|A,R) u Pr(Ru,i|P,Q,F)Pr(Y,P,Q,F|A,R) • Likes Pr(Pr(Ru,i|A,R) Pr(Au,v|A,R) fish • Enjoys rock music • No smoker Au,v | P,F, " ⇠ N(P0uFv, "−1) • Member of “Slow Food” Pr(Ru,i|A,R) = Z X Y Au,v|A,R) Z X Y v
- 12. The Bayesian Genera3ve Model W0, ⌫0 μ0, "0 ⇤P ⇤F μP μF μQ ⇤Q F P Q N M " a r ↵ N ⇥ N N ⇥M Fig. 1. Graphical representation of the proposed Bayesian hierarchical model. Sample ⇥P ⇠NW(⇥0) ⇥Q ⇠NW(⇥0) ⇥F ⇠NW(⇥0) For each item i 2 I sample Qi ⇠ N(μQ,⇤−1 Q ) W0, ⌫0 μ0, "0 ⇤P ⇤F μP μF μQ ⇤Q F P Q " N M a r ↵ N ⇥ 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,⇤−1 Q ) 3. For each user u 2 N sample Pu ⇠N(μP,⇤−1 P ) Fu ⇠N(μF,⇤−1 F ) 4. For each pair hu, vi 2 N ⇥ N sample Au,v ⇠ N( ! P0uFv " , "−1) 5. For each pair hu, ii 2 N ⇥ I sample Ru,i ⇠ N((Pu + Fu)Q0j , ↵−1)
- 13. Ru,i | P,Q,F, ↵ ⇠ N((Pu + Fu)0 Qi, ↵−1) Inference and Predic3on • Given Au,v | P,F, " ⇠ N(P0uFu, "−1) observed trust rela3onships (A) and item adop3ons (R) we want to infer Pr(Ru,i|A,R) Pr(Au,v|A,R) • Problem: trust bias – Observed rela3onships in a social network are rarely nega3ve: people only make posi3ve connec3ons explicit
- 14. Inference and Predic3on • Solu3on: latent variable modeling • Yu,v u represents a (bernoulli) latent variable sta3ng whether a nega3ve trust rela3onship exists between u and v v
- 15. Ru,i | P,Q,F, ↵ ⇠ N((Pu + Fu)0 Qi, ↵−1) Inference, model learning • Inference Au,v | P,F, " ⇠ N(P0uFu, "−1) by averaging on latent variables Pr(Ru,i|A,R) = Pr(Au,v|A,R) • Posteriors Pr(Ru,i|A,R) Pr(Au,v|A,R) Z X Y Pr(Ru,i|P,Q,F)Pr(Y,P,Q,F|A,R) dPdFdQ Z X Y Pr(Au,v|P,Q,F)Pr(Y,P,Q,F|A,R) dPdFdQ sampled through Gibbs sampling
- 16. ✓ ⇥ ! v62 2: P ⇠ NW(⇥n) where ⇥n is computed by updating ⇥0 with P, SP; Evalua3on Initialize P(0), F(0), Q(0), Y(0); 3: for h = 1 to H do 4: Sample ⇥(h) • Two datasets – Product evalua3on, trust rela3onships – 5-‐star ra3ng system 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 Frequency ◆ 10 100 1000 10000 ; 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 ● ● ● ● InDegree − Epinions ● ● ● ●● ● ● ●● ●● ● ● ● ● ● ● ● ●●● ●● ● ● ● ● ●● ● ● ●● ● ●● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●●●●●● ● ●●● ● ●● ●●● ● ● ● ● ● ●●●●●● ● 1 10 100 1000 InDegree ● ● ● OutDegree − Epinions ● ● ● ● ● ●● ● ●● ● ●● ● ● ●● ● ● ●●●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ●●●●● ●●●●●●●●●● ● ●● ●● 1 10 100 1000 OutDegree Frequency 10 100 1000 10000 ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ●●● ● ● ●● ● ● ● Fig. 4. The scheme of Gibbs sampling algorithm in pseudo code Ciao Epinions Users 7,375 49,289 Trust Relationships 111,781 487,181 Items 106,797 139,738 Ratings 282,618 664,823 InDegree (Avg/Median/Min/Max) 15.16/6/1/100 9.8/2/1/2589 OutDegree (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 4.75/1/1/2026 Ratings 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. ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ●● ● ●● ● ●●● ●● ●●●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ●● ● ● ●● ●● ● ● ●● ●● ● ● ●● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●●●●●● ●●●●●●●●●●● ● ●● ● ItemRatings − Epinions ItemRatings Frequency 1 10 100 1000 10 100 1000 10000 1e+05 ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ●● ●● ● ● ● ●● ● ● ● ●●●● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●●● ● ● ●● ●●●●●●● ● ● ●●●●●● ●●● ●●● UserRatings − Epinions UserRatings Frequency 1 10 100 1000 10 100 1000 10000 ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● InDegree − Ciao InDegree Frequency 1 10 100 10 100 1000 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ●●●●● ●●●●●● ●● ● ●● ● OutDegree − Ciao OutDegree Frequency 1 10 100 10 100 1000 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ●● ● ●●●●● ● ●●●●●● ●●● ● ●●● ● ItemRatings − Ciao ItemRatings Frequency 1 10 100 10 100 1000 10000 1e+05 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ●●●●● ● ●●●●● ●●●●●●●●●● ● ●●●●●●●●●●●●●● ● ● UserRatings − Ciao UserRatings Frequency 10 100 1000 10 100 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 the AUC loss. We exploited the implementation made available by the authors http://cseweb.ucsd.edu/ akmenon/code. We refer to this method as AUC-MF in the following. In addition, we considered a further comparison in terms both RMSE and AUC against a basic matrix factorization approach based on SVD named Joint SVD (JSVD) [11]. We computed a low-rank factorization the 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 the K greatest eigenvalues of XTX. The matrices U and V resemble the roles
- 17. 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)
