Online Social Networks and Media 7 (2018) 1–11

Contents lists available at ScienceDirect

Online Social Networks and Media

journal homepage: www.elsevier.com/locate/osnem

Social trust model for rating prediction in recommender systems:

Effects of similarity, centrality, and social ties

Anahita Davoudi ∗, Mainak Chatterjee

University of Central Florida, Orlando, FL 32816, United States

a r t i c l e i n f o

Article history:

Received 2 October 2017

Revised 22 May 2018

Accepted 25 May 2018

a b s t r a c t

The success of e-commerce companies is becoming increasingly dependent on product recommender sys-

tems which have become powerful tools that personalize the shopping experience for users based on user

interests and interactions. Most modern recommender systems concentrate on finding the relevant items

for each user based on their interests only, and ignore the social interactions among users. Some recom-

mender systems, rely on the ‘trust’ of users. However in social science, trust, as a human characteristic,

is a complex concept with multiple facets which has not been fully explored in recommender systems.

In this paper, to model a realistic and accurate recommender system, we address the problem of so-

cial trust modeling where trust values are shaped based users characteristics in a social network. We

propose a method that can predict rating for personalized recommender systems based on similarity,

centrality and social relationships. Compared with traditional collaborative filtering approaches, the ad-

vantage of the proposed mechanism is its consideration of social trust values. We use the probabilistic

matrix factorization method to predict user rating for products based on user-item rating matrix. Similar-

ity is modeled using a rating-based (i.e., Vector Space Similarity and Pearson Correlation Coefficient) and

connection-based similarity measurements. Centrality metrics are quantified using degree, eigen-vector,

Katz and PageRank centralities. To validate the proposed trust model, an Epinions dataset is used and the

rating prediction scheme is implemented. Comprehensive analysis shows that the proposed trust model

based on similarity and centrality metrics provide better rating prediction rather than using binary trust

values. Based on the results, we find that the degree centrality is more effective compared to other cen-

tralities in rating prediction using the specific dataset. Also trust model based on the connection-based

similarity performs better compared to the Vector Space Similarity and Pearson Correlation Coefficient

similarities which are rating based. The experimental results on real-world dataset demonstrate the ef-

fectiveness of our proposed model in further improving the accuracy of rating prediction in social rec-

ommender systems.

© 2018 Elsevier B.V. All rights reserved.

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. Introduction

Use of the Internet has enhanced the quality of life for us; tasks

hat were once done mostly through human interactions, such as

anking, shopping, or communication, can now be done online

hich is a simpler and a better alternative. Meanwhile, the amount

f information that is available on the web is rapidly growing,

hich makes it difficult to find tailored information quickly and

fficiently. Recommender systems help users with personalized in-

ormation for item selection and purchasing decisions based on

sers’ tastes and preferences using a variety of information gath-

ring techniques. The user’s information is gathered either explic-

∗ Corresponding author.

E-mail addresses: anahita@cs.ucf.edu , anahita@eecs.ucf.edu (A. Davoudi),

ainak@eecs.ucf.edu (M. Chatterjee).

m

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ttps://doi.org/10.1016/j.osnem.2018.05.001

468-6964/© 2018 Elsevier B.V. All rights reserved.

tly by mining user’s ratings for the products they purchased, or

mplicitly by monitoring user’s behavior. These systems, offering a

ersonalized experience based on social interactions or user prefer-

nces, are fantastic opportunities for e-commerce and, hence many

uch techniques have been adopted for a variety of applications

ike shopping, movies, book reviews, etc [23,41] .

Online platforms allow users the opportunity to interact with

ach other. Users create and read reviews, form opinion based on

xperience of other users, and make informed decisions on pur-

hasing a specific item. These social-centered platforms are called

social recommender systems” [45] . In real world, our friends help

s discover new things and advise us about interesting items,

ovies or restaurants; thereby collaborating with us in a selec-

ion process. Being aware of this form of collaboration in real life,

esearchers have invested on the development of recommender

2 A. Davoudi, M. Chatterjee / Online Social Networks and Media 7 (2018) 1–11

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systems [39] which recommend products to a user based on the

information extracted from or suggested by other users.

Despite the fact that many studies have been done on how to

improve recommender systems, there is still a great potential in

using the social relationships in furnishing and harnessing the rec-

ommender systems. Traditional recommender systems assume that

users are independent and identically distributed which ignore the

social interactions and trust relationships between users. However,

user’s social relationships play an important role in the behavior of

users regarding future ratings. Since most of the similarities within

a social network are caused by the influence and interactions of

its users, it is reasonable to develop a social recommender sys-

tem based on the user connections and interactions. Retailing plat-

forms usually do not consider social factors and social networking

platforms generally do not consider online shopping related factors

such as purchase history and product rating.

In addition to social connections, trust relationships also influ-

ence one’s decisions and ought to be considered for recommenda-

tions [43] . In a social network, trust relationships and social rela-

tionships are two different concepts. Two socially connected users

would not necessary trust each other. Also, multiple connections

of a user would not have equal impact on user’s opinions and de-

cisions. Trust has been extensively researched for improving the

predictive accuracy of recommender systems [10,15,29] . Trust pro-

vides beneficial information from which user preferences can be

better extracted, an alternative to rating-based similarity. In addi-

tion to trust relationships, users with similar taste in purchasing

would show similar behavior when rating a product as well.

