Circle-based Recommendation in Online Social Networks
Xiwang Yang
ECE Department
Polytechnic Institute of NYU
Brooklyn, New York
xyang01@students.poly.edu
Harald Steck
∗
Bell Labs
Alcatel-Lucent
Murray Hill, New Jersey
hsteck@gmail.com
Yong Liu
ECE Department
Polytechnic Institute of NYU
Brooklyn, New York
yongliu@poly.edu
ABSTRACT
Online social network information promises to increase rec-
ommendation accuracy beyond the capabilities of purely
rating/feedback-driven recommender systems (RS). As to
better serve users’ activities across different domains, many
online social networks now support a new feature of“Friends
Circles”, which refines the domain-oblivious “Friends” con-
cept. RS should also benefit from domain-specific “Trust
Circles”. Intuitively, a user may trust different subsets of
friends regarding different domains. Unfortunately, in most
existing multi-category rating datasets, a user’s social con-
nections from all categories are mixed together. This paper
presents an effort to develop circle-based RS. We focus on
inferring category-specific social trust circles from available
rating data combined with social network data. We out-
line several variants of weighting friends within circles based
on their inferred expertise levels. Through experiments on
publicly available data, we demonstrate that the proposed
circle-based recommendation models can better utilize user’s
social trust information, resulting in increased recommenda-
tion accuracy.
1. INTRODUCTION
Recommender Systems (RS) deal with information over-
load by suggesting to users the items that are potentially of
their interests. Traditional collaborative filtering approaches
predict users’ interests by mining user rating history data
[1], [2], [4], [6], [15], [22] and [23]. The increasingly popu-
lar online social networks provide additional information to
enhance pure rating-based RSes. Several social-trust based
RSes have recently been proposed to improve recommenda-
tion accuracy, to just name a few, [9], [10], [11], [12], [14],
[16], [17], and [18]. The common rationale behind all of
them is that a user’s taste is similar to and/or influenced
by her trusted friends in social networks. Meanwhile, an-
other obvious fact is that users’ social life, being online or
offline, is intrinsically multifaceted. To better serve a user’s
∗
Now at Netflix Inc. Work was done while at Bell Labs.
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activities across different domains, many online social net-
works now support a new feature of “Friends Circles”, which
refines the domain-oblivious “Friends” concept. Google+ is
the first to introduce “Circles”, the function that let users
assign classmates, family members, colleagues and others to
different groups. Facebook, which has long had Friend lists,
also launched its “Groups” feature to assign users to groups
for finer granular information sharing: a user can share dif-
ferent information with different groups. In Twitter, users
can organize people who they follow (followees) into “lists”.
When a user clicks to view a list, she will see a stream of
Tweets from all her followees in that list.
RSes should also benefit from domain-specific “Trust Cir-
cles”. Intuitively, a user trusts different subsets of friends in
different domains. For example, in the context of multi-
category recommendation, a user u may trust user v in
Cars category while not trust v in Kids’ TV Show category.
Therefore, u should care less about v’s ratings in Kids’ TV
Show category than in Cars category. Ideally, if we know
users’ trust circles in different categories, to predict ratings
in one category, we probably should only use trust circles
specific to that category. We call it circle-based recommen-
dation. Unfortunately, in most existing multi-category rat-
ing datasets, a user’s social connections from all categories
are mixed together. So if we use all social trust informa-
tion for rating prediction in a specific category, we mis-
use social trust information from other categories, which
compromises the rating prediction accuracy. Apart from
that, even if the circles were explicitly known, e.g. Circles
in Google+ or Facebook, they may not correspond to par-
ticular item categories that a recommender system may be
concerned with. Therefore, inferred circles concerning each
item-category may be of value by themselves, besides the
explicitly known circles.
This paper presents an effort to develop circle-based RS.