- 18. minimum RMSE on both datasets. There is a tendency decrease. However, this tendency is more evident the other two methods exhibit negligible di↵erences. RMSE 4 8 16 32 64 128 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Epinions N. of factors RMSE HBPMF JSVD SocialMF 4 8 16 32 64 128 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Ciao N. of factors RMSE HBPMF JSVD SocialMF 0.2 0.4 0.6 0.8 1.0 1.2 Epinions 4 8 16 32 0.0 N. of factors AUC HBPMF JSVD AUC−MF Fig. 6. Prediction results. The opposite trend is observed in trust prediction. prefer a low number of factors, as the best results are
- 19. There is a tendency of the RMSE to pro-gressively tendency is more evident on SocialMF, while The opposite trend is observed in trust prediction. Here, all prefer a low number of factors, as the best results are achieved devised HBPMF model AUC achieves the maximum AUC on the and results comparable to JSVD on Ciao. The detailed results Fig. 7, where the ROC curves are reported. In general, the predictive of the Bayesian hierarchical model is stable with regards to the This is a direct result of the Bayesian modeling, which makes to the growth of the model complexity. Fig. 8 also shows varies according to the distributions which characterize the data. a correlation between accuracy and node degrees, as well as the provided by a user or received by an item. negligible di↵erences. 4 8 16 32 64 128 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Epinions N. of factors AUC HBPMF JSVD AUC−MF 4 8 16 32 64 128 0.0 0.2 0.4 0.6 0.8 1.0 Ciao N. of factors AUC HBPMF JSVD AUC−MF Prediction results. Epinions trust prediction. Here, all methods tend to best results are achieved with K = 4. The maximum AUC on the Epinions dataset, False positive rate on Ciao. The detailed results are shown in True positive 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 HBPMF JSVD AUC−MF Ciao False positive rate True positive 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 HBPMF JSVD AUC−MF 4 factors Fig. 7. ROC curves on trust prediction for K =
- 20. Cold/Warm start effects Epinions False positive rate True positive 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 InDegree 10 10 InDegree 100 InDegree 100 Epinions False positive rate True positive 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 OutDegree 10 10 OutDegree 100 OutDegree 100 Ciao False positive rate True positive 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 InDegree 10 10 InDegree 100 Ciao False positive rate True positive 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 OutDegree 10 10 OutDegree 100 OutDegree 100 Epinions False positive rate True positive 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 UserRatings(src) 10 10 UserRatings(src) 100 UserRatings(src) 100 Epinions False positive rate True positive 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 UserRatings(dst) 10 10 UserRatings(dst) 100 UserRatings(dst) 100 Ciao False positive rate True positive 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 UserRatings(src) 10 10 UserRatings(src) 100 UserRatings(src) 100 Ciao False positive rate True positive 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 UserRatings(dst) 10 10 UserRatings(dst) 100 UserRatings(dst) 100 0.0 0.2 0.4 0.6 0.8 1.0 Epinions RMSE ItemRatings 10 10ItemRatings 100 100 ItemRatings 1000 ItemRatings 1000 UserRatings 10 10UserRatings 100 UserRatings 100 0.0 0.2 0.4 0.6 0.8 1.0 epinions RMSE InDegree 10 10InDegree 100 OutDegree 10 10OutDegree 100 OutDegree 100 0.0 0.2 0.4 0.6 0.8 1.0 Ciao RMSE ItemRatings 10 10ItemRatings 100 ItemRatings 100 UserRatings 10 10UserRatings 100 UserRatings 100 0.0 0.2 0.4 0.6 0.8 Ciao RMSE InDegree 10 10InDegree 100 OutDegree 10 10OutDegree 100 OutDegree 100 Fig. 8. Data distribution vs. AUC and rating prediction. More precisely, we performed a simple BPMF (as described in [25]). Dually, we
- 21. Joint modeling • Significant on RMSE RMSE (1) AUC (1) RMSE (2) AUC (2) 0.0 0.2 0.4 0.6 0.8 1.0 Metric (RMSE/AUC) Full Model Partial Model 1000 2000 3000 4000 5000 6000 Epinions 4 8 16 32 0 N. of factors Time (secs.) HBPMF JSVD SocialMF AUC−MF Fig. 9. (a) E↵ects of the joint modeling. (1 denotes Average running time for iteration (JSVD reports
- 22. Computa3onal cost 4 8 16 32 64 128 0 1000 2000 3000 4000 5000 6000 Epinions N. of factors Time (secs.) HBPMF JSVD SocialMF AUC−MF 4 8 16 32 64 128 0 50 100 150 200 250 300 350 Ciao N. of factors Time (secs.) HBPMF JSVD SocialMF AUC−MF modeling. (1 denotes Epinions, and 2 denotes Ciao).
- 23. 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?
- 24. Thank you Ques3ons [email protected] @beman