In this paper, we combine the features of social networks and

e-commerce platforms to design a social recommendation mecha-

nism to increase the prediction accuracy of product ratings by con-

sidering similarity, user importance in the network, and social trust

relationships. We argue that users are influenced by social interac-

tions, in particular, by the set of trusted friends and their respec-

tive importance. To that end, we combine social trust connections

and user-item matrix to predict the rating that a user would assign

to a product. We use matrix factorization to factor the user-item

rating matrix into two low-dimensional matrices consisting of user

latent matrix and item latent matrix. For the social connections,

we consider both user importance and user similarity to build the

social trust model between users. We use Vector Space Similarity

(VSS) [7] and Pearson Correlation Coefficient (PCC) [38] to obtain

the similarity between users. Using centrality measures such as de-

gree, eigen-vector, Katz and PageRank, we quantify the importance

of users in the network. We use a linear combination of similar-

ity and centrality to model the trust parameter between users. The

proposed method captures the balance between user taste and her

friends’ taste and adjusts the share of centrality and similarity in

the trust values using two parameters. The low-dimensional latent

user-specific and item-specific matrices are estimated by perform-

ing gradient descent on the objective function. We use a dataset

from Epinions to validate the proposed model. We estimate the

accuracy of the proposed method in terms of the mean absolute

error by comparing the predicted and the actual user ratings of

products. Results reveal that there is a high correlation between

the predicted and the actual ratings. The proposed method is also

compared using binary trust values as well as considering the four

different centrality measures. In summary, our experiment results

show that the proposed model could enhance the recommendation

accuracy.

The remainder of the paper is organized as follows. In Section 2 ,

we discuss the existing literature related to recommender systems.

We formally propose a social-based trust model for rating predic-

tion in recommender systems in Section 3 . In Section 4 , we present

the social trust model using matrix factorization. In Section 5 ,

we introduce the Epinions dataset that we used and also the er-

or metrics. We evaluate the performance of the proposed model

n the Epinions dataset in Section 5 . Conclusions are drawn in

ection 7 .

. Related work

In recent years, different types of recommender systems have

een developed, most of which use content-based filtering, col-

aborative filtering, or a mix of both [6] . Content-based systems

se items’ characteristics and the ratings that users have given to

enerate recommendations. Collaborative systems identify similar

sers and analyze their preferences to generate recommendations.

ybrid methods, such as the content-boosted collaborative filter-

ng algorithm [33] , combines these two techniques, hoping to avoid

he limitations of either approach and improve the recommenda-

ion performance. In Section 2.1 , we discuss collaborative filtering

s the techniques used in this paper are motivated by those for

ollaborative filtering. In Section 2.2 , we discuss the related re-

earch with respect to the important factors that govern the design

f an efficient recommender system.

.1. Collaborative filtering

Collaborative filtering methods have proved to be useful and

ake advantage of the collaborative world especially when com-

ined with hybrid methods [6] . Collaborative filtering methods are

urther divided into three categories: memory-based, model-based,

nd hybrid of both. An example of an algorithm which is a hybrid

etween memory-based and model-based methods is personality

iagnosis [36] .

Memory-based methods utilize users’ past behavior and rec-

mmend products that other users with similar interests have

elected in the past [41] . They have been widely used in com-

ercial recommender systems [38] . Memory-based algorithms are

ither user-based [7] or item-based [26] . User-based algorithms

redict rating given by a user to an item based on the ratings

y similar users, whereas, item-based algorithms estimate the

ating based on the ratings of similar items previously chosen

y the user. Methods used in traditional recommender systems

re mostly based on user-item rating matrix. However, these

lgorithms usually fail to find similar users since density of ratings

n user-item rating matrix is often less than 1 percent [26] .

Model-based methods utilize available data to train a prede-

ned model for rating prediction. Some of the commonly used:

lustering [22] and Matrix Factorization model [29] . Model-based

pproaches can handle problems with limited data using hierarchi-

al clustering to enhance the accuracy of the prediction [22] . Ma-

rix factorization factorizes the user-item rating matrix using low-

ank representation. Although model-based methods mitigate the

parsity problem, handling users who have never rated any item

s a challenging problem in both memory-based and model-based

pproaches.

.2. Factors governing rate prediction

Let us now discuss some related research on trust, similarity,

reference, and social influence which we argue are the most im-

ortant factors that govern the design of efficient recommender

ystems.