We focus on inferring category-specific social trust circles
from available rating data combined with social network
data where social trust links across all categories are mixed
together. We propose a set of algorithms to infer category-
specific circles of friends and to infer the trust value on each
link based on user rating activities in each category. To infer
the trust value of a link in a circle, we first estimate a user’s
expertise level in a category based on the rating activities of
herself as well as all users trusting her. We then assign to
users trust values proportional to their expertise levels. The
reconstructed trust circles are used to develop a low-rank
matrix factorization type of RS. Through experiments on
publicly available data, we demonstrate that the proposed
circle-based RSes can better utilize user’s social trust infor-
mation and achieve more accurate recommendation than the
traditional matrix factorization approaches that do not use
any social trust information, and the existing social-trust
based RSes that use mixed social trust information across
all categories.
The rest of the paper is organized as follows. Section 2
presents the related work. In Section 3, we first introduce
the concept of trust circle, then propose three variants of
assigning weights to users within each circle. Finally, we
present circle-based training models, based on either ratings
from one category or ratings from all categories. Experimen-
tal results are presented in Section 4. The paper is concluded
in Section 5.
2. RELATED WORK
In this paper, we focus on low-rank matrix factorization
models, as they were found to be one of the most accurate
single models for collaborative filtering [5,7,8,19,20]. In the
following, we briefly review the ones relevant to this paper.
2.1 Matrix Factorization (MF)
While there are various sophisticated approaches (e.g. [5,
7, 8, 19, 20]), we here briefly review the basic low-rank ma-
trix factorization (MF) approach, which will be extended
towards social network information in the remainder of this
paper. The matrix of predicted ratings
ˆ
R ∈ R
u
0
×i
0
, where
u
0
denotes the number of users, and i
0
the number of items,
is modeled as:
ˆ
R = r
m
+ QP
>
, (1)
with matrices P ∈ R
i
0
×d
and Q ∈ R
u
0
×d
, where d is the
rank (or dimension of the latent space), with d i
0
, u
0
,
and r
m
∈ R is a (global) offset value.
This model is trained on the observed rating data by min-
imizing the square error (with the usual Frobenius/L2-norm
regularization) (see also [5, 19]):
1
2
X
(u,i)obs.
R
u,i
−
ˆ
R
u,i
2
+
λ
2
||P ||
2
F
+ ||Q||
2
F
, (2)
where
ˆ
R
u,i
denotes the ratings predicted by the model in
Eq. (1); and R
u,i
are the actual rating values in the training
data for item i from user u. This objective function can be
minimized efficiently using gradient descent method [12].
Once the low-rank matrices P and Q have been learned,
rating values can be predicted according to Eq. (1) for any
user-item pair (u, i).
2.2 MF and Social Networks
The usage of social network data has been found to im-
prove the prediction accuracy of rating values, and various
models for integrating these two data sources have been
proposed, like Social Recommendation (SoRec) [10], Social
Trust Ensemble (STE) [9], Recommender Systems with So-
cial Regularization [11], Adaptive social similarities for rec-
ommender systems [13], among which the SocialMF model
[12] was found to achieve a particularly low RMSE value,
and is hence used as a baseline model in our experimental
comparison study.
2.2.1 SocialMF Model
The SocialMF model was proposed in [12], and was found
to outperform SoRec and STE with respect to RMSE. The
social network information is represented by a matrix S ∈
R
u
0
×u
0
, where u
0
is the number of users. The directed and
weighted social relationship of user u with user v (e.g. user
u trusts/knows/follows user v) is represented by a positive
value S
u,v
∈ (0, 1]. An absent or unobserved social relation-
ship is reflected by S
u,v
= s
m
, where typically s
m
= 0. Each
of the rows of the social network matrix S is normalized to
1, resulting in the new matrix S
∗
with S
∗
u,v
∝ S
u,v
, and
P
v
S
∗
u,v
= 1 for each user u.
The idea underlying SocialMF is that neighbors in the
social network may have similar interests. This similarity
is enforced by the second term in the objective function in
equation (3), which says that user profile Q
u
should be sim-
ilar to the (weighted) average of his/her friends’ profiles Q
v
(measured in terms of the square error):
1
2
X
(i,u)observed
R
u,i
−
ˆ
R
u,i
2
+
β
2
X
all u
(Q
u
−
X
v
S
∗
u,v
Q
v
)(Q
u
−
X
v
S
∗
u,v
Q
v
)
>
!
