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.”
– 4
Writes
on
“Rolling
Stone”
Overall
ra3ng:

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

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:

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
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)

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

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

✓ ⇥ ! 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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OutDegree − Epinions
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1 10 100 1000
OutDegree
Frequency
10 100 1000 10000
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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
1 10 100 1000
10 100 1000 10000
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InDegree − Ciao
InDegree
Frequency
1 10 100
10 100 1000
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OutDegree − Ciao
OutDegree
Frequency
1 10 100
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

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)

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

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 =

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

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

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).

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
manco@icar.cnr.it
@beman

A Bayesian Model for Recommendation in Social Rating Networks with Trust Relationships

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A Bayesian Model for Recommendation in Social Rating Networks with Trust Relationships