Trust : Since in online environments users do not have enough

nformation about other users or items being offered, online in-

eractions involve taking some risks as doing business with peo-

le we never met before requires a great deal of trust [19] . Trust

elps users to assign a value to other users based on their will-

ngness to interact with them [5] . Trust between users can be of

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wo types: implicit [35] and explicit [31,37] . Implicit trust is usu-

lly obtained from user-item interactions (i.e., ratings), and explicit

rust is extracted from the user relationships (who they trust and

pto what extent). Collaborative filtering methods are most effec-

ive when users have expressed enough ratings. Since these meth-

ds need users to have mutually rated items, they perform poorly

ith respect to cold start users. Also, similarity metrics would not

e helpful with cold start users. However the trust-based recom-

enders can make better recommendations since users can benefit

rom their trust relationships as well. Some methods use random

alks, so to use enough ratings without suffering from noisy data

ue to being far from source, TrustWalker proposed in [17] a ran-

om walk model which combines trust-based and item-based rec-

mmendations. There are some algorithms such as Eigentrust [20] ,

ppleseed [49] and another algorithm in [40] which use principal

igenvector to make trust computations. However, these methods

roduce ranks of trustworthiness of users, so they would be suit-

ble for systems where ranks are considered. The TidalTrust model

nds all raters with the shortest path from the source user and

ggregates their ratings weighted by the trust between them. An-

ther method is MoleTrust [3] where computation of trust value

etween two users is based on backward exploration. Also, trust

alues in recommender systems help to predict the behavior of

hose users who have rated fewer products [28] . Another trust

etric in [4] has been proposed in order to discover which users

re trusted by members of an online network. Each user is as-

igned a capacity, where trust values will need to be normalized

ithin that capacity, and for computing the trust value knowledge

f the whole structure of the network is required. Moreover, it only

roduces the nodes to trust not the value of the trust. Since there

s no distinction between trusted users, and number of users to

rust is independent of users and items, this method is not appro-

riate for trust-based recommendation systems. Other work such

s [32] uses similarity measures, however it is only designed to be

sed in systems with binary trust ratings.

Similarity : Users with similar preferences or behavior tend to

e interested in the same products, even though they may not

now each other [13] . The preference similarity of two customers

an be estimated according to their product purchases or rating

ecords. The similarity measures (i.e., VSS and PCC) have been in-

orporated in social recommender systems [7,28] . Trust relations

re typically bidirectional and equal in both directions. However,

his is not true in real world relationships where trust relationships

re non-transitive [29] . Also in order to provide meaningful recom-

endation, trust must reflect user similarity to some extent; rec-

mmendations only make sense when obtained from like-minded

eople exhibiting similar taste [1,18] .

Preference : To provide personalized recommendation, there are

wo ways to capture users’ preferences [16] : implicit and explicit.

n implicit feedback [9] , system infers userâs preferences by moni-

oring different actions of users such as purchasing history, brows-

ng history, clicks, email contents, etc., so this type of feedback

educes the burden from user. In explicit feedback [42] , recom-

ender systems prompt users to provide ratings for items in or-

er to reconstruct and improve its model. The drawback with this

ethod is that it requires some effort s from users. However, it

eems that explicit feedback still provide more reliable data, since

t does not involve extracting preferences from actions [2,8] . How-

ver, an implicit feedback system lacks these characteristics, at it

bserves the userâs actions and makes inferences about the userâs

nterests based on these actions. Matrix factorization models can

se both implicit and explicit feedbacks from the system [23] .

n [13] , a framework has been developed to recommend similar

sers and resources based on social network analysis. The work

n [47] use social network to develop a recommender system for

eer-to-peer knowledge sharing.

Social influence : Users with closer social ties to others are much

orth to be believed and are more powerful in influencing oth-

rs [24] . In [27] , user’s opinion is modeled based on her own and

er friends’ opinions which reflect real life social interactions. Also,

ocial influence might create shopping intention for people to con-

ume a product [21] and is thus one of the important factors for

redicting the potential purchasing intention of a customer [24] .

. Proposed social trust model

We model a social recommender system as a social network

epresented as a weighted directed graph with M users. In this so-

ial network, edges represent the social trust relationship between

sers. The users rate their items of interests on a scale of 1 to

. The social relationships (connections) between users are built

nto the adjacency matrix A M × M

. The rating assigned by each user

o each item is represented by the user-item rating matrix R M × N ,

here N represents number of items (products). Our objective is to

redict the rating that user i would assign to product j , when the

ocial relationship graph and the user-item rating matrix are given.

.1. Similarity-based trust

A critical part of collaborative filtering is to compute similari-

ies among users by building a user-item rating matrix. However,

ollaborative filtering methods suffer from various issues such as

ata sparsity and cold start users. To address this issue, some stud-

es have incorporated user similarity in trust models. In [25] , user

imilarity and weighted trust propagation are used to reconstruct

rust matrix which helps with the cold start problem. In [14] , the

uthors proposed an algorithm for trust score which combines

he number of items with the similarity score between users, and

uild a trust relationship matrix. Another study [48] , proposed a

rust model which is based on using propagated trust and simi-

arity of users rating habits. A novel algorithm based on the trust

odel combined with the user similarity factor has been proposed

n [46] . Our method assumed that the trust between users is im-

acted by similarity between two users and importance of each

ser. Similarity between users is one of the most important fac-

ors that affect the value of trust between users since two users

ith the same taste are more likely to trust each other. Here we

pply both rating-based and connection-based methods to capture

he similarity between two users.

.1.1. Rating similarity

We apply similarity algorithms to identify the similarity be-

ween users. The VSS algorithm utilizes the common items that

ave been rated by both users i and f to compute similarity which

s given by:

im (i, f ) =

∑

j∈ I (i ) ∩ I ( f )

R i, j · R f, j √ ∑

j∈ I (i ) ∩ I ( f )

R

2 i, j

·√ ∑

j∈ I (i ) ∩ I ( f )

R

2 f, j

(1)

here j is an item that both users i and f have rated and R i, j is the

ating that user i assigned to item j. I ( i ) represents the set of items

ated by user i . VSS is defined in [0, 1]; a larger value implies more

imilarity between user i and user f .