+
λ
2
||P ||
2
F
+ ||Q||
2
F
, (3)
where the ratings
ˆ
R
u,i
predicted by this model are obtained
according to Eq. (1). Note that we omitted the logistic
function from the original publication [12], as we found its
effect rather negligible in our experiments. The trade-off
between the feedback data (ratings) and the social network
information is determined by a weight β ≥ 0. Obviously,
the social network information is ignored if β = 0, and in-
creasing β shifts the trade-off more and more towards the
social network information.
Eq. (3) can be optimized by the gradient descent approach
(see the update equations (13) and (14) in [12]).
Once the model is trained, the soft constraint that neigh-
bors should have similar user profiles is captured in the user
latent feature matrix Q. The rating value for any user con-
cerning any item can be predicted according to Eq. (1).
3. CIRCLE-BASED RECOMMENDATION
MODELS
Our proposed Circle-based Recommendation (CircleCon)
models may be viewed as an extension of the SocialMF
model [12] to social networks with inferred circles of friends.
3.1 Trust Circle Inference
We infer the circles of friends from rating (or other feed-
back) data concerning items that can be divided into different
categories (or genres etc.). The basic idea is that a user may
trust each friend only concerning certain item categories but
not regarding others. For instance, the circle of friends con-
cerning cars may differ significantly from the circle regarding
kids’ TV shows.
To this end, we divide the social network S of all trust
relationships into several sub-networks S
(c)
, each of which
concerning a single category c of items.
Definition (Inferred Circle): Regarding each category
c, a user v is in the inferred circle of user u, i.e., in the set
C
(c)
u
, if and only if the following two conditions hold:
1
u
2
u
3
u
4
u
5
u
1 2 3
( , , )c c c
12
( , )cc
23
( , )cc
13
( , )cc
13
( , )cc
(a)
1
u
2
u
3
u
4
u
5
u
1 2 3
( , , )c c c
12
( , )cc
23
( , )cc
13
( , )cc
13
( , )cc
(b)
1
u
2
u
3
u
4
u
5
u
1 2 3
( , , )c c c
12
( , )cc
23
( , )cc
13
( , )cc
13
( , )cc
(c)
1
u
2
u
3
u
4
u
5
u
1 2 3
( , , )c c c
12
( , )cc
23
( , )cc
13
( , )cc
13
( , )cc
(d)
Figure 1: Illustration of inferred circles, each user is labeled with the categories in which she has ratings. a):
the original social network; b), c) and d): inferred circles for categories c
1
, c
2
and c
3
respectively.
u
1
v
2
v
3
v
4
v
1
()c
v
E
2
()c
v
E
3
()c
v
E
4
()c
v
E
1
()
,
c
uv
S
2
()
,
c
uv
S
3
()
,
c
uv
S
4
()
,
c
uv
S
Inferred circle of in category
u
c
Figure 2: Illustration of expertise-based trust-
assignment in category c.
• S
u,v
> 0 in the (original) social network, and
• N
(c)
u
> 0 and N
(c)
v
> 0 in the rating data,
where N
(c)
u
denotes the number of ratings that user u has
assigned to items in category c. Otherwise, user v is not in
the circle of u concerning category c, i.e., v 6∈ C
(c)
u
.
This is illustrated for a toy example in Figure 1.
3.2 Trust Value Assignment
The trust values between friends in the same inferred circle
(based on item category c) are captured in a social network
matrix S
(c)
, such that S
(c)
u,v
= 0 if v 6∈ C
(c)
u
, S
(c)
u,v
> 0 if
v ∈ C
(c)
u
. In the following, we consider three variants of
defining the positive values S
(c)
u,v
> 0 when user v is in the
inferred circle of user u regarding category c. They are then
experimentally evaluated in Section 4.
3.2.1 CircleCon1: Equal Trust
We start with the simplest variant of defining trust val-
ues S
(c)
u,v
> 0 within inferred circles regarding item cate-
gory c: each user v in the inferred circle of user u gets as-
signed the same trust value, i.e., S
(c)∗
u,v
= const if v ∈ C
(c)
u
.