The trust values enforced by similarity can be modeled by

eighted average rating of the users using the similarity scores as

he weights. Consequently, a connection with high similarity will

ave more impact on the user’s rating. When calculating the VSS

alue, the difference in user’s rating style is not considered (e.g., al-

ays high rating or always low rating). The PCC method can obtain

etter performance than the VSS approach, since the PCC method

4 A. Davoudi, M. Chatterjee / Online Social Networks and Media 7 (2018) 1–11

C

C

10-4

10-2

100

10-2

10-1

100

Normalized Centrality

P(c>

C)

DegreeEigenKatzPagerank

Fig. 1. Distribution of centralities.

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considers the differences of user ratings. So we apply the PCC al-

gorithm to identify the similarity between users. The similarity be-

tween users that have been rated by both users i and f to compute

similarity which is given by:

Sim (i, f ) =

∑

j∈ I (i ) ∩ I ( f )

( R i, j − R i ) · (R f, j − R f ) √ ∑

j∈ I (i ) ∩ I ( f )

( R i, j − R i ) 2 ·√ ∑

j∈ I (i ) ∩ I ( f )

( R f, j − R f ) 2

(2)

where R i is the average rating of user i . We use the mapping func-

tion, f (x ) = (x + 1) / 2 , to map PCC values to [0,1]. It is important

to note that the value of similarity could be negative and its mag-

nitude signifies the dissimilarity degree.

3.1.2. Connection similarity

There are some drawbacks with using the rating-based simi-

larity methods. These methods (VSS and PCC) are rating-based, so

they would not be applicable if two users have not mutually rated

the same product. Also these similarity measures are restricted to

symmetric ones such that the similarity between users u and v

are the same for v and u , although the symmetry may not hold in

many real world applications specifically in a social network mod-

eled by a directed graph.

The similarity between two users can be measured by the con-

nections they have in common. This can be done using each user’s

list of connections. A larger value is an indication of the users

having more similarity which shows that their connection is more

valid in shaping the trust [12] . The list of friends for each user i is

defined as F ( i ). The proportion of mutual friends to the total num-

ber of friends is defined as follows:

Sim (i, f ) =

F (i ) ∩ F ( f )

F ( i ) (3)

3.2. Centrality-based trust

A user with high importance (i.e., high impact) is more likely

to be followed by her friends regardless of their similarities. This

aspect of trust relationship is modeled by considering the impor-

tance of users which can be quantified using centrality measures.

To obtain the importance of users, we use degree centrality, eigen-

vector centrality, Katz centrality and PageRank [34] . We choose

these centrality measures since they consider the connections and

also the importance of each connection by adding the free initial

centrality to deal with special cases.

Degree centrality is used as the basic indication of a user’s im-

portance which can be defined as the number of connections . In our

case, it is the number of incoming edges (in-degree) in the social

graph. We define the degree centrality C l of a user l as:

l =

∑

∀ m,l � = m

A l,m

(4)

where A l, m

is the element of the adjacency matrix which repre-

sents the connection between user l and user m . Thus, with all

connections treated equally, a user with more incoming edges has

higher importance in the network.

Eigen-vector centrality gives each node a value which is propor-

tional to the sum of values of its neighbors. Eigen-vector centrality

has a property: it can be large either because a node has many

neighbors or because it has important neighbors (or both). Eigen-

vector centrality of user l at time t is the defined as sum of the

centrality of all connections of user l which is given as:

l (t) =

∑

∀ m

A l,m

(t) × C l (t − 1) (5)

here C l (t − 1) is the centrality of user l at time t − 1 . In contrast

o the degree centrality, the eigen-vector centrality considers both

he number of incoming edges and also the centrality of the neigh-

oring users. The eigen-vector centrality is computed iteratively by

etting all initial values to 1 i.e., C l (0) = 1 for all user l .

Katz centrality is similar to eigen-vector centrality except that

t adds a free centrality value to each node. In this centrality, we

onsider a value which is called free centrality. We add the free

entrality to account for users that do not have any outgoing edges.

he Katz centrality of user l at time t is defined as:

l (t) = α ×∑

∀ m

A l,m

(t) × C l (t − 1) + ε (6)

here ε is the free centrality value. By adding this second term,

ven nodes with zero in-degree still get centrality ε, and once

hey have a non-zero centrality, then the nodes they point to de-

ive some advantage from being pointed to. This means that any

ode that is pointed to by many others will have a high centrality,

lthough those that are pointed to by others with high centrality

hemselves will still do better.

PageRank centrality A problem with with Katz centrality is that

f a node with high Katz centrality points to many others then

hose others also get high centrality. The centrality gained by

irtue of receiving an edge from a prestigious node is diluted by

eing shared with so many others. The PageRank centrality fixes

his by defining a variation of the Katz centrality in which the cen-

rality a node derives from others is proportional to their centrality

ivided by their out-degrees ( k out � = 0). Nodes that point to many

thers pass only a small amount of centrality to each of those oth-

rs, even if their own centrality is high. In mathematical terms, we

efine this centrality by:

l (t) = α ×∑

∀ m

A l,m

(t) × C l (t − 1)

k out (t − 1) + ε (7)

.3. Linearsocial trust ensemble

To model the social trust between users in a social recom-

ender system, we use a linear combination of similarity and cen-

rality to represent the trust of user i on user k as [11] :

i,k = βSim (i, k ) ∑

l∈T (i )

Sim (i, l) + (1 − β)

C k ∑

l∈T (i )

C l (8)

ere, β is the parameter that defines the contribution of similarity

nd centrality to the overall trust. β = 0 implies purely centrality

A. Davoudi, M. Chatterjee / Online Social Networks and Media 7 (2018) 1–11 5

10-2

10-1

100

10-2

10-1

100

Normalized Similarity

P(s>

S)

PCCVSSConnection

Fig. 2. Distribution of similarity.

10-4

10-2

100

10-2

100

Normalized Trust

P(t>T

)

DegreeEigenKatzPagerank

Fig. 3. Distribution of trust values for PCC similarity.

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10-4

10-2

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10-2

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Normalized Trust

P(t>T

)

DegreeEigenKatzPageRank

Fig. 4. Distribution of trust values for VSS similarity.

Fig. 5. Distribution of trust values for connection similarity.

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nforced trust while β = 1 refers pure similarity-based trust val-

es. T (i ) refers to the set of trusted friends of user i. C k refers to

he centrality (i.e., measured using either degree or eigen-vector

entrality) of user k .

. Social trust model using matrix factorization

Matrix factorization has been widely used to develop social rec-

mmender systems as it helps to estimate either the user-item rat-

ng or user-trust matrix [28] using low-dimensional representative

atent matrices. Here, matrix factorization for social recommenda-

ion proposed in [29] is employed to examine the performance of

he proposed trust relationship.

The user-item rating matrix is factorized to learn two

−dimensional feature representation of users U and items V ma-

rices. The user-item rating matrix R consists of M users and N

tems with rating values in the range [0, 1]. U i and V j repre-

ent the l−dimensional user-specific and item-specific latent fea-

ure vectors of user i and item j . A low-rank matrix factorization

pproach seeks to approximate the matrix R by multiplication of

−dimensional factor R ≈ U

T V , where U ∈ R l × M and V ∈ R l × N with

≤ min ( M, N ). In real datasets, matrix R is usually very sparse.

The conditional distribution for R , given �, U, V and σ 2 R

is de-

ned as [29] :

p(R | �, U, V, σ 2 R ) =

M ∏

i =1

N ∏

j=1

[ N (R i j | g( ∑

k ∈T (i )

�ik U

T k V j ) , σ

2 � )] I

R i j (9)

here N (R i, j | μ, σ 2 �) is probability density function of the Gaus-

ian distribution with mean μ and variance σ 2 �

. Here, � is the pro-

osed trust parameter given by Eq. (8) , �i, k is the trust value be-

ween users i and k. R i, j is the rating given to item j by user i , and2 R

is the rating variance. I R i j

is an indicator function representing

hether user i rated item j . Based on the Bayesian inference and

ssuming � is independent of U and V , the conditional probability

f U and V , given R , �, σ 2 R

, σ 2 U

, and σ 2 V

, is defined as:

p(U, V | R, �, σ 2 �, σ 2

U , σ2

V )

=

M ∏

i =1

N ∏

j=1

[

N

(

R i, j | g (

αU

T i V j + (1 − α)

∑

k ∈T (i )

�i,k U

T k V j

)

, σ 2 �

) ] I R i, j

×M ∏

i =1

N (U i | 0 , σ 2 U I ) ×

M ∏

i =1

N (V j | 0 , σ 2 V I ) (10)

here σ 2 U and σ 2

V are the variances of user and item feature ma-

rices. I is the identity matrix. The function g(x ) = 1 / (1 + exp(−x ))

6 A. Davoudi, M. Chatterjee / Online Social Networks and Media 7 (2018) 1–11

e

[

5

p

m

5

b

c

c

r

l

u

o

q

a

f

i

m

n

u

d

i

a

r

e

f

u

5

M

s

5

v

t

i

M

u

v

R

5

C

is a mapping function whose range is within [0, 1]. The set T (i )

contains user i ’s trusted friends.

The proposed social recommender system is based on the idea

that user’s ratings are impacted by her own taste and her imme-

diate friends’ tastes. The parameter α is used to balance between

these two factors. The term U

T i

V j represents the estimated taste of

user i of item j , while ∑

k ∈T (i ) �i,k U

T k

V j term reflects her immediate

friends’ taste, given as the weighted average of their taste using

the trust value as weights.