The constant is determined by the normalization constraint
P
v∈C
c
u
S
(c)∗
u,v
= 1. In other words, S
(c)∗
u,v
= 1/|C
(c)
u
|, ∀v ∈ C
(c)
u
.
3.2.2 CircleCon2: Expertise-based Trust
In this section, we outline two variants of assigning differ-
ent trust values to friends within a trust circle. The goal is
to assign a higher trust value or weight to the friends that
are experts in the circle / category. As an approximation to
their level of expertise, we use the numbers of ratings they
assigned to items in the category. The idea is that an expert
in a category may have rated more items in that category
than users who are not experts in that category.
We formalize this as follows. We consider directed trust
relationships; undirected/mutual trust relationships (e.g.,
friendship) can be viewed as a special case of directed trust.
If user u trusts user v in category c, we say u follows v in
category c, i.e., u is the follower of v, and v is a followee
of u. All of user u’s followees in category c form the trust
circle C
(c)
u
of u in c. We also denote u’s followers in category
c as F
(c)
u
. Finally, a user u’s expertise level in category c is
denoted as E
(c)
u
. This is illustrated in Figure 2.
We assign trust values to u’s followees in circle C
(c)
u
to be
proportional to their expertise levels in category c. Based
on this idea, we consider two variants in the following:
• Variant a: In this case, user v
0
s expertise level in
category c is equal to the number of ratings that v
assigned in category c, i.e., E
(c)
v
= N
(c)
v
. Thus,
S
(c)
u,v
=
N
(c)
v
if v ∈ C
c
u
0 otherwise.
We then normalize each row of S
(c)
matrix as follows
S
(c)∗
u,v
=
S
(c)
u,v
P
v∈C
(c)
u
S
(c)
u,v
, (4)
which ensures that, for each user u, the weights across
all users v in each circle are normalized to unity:
X
v∈C
(c)
u
S
(c)∗
u,v
= 1.
• Variant b: In this case, the expertise level of user v in
category c is the product of two components: the first
component is the number of ratings that v assigned
in category c, the second component is some voting
value in category c from all her followers in F
(c)
v
. The
intuition is that if most of v’s followers have lots of
ratings in category c, and they all trust v, it is a good
indication that v is an expert in category c.
We denote the voting value from followers of v in cate-
gory c by φ
(c)
v
. For each follower w ∈ F
(c)
v
, we compute
the distribution of her ratings in each individual cate-
gory. We denote D
w
as a distribution vector over all
the categories,
D
w
=
N
w
(1)
N
w
,
N
w
(2)
N
w
, ...,
N
w
(m)
N
w
, (5)
where m is the number of categories, and N
w
(c) with
c = 1, ..., m is the number of ratings assigned by user
w in category c; N
w
is the total number of ratings as-
signed by user w, N
w
=
P
c
N
w
(c). Thus, D
w
records
the proportions of ratings user w assigned in all cat-
egories. It reflects the interest distribution of w cross
all categories.
The second component, namely the voting value from
all followers is defined as φ
(c)
v
=
P
w∈F
(c)
v
D
w
(c).
Combining both components, we have the following
expression for v’s expertise level:
E
(c)
v
= N
(c)
v
·
X
w∈F
(c)
v
D
w
(c)
which results in the trust values
S
(c)
u,v
=
(
N
(c)
v
·
P
w∈F
(c)
v
D
w
(c) if v ∈ C
(c)
u
0 otherwise.
As in each of the above cases, also here we finally nor-
malize each row of the S
(c)
matrix (across v):
S
(c)∗
u,v
= S
(c)
u,v
/
X
v∈C
(c)
u
S
(c)
u,v
.
3.2.3 CircleCon3: Trust Splitting
The previous circle inference and trust value assignment
essentially assume that if u issues a trust statement towards
v, and u and v simultaneously have ratings in a category c,
then u trusts v in c. The trust value assignment is done in
each circle separately. In practice, user u might issue a trust
statement towards v just because of v’s ratings in a subset of
categories in which they simultaneously have ratings. The
trust value of u towards v in category c should reflect the
likelihood that u issues the trust statement towards v due
to v’s ratings in c. One simple heuristic is to make the like-
lihood proportional to the number of v’s ratings in category
c. In other words, given that u trusts v, if v has more ratings
in category c
1
than in c
2
, it is more likely that u trusts v
because of v’s ratings in c
1
than v’s ratings in c
2
. Now if
u and v simultaneously have ratings in multiple categories,
the trust value of u towards v should be split cross those
commonly rated categories.