4.1. User-specific and item-specific matrices

In this section, we seek to find the U and V matrices. The log of

posterior distribution for the recommendation is given by:

ln p(U, V | R, �, σ 2 �, σ 2

U , σ2

V )

= − 1

2 σ 2 �

M ∑

i =1

N ∑

j=1

I R i, j

(

R i, j − g(αU

T i V j + (1 − α)

∑

k ∈T (i )

�i,k U

T k V j )

) 2

− 1

2 σ 2 U

M ∑

i =1

U

T i U i −

1

2 σ 2 V

N ∑

j=1

V

T j V j −

1

2

(

M ∑

i =1

N ∑

j=1

I R i, j

)

ln �2

−1

2

(Ml ln σ 2 U + Nl ln σ 2

V ) + C (11)

Here C is a constant independent of other parameters. Maxi-

mizing the log-posterior over the two latent features is equivalent

to minimizing the sum-of-squared-errors objective functions with

quadratic regularization terms to derive U and V :

L (R, �, U, V )

=

1

2

M ∑

i =1

N ∑

j=1

I R i, j

(

R i, j − g

(

αU

T i V j + (1 − α)

∑

k ∈T (i )

�i,k U

T k V j

) ) 2

+

λU

2

|| U|| 2 F +

λV

2

|| V || 2 F (12)

Here λU =

σ 2

σ 2 U

, λV =

σ 2

σ 2 V

and || . || 2 F is the Frobenius norm. λU and

λV are user and item latent variance ratios.

The gradient decent approach can be used to solve the mini-

mization problem given in Eq. (11) for finding U and V . Gradient

decent is a local optimization method based on the partial deriva-

tive of the objective function with respect to the decision variables

(i.e., U and V ). The partial derivatives of L with respect to U and V

are given in Eqs. (13) and (14) .

∂L

∂U i

= αN ∑

j=1

I R i, j g ′ (αU

T i V j + (1 − α)

∑

k ∈T (i )

�i,k U

T k V j ) V j

×(

g

(

αU

T i V j + (1 − α)

∑

k ∈T (i )

�i,k U

T k V j − R i, j

)

+(1 − α) ∑

p∈ φ(i )

N ∑

j=1

I R p, j g ′ (

αU

T p V j + (1 − α)

∑

k ∈T (p)

�p,k U

T k V j

)

×(

g

(

αU

T p V j + (1 − α)

∑

k ∈T (p)

�p,k U

T k V j

)

− R p, j

)

�p,i V j + λU U i

(13)

∂L

∂V j

=

M ∑

i =1

I R i, j g ′ (

αU

T i V j + (1 − α)

∑

k ∈T (i )

�i,k U

T k V j

)

×(

g

(

αU

T i V j + (1 − α)

∑

k ∈T (i )

�i,k U

T k V j − R i, j

)

×(

αU i + (1 − α) ∑

k ∈T (i )

�i,k U

T k

)

+ λV V j (14)

Here g ′ ( x ) is the derivative of logistic function where g ′ (x ) =xp (x ) / (1 + exp(x )) 2 . φ( i ) is the set of the users who trust user i

27] .

. Accuracy measures

In order to test the validity and accuracy of the proposed rate

rediction framework, we conduct extensive simulation experi-

ents with data from Epinions [44] .

.1. Data source

We base our experimental analysis on a dataset based on trust-

ased product review website Epinions.com which is a product

omparison website that features products reviews with a social

omponent. It allows users to post reviews about products with a

ating from 1 to 5. It also allows users to create directional social

inks that can be defined as trust and distrust links towards other

sers. Since the distrust links are not publicly available, we study

nly the trust links. Also users can provide feedback about the

uality of product reviews written by other users. Each review has

helpfulness score summarized as very helpful, somewhat help-

ul, helpful, not helpful, or no feedback. The Epinions website takes

nto account the trust links in order to make personalized recom-

endations.

The social connections in this dataset are binary values and do

ot represent the actual trust values. The dataset includes 22166

sers and 355754 social connections, leading to 0.0724 percent

ensity in the user social relationship matrix. The total number of

tems is 296277, with a total of 922267 ratings, which results in

very sparse item-rating matrix with 0.0140 percent density. As a

esult, the user-item rating matrix is also relatively sparse. On av-

rage, users have 16.05 trusted friends. The maximum number of

riends for a user is 1551 and the most trusted user has 2023 other

sers trusting her.

.2. Evaluation metrics

In this section we use predictive accuracy measures (such as

AE, RMSE) and Classification accuracy measures (such as Preci-

ion, Recall, F1) for performance evaluation purposes.

.3. Predictive accuracy metrics

Different types of error metrics are defined as follows.

Mean Absolute Error (MAE) : This metric measures the average

ariation in the predicted rating vs. the actual rating. Let R pre i, j

be

he predicted rating and R act i, j

be the actual rating given by the user

to the product j . The MAE is defined as follows:

AE =

∑

i, j | R

pre i, j

− R

act i, j

| M

(15)

Root Mean Squared Error (RMSE) : This metric is the most pop-

lar metric used in evaluating accuracy of predicted rating. It is a

ariant of mean square error and is defined as follows:

MSE =

√ ∑

i, j | R

pre i, j

− R

act i, j

| 2 M

(16)

.4. Classification accuracy metrics

Other popular parameters used for performance evaluation are:

overage, Precision, and F-measure. Let us define these metrics.

A. Davoudi, M. Chatterjee / Online Social Networks and Media 7 (2018) 1–11 7

i

i

o

P

S

R

F

6

t

l

c

i

T

t

t

6

t

a

c

t

u

a

w

t

s

β

i

7

a

t

f

n

s

t

o

a

l

6

u

F

p

p

e

m

s

t

i

m

Fig. 6. MAE using binary trust and the proposed trust model for PCC similarity.