Essentially, we now normalize trust values across c,
S
(c)
u,v
=
N
(c)
v
P
c:v∈C
(c)
u
N
(c)
v
if v ∈ C
(c)
u
0 otherwise.
To illustrate this trust splitting, let us look at Figure 1:
user u
2
trusts user u
1
and both of them have ratings in
category c
1
and c
2
. Assume the number of ratings u
1
issued
in category c
1
and c
2
are 9 and 1 respectively. The trust
value in original social network is S
u
2
,u
1
= 1. Now after
trust splitting, we get S
(c
1
)
u
2
,u
1
= 0.9 and S
(c
2
)
u
2
,u
1
= 0.1.
Like before, we then also normalize each row of S
(c)
matrix
(across v), as to make the trust values independent of the
activity levels of the users in each circle:
S
(c)∗
u,v
= S
(c)
u,v
/
X
v∈C
(c)
u
S
(c)
u,v
.
We note that the normalizations across c and then v may
also be viewed as the first step of an iterative procedure
called iterative proportional fitting [24]. In short, when this
procedure is iterated until convergence, it results in an exact
joint normalization regarding both c and v:
P
c
P
v
S
(c)
u,v
=
const, where
P
c
S
(c)
u,v
= 1 for each v, and
P
v
S
(c)
u,v
= const
for each c. While the iterative procedure yields exact nor-
malization, it is computationally expensive. For the latter
reason, the reported results in our experiment section are
obtained after only one iteration.
3.3 Model Training
3.3.1 Training with ratings from each category
Using the (normalized) trust network S
(c)∗
, as defined
above, we train a separate matrix factorization model for
each category c. For each kind of inferred circles of friends,
we obtain a separate user profile Q
(c)
and item profile P
(c)
for each c. Similar to the SocialMF model [12], but with the
crucial difference of using inferred social circles of friends,
we use the following training objective function
L
(c)
(R
(c)
, Q
(c)
, P
(c)
, S
(c)∗
) =
1
2
X
(u,i)obs.
R
(c)
u,i
−
ˆ
R
(c)
u,i
2
+
β
2
X
all u
(Q
(c)
u
−
X
v
S
(c)∗
u,v
Q
(c)
v
)(Q
(c)
u
−
X
v
S
(c)∗
u,v
Q
(c)
v
)
>
!
+
λ
2
||P
(c)
||
2
F
+ ||Q
(c)
||
2
F
, (6)
where we only use ratings R
(c)
u,i
in category c;
ˆ
R
(c)
u,i
is the
predicted rating of item i in category c,
ˆ
R
(c)
u,i
= r
(c)
m
+ Q
(c)
u
P
(c)
i
>
, (7)
where we define the global bias term r
(c)
m
as the average value
of observed training rating in category c (see also Table 4).
The summation in Eq. (6) extends over all observed user-
item pairs (u, i) where item i belongs to category c. Note
that this model only captures user and item profiles in cat-
egory c, i.e., Q
(c)
and P
(c)
. P
(c)
∈ R
i
(c)
0
×d
, where i
(c)
0
is the
number of items in category c and Q
(c)
∈ R
u
0
×d
. Eq. (6)
can be minimized by the gradient decent approach, analo-
gous to [12]:
∂L
(c)
∂Q
(c)
u
=
X
i:cat(i)=c
I
R
(c)
u,i
r
(c)
m
+ Q
(c)
u
P
(c)
i
T
− R
(c)
u,i
P
(c)
i
+ λQ
(c)
u
+ β
Q
(c)
u
−
X
v∈C
(c)
u
S
(c)∗
u,v
Q
(c)
v
− β
X
v:u∈C
(c)
v
S
(c)∗
v,u
Q
(c)
v
−
X
w∈C
(c)
v
S
(c)∗
v,w
Q
(c)
w
,
(8)
where cat(i) is the category of item i.
Table 1: Epinions Data: Top-10 Category Statistics.
Category User Count Item Count Rating Count Sparsity
Trust Original Degree
Fraction Degree in Circle
Videos & DVDs 17,312 10,065 94,261 0.999459 44.64% 29.4 13.1
Books 11,296 21,662 47,889 0.999804 36.09% 45.1 16.3
Music 10,188 14,905 43,079 0.999716 21.48% 50.0 10.7
Video Games 9,124 2,389 29,661 0.998639 13.32% 55.8 7.43
Toys 6,373 3,344 26,789 0.998743 21.49% 79.9 17.2
Online Stores& Services 8,074 973 22,661 0.997115 28.55% 63.0 18.0
Software 8,290 1,624 19,400 0.998559 22.05% 61.4 13.5
Destinations 7,438 1475 19,395 0.998232 23.12% 68.4 15.8
Cars 10,847 3,108 17,604 0.999478 19.35% 46.9 9.1
Kids’ TV Shows 4,784 259 11,203 0.990958 10.85% 106.4 11.6
∂L
(c)
∂P
(c)
i
=
X
all u
I
R
(c)
u,i
r
(c)
m
+ Q
(c)
u
P
(c)
i
T
− R
(c)
u,i
Q
(c)
u
+ λP
(c)
i
,
(9)
where I
R
(c)
u,i
is the indicator function that is equal to 1 if u
has rated i in category c, and equal to 0 otherwise. The
initial values of Q
(c)
and P
(c)
are sampled from the normal
distribution with zero mean. In each iteration, Q
(c)
and P
(c)
are updated based on the latent variables from the previous
iteration.
Once Q
(c)
and P
(c)
are learned for each category c, this
model can be used to predict ratings for user-item pairs (u, i)
according to Eq. (7), where the category c of item i deter-
mines the matrices Q
(c)
u
and P
(c)
i
to be used.
3.3.2 Training with ratings for all categories.
As an alternative training objective function, we also con-
sidered using all ratings in the data, instead of only the
ratings in category c. The only difference to Eq. (6) is that
the first line is replaced by
1
2
X
(u,i)obs.
R
u,i
−
ˆ
R
u,i
2
, (10)
where the summation extends over all observed user-item
pairs (u, i) from all categories. As before, we train a separate
model for each category c, i.e., Q
(c)
and P
(c)
, with P
(c)
∈
R
i
0
×d
, and Q
(c)
∈ R
u
0
×d
.
4. EXPERIMENTS
In this section, we evaluate our different variants of Circle-
based recommendation and compare them to the existing
approaches using the Epinions dataset
1
.
4.1 Dataset
Epinions is a consumer opinion website where users can
review items (such as cars, movies, books, software,...) and
also assign them numeric ratings in the range of 1 (min) to
1
http://www.epinions.com/
Figure 7: Distribution of Number of Ratings across
Categories.
5 (max). Users can also express their trust to other users,
such as reviewers whose reviews and ratings they have con-
sistently found to be valuable. Each user has a list of trusted
users. A user issues a trust statement to another user by
adding the user to her trust list. In the Epinions dataset,
the trust values between users are binary: if user B is in
user A’s trust list, then user A’s trust value towards B is 1,
otherwise it is 0.
We use the version of the Epinions dataset
2
published
by the authors of [3]. It consists of ratings from 71, 002
users who rated a total of 104, 356 different items from 451
categories. The total number of ratings is 571, 235. The
distribution of the ratings cross all categories is plotted in
Figure 7. A large number of ratings fall into a small number
of large categories. The distribution of users and items in
the top-10 categories is presented in Table 1.
The total number of issued trust statements is 508, 960.
We apply the circle inference algorithms presented in Sec-
tion 3.1 to the Epinions dataset. The fraction of trust links
with S
(c)
u,v
> 0 for each category c, out of all trust links in
the entire original social network is shown in Table 1 column
Trust Fraction. We can see that the inferred social network
of each category is much smaller than the original one. A
pair of users connected by a trust link in the original social
2
http://alchemy.cs.washington.edu/data/epinions/