Fig. 7. MAE using binary trust and the proposed trust model for VSS similarity.

b

s

t

p

b

p

e

t

m

s

p

a

m

6

t

t

c

h

t

l

T

Coverage : Coverage is defined as the percentage of < user,

tem > pairs which the method has been able to predict the rat-

ng for.

Precision : In this context, precision refers to a normalized form

f RMSA and is defined as:

recision = 1 − RMSE

RMSE max (17)

ince the ratings in our dataset are in range of [1,5], we use

MSE max = 4 as the maximum possible value for the error.

F-measure : F-measure is defined as:

− measure =

2 × P recision × Cov erage

P recision + Cov erage (18)

. Results and discussions

We present how our trust models perform with the data ob-

ained from Epinions. Based on the proposed model, the trust re-

ationships between users are built on the two components of

entrality and similarity measures. We demonstrate the probabil-

ty density function of centrality, normalized similarity, and trust.

hese distributions reveal what and how much impact each of

hese parameters have for various values of the parameter in ques-

ion.

.1. Distribution analysis

Fig. 1 shows the distribution of different centrality measures

hat have been analyzed in our model: degree, eigen-vector, Katz

nd PageRank centrality.

In Fig. 2 , the distribution of rating-based (i.e., VSS and PCC) and

onnection-based similarity are shown. VSS and PCC have a rela-

ively sparse distribution due to the lack of mutually rated prod-

cts by two friends in many cases. The trust values are calculated

s the weighted summation of centrality of similarity using the

eight constant β .

Figs. 3–5 show the distribution of trust values for different

ypes of similarity being applied; PCC, VSS, and connection-based

imilarity. These figures show the distribution of trust values using

= 0 . 5 . The proposed trust model is used to predict users’ rat-

ng based on the discussed matrix factorization technique using

5 percent of the data as the training set. According to Eq. (10) ,

user’s opinion about a particular product would be a linear func-

ion of her connections’ taste and her own taste using a weighting

actor α. Smaller values of α is an indication of less impact from

eighbors. As previously defined in Eq. (8) , the trust model is pre-

ented as the linear combination of centrality and similarity using

he weighting factor β . Higher values of β indicate higher impact

f similarity rather than centrality on the trust values. Here, user

nd item latent variance ratio ( λU and λV ) are set to 0.001. The

atent size is L = 4 , α = 0 . 4 , and the number of iterations is 300.

.2. Performance analysis

The performance of the proposed trust model for different val-

es of β in terms of MAE is shown in Fig. 6 for PCC similarity,

ig. 7 for VSS similarity, and Fig. 8 for connection-based similarity.

The same is shown for RMSE for the performance of the pro-

osed trust model for different values of β Figs. 9 –11 .

Compared to the binary trust model (dashed black lines), the

roposed trust model has better performance. Comparing differ-

nt definitions of trust reveals that degree centrality is the better

easure to model trust compared to using other centrality mea-

ures. The same is true for connection-based similarity compared

o rating-based. An interesting observation is that, although includ-

ng centrality in trust model enhances the recommendation perfor-

ance compared to the binary trust model, the trust models solely

ased on similarity (i.e., β = 1 ) show the best performance for the

tudied network.

The probability distribution of rating estimation error (i.e., es-

imated rating minus actual rating) for the binary trust and pro-

osed trust model is shown in Fig. 14 . Both probability distri-

utions are right skewed, implying over-estimation. However, the

roposed trust model seems to have relatively better performance

specially for errors between 0.5 and 2, since it estimates more be-

ween 0.5 and 1 and less between 1 and 2 compared to the binary

odel. The probability distribution of absolute error ratio (i.e., ab-

olute error divided by the actual rating) is shown in Fig. 15 . The

roposed trust model leads to lower error ratio between 1 and 2

nd more between 0 and 1 which implies relatively better perfor-

ance.

.3. Error analysis

The performance of the trust model (the definition which had

he best performance in Figs. 3 –5 ) for different latent sizes and

raining percentages are shown in Figs. 12 and 13 . Generally, in-

reasing the latent size as well as using more training data en-

ance the performance of the recommender system.

In Figs. 16 and 17 show the estimated versus actual ratings for

he proposed and the binary trust models. The boxes illustrate the

ower, upper, and inter quartiles, while the red line is the medium.

he height of the boxes represents the variation of the estimated

8 A. Davoudi, M. Chatterjee / Online Social Networks and Media 7 (2018) 1–11

Fig. 8. MAE using binary trust and the proposed trust model for connection simi-

larity.

Fig. 9. RMSE using binary trust and the proposed trust model for PCC similarity.

Fig. 10. RMSE using binary trust and the proposed trust model for VSS similarity.

Fig. 11. RMSE using binary trust and the proposed trust model for connection sim-

ilarity.

Fig. 12. Errors for different latent sizes using degree centrality and connection-

based similarity.

rating. Comparing Figs. 16 and 17 , it is observed that the proposed

trust model produces better estimations for low ratings (1 and 2)

by slightly undermining the estimation. In addition, for high rat-

ings, the proposed trust model reduces the variation of estima-

tions, i.e., the height of the quartile boxes.

6.4. Comparing with other methods

We compare our method with other trust models such as

TrustWalker [17] , SoRec (Matrix Factorization) [30] , Item-based

(model-based method) [41] , Similarity-based [11] , and Centrality-

based [11] . Trustwalker is a trust-based recommender system

which follows multiple random walks through the network start-

ing from a specific user and makes a rating prediction based on

the similar items observed throughout the walks. In SoRec, the

trust values are binary and the social recommendation is done us-

ing probabilistic matrix factorization. Item-based method is imple-

mented using Pearson Correlation as the item similarity. We also

tried two derivatives of our proposed model by using pure similar-

ity which ignores centrality parameter (similarity-based method)

to build the trust factor and the centrality-based method which

uses pure centrality when building the trust factor and ignores the

similarity between users.

Table 1 shows the values for the various measures for different

methods. 75% of rating information was used to estimate the re-

A. Davoudi, M. Chatterjee / Online Social Networks and Media 7 (2018) 1–11 9

Fig. 13. Errors for various training set sizes using degree centrality and connection-

based similarity.

Fig. 14. The probability distribution of error for rating estimation using binary trust

and the proposed trust model.

Fig. 15. Absolute error ratio for rating estimation using binary trust and the pro-

posed trust model.

Fig. 16. The quartile plot of actual versus estimated rating for the proposed trust

model.

Fig. 17. The quartile plot of actual versus estimated rating for the binary model.

Table 1

Comparison with other methods.

Methods RMSE Coverage (%) Precision F-measure

Our method 1.144 100 0.714 0.833139

TrustWalker 1.16 82 0.71 0.761045

SoRec 1.229 100 0.69275 0.818491

Item-based 1.23 77 0.6925 0.729197

Similarity-based 1.209 100 0.69775 0.82197

Centrality-based 1.217 100 0.69575 0.820581

m

m

a

s

7

t

i

u

a

aining 25% of the user’s ratings for performance evaluation. Our

ethod outperforms all other methods in terms of RMSE, Cover-

ge, Precision and F-measure. For our method we use β = 0 . 4 , for

imilarity-Based β = 1 , and for centrality-Based, β = 0 .

. Conclusions

With emerging applications of social networks and considering

he role of social interactions in our daily life decisions, extract-

ng information from user’s social relationships is becoming a pop-

lar method for predicting user’s behavior. To consider and bal-

nce these factors, this paper proposes a social trust model that

10 A. Davoudi, M. Chatterjee / Online Social Networks and Media 7 (2018) 1–11

[

incorporates the preference similarity, user’s centrality, and social

relation in order to predict the rating for the social recommender

system. We capture the trust relationships between users consider-

ing users with similar profile and their importance. We argue that

users with more similarity would trust each other more; also users

with higher importance would be trusted more. Similarity is quan-

tified by using rating-based approaches and a connection-based

centralities. The importance of users is modeled by degree, eigen-

vector centrality, Katz and PageRank centralities. We define trust

as a linear combination of similarity and centrality using a weight-

ing parameter. The proposed framework is validated using real data

from Epinions. Our result indicates that the proposed trust model

produces better rating estimation in terms of the mean absolute

error (MAE), the root mean squared error (RMSE) and error dis-

tribution, compared to the traditional binary trust model which

is widely used in recommender systems. Trust enforced by degree

centrality shows better performance compared to other centrality

methods. The same conclusion is valid for connection-based sim-

ilarity compared to rating-based. The trust relationships are also

observed to be more dependent on the similarity rather than cen-

trality. The proposed framework can thus be effectively applied to

electronic retailers in promoting their products and services.

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A. Davoudi, M. Chatterjee / Online Social Networks and Media 7 (2018) 1–11 11

I

P

H

s

c

s

Anahita Davoudi received the B.S. degrees in Computer

Engineering from the Amirkabir University of Technol-ogy (Tehran Polytechnic), Tehran, Iran, in 2008, and mas-

ter in Electrical Engineering from University of Texas Ar-

lington in 2012. She received the Ph.D. degree in Com-puter Science from the University of Central Florida (UCF),

Orlando, FL, USA. Her research interests include SocialRecommender Systems, Social Data Science and Compu-

tational Social Science.

Mainak Chatterjee is an Associate Professor in the de-

partment of Electrical Engineering and Computer Science,University of Central Florida, Orlando. He received the BSc

degree in physics (Hons.) from the University of Calcutta,

the ME degree in electrical communication engineeringfrom the Indian Institute of Science, Bangalore, and the

Ph.D. degree from the Department of Computer Scienceand Engineering from the University of Texas at Arlington.

His research interests include economic issues in wirelessnetworks, applied game theory, cognitive radio networks,

dynamic spectrum access, and mobile video delivery. He

has published over 200 conferences and journal papers.He got the Best Paper Awards in IEEE Globecom 2008 and

EEE PIMRC 2011. He is the recipient of the AFOSR sponsored Young Investigatorrogram (YIP) Award. He co-founded the ACM Workshop on Mobile Video (MoVid).

e serves on the editorial board of Elsevier Computer Communications and Perva-ive and Mobile Computing Journals. He has served as the TPC Co-Chair of a dozen

onferences. He also serves on the executive and technical program committee of

everal international conferences.