Journal of Computer and Communications, 2013, 1, 1-8
http://dx.doi.org/10.4236/jcc.2013.15001 Published Online October 2013 (http://www.scirp.org/journal/jcc)
1
Improving Personal Product Recommendatio n via
Friendships’ Expansion
Chunxia Yin, Tao Chu
The First Aircraft Institute, Aviation Industry Corporation of China, Xi’an, China.
Received July 2013
ABSTRACT
The trust as a social relationship captures similarity of tastes or interests in perspective. However, the existent trust in-
formation is usually very sparse, which may suppress the accuracy of our personal product recommendation algorithm
via a listening and trust preference network. Based on this thinking, we experiment the typical trust inference methods
to find out the most excellent friend-recommending index which is used to expand the current trust network. Experi-
mental results demonstrate the expanded friendships via superposed random walk can indeed improve the accuracy of
our perso na l product re c ommendation.
Keywords: Personal Product Recommendation; Trust Inference; Listening and Trust Preference Network
1. Introduction
Nowadays some web sites, su ch as last.fm and delicious,
have taken the lead in guiding users to build trust rela-
tionships, that is, a user on such a website can select
some other users with similar tastes as his/her trusted
friends. The effectiveness of a recommendation algo-
rithm rests with the effective use of personal information
from net users. Personal trust information obviously
contains reliable personal friend relations and therefore
has the potential to help provide better recommendations.
Based on this thinking, we proposed the LTPN algorithm
[1], namely the artist recommendation algorithm via the
listening and trust preference network. However, the
provided user-trusted friend relations by web users are
only a small fraction of the potential pairings between
users so that the existent trust data is very sparse.
The trust as a social relationship captures similarity of
tastes or interests in perspective. Obviously, it may be
difficult to know tastes or interests of a person from his
sparse friendships. Instinctively, if the friendships can be
accurately inferred, the sparse problem may be relieved
so that the expanded friendships are expected to improve
the accuracy of our LTPN algorithm. That is, th e sparsity
of trust data may severely influence the behavior of our
LTPN algorithm. In a word, if trust is used to support
decision making, it is important to have an accurate es-
timate of trust when it is not directly av ailable, as w ell as
a measure of confidence in that estimate [2].
Many biological, technological, social systems can be
well described by complex networks with nodes repre-
senting individuals or items and links denoting the rela-
tions or interactions between nodes [3-6]. As part of the
surge of research on complex networks, a considerable
amount of attention has been devoted to the computa-
tional analysis of social networks [7]. An important
scientific issue relevant to social network analysis is per -
sonal trust inference, which aims at estimating the like-
lihood of existence of trust with respect to a particular
topic between two individuals. A number of algorithms
for computing trust from social networks rely on the
knowledge of the network topology, which are the main-
streaming class of trust inference algorithms. They pro-
vide us an entry of inferring friendships for online web
users.
In this paper, we investigate the use of trust inference
for improving personal product recommendation. To this
end, we review and discuss typical trust inference me-
thods to find out the most effective friendship-expanding
method, and then explore whether the expanded user-
trusted friend relations can improve the accuracy of the
LTPN algorithm. The numerical results indicate that the
inferred friendships via superposed random walk can
improve our existent trust-aware product recommenda-
tion.
2. Method
In this paper, we still discuss the artist as the special
product. We assume that a music artist recommendation
system consists of a user-set U = {u1, u2, …, un} and an
artist-set A = {a1, a2, …, am}. Each user has listened to

Improving Personal Product Recommendation via Friendships’ Expansion
2
some artists and selected some other users with similar
musical tastes as his/her trusted friends . The user -listened
artist and user-trusted friend relations can uniformly be
described by a listening and trust relation network
(LTRN for short), G (U, A, E1, E2), where E1 is the set
of user-trusted friend relations and E2 is the set of user-
listened artist relations. Note that multiple links and
self-connections are not allowed in the network. Clearly,
G (U, A, E1, E2) can seen as a combination of the trust
network G (U, E1) and bipartite network G (U, A, E2).
Figure 1 gives a simple example of the listening and
trust relation network. Each black solid line represents a
listening count for a [user, artist] pair. For example, u1
has listened to a1 80 times. The red dashed curves show
the mutual (undirected) trust relation s between use rs. The
listening and trust relation network (a) can be seen as a
combination of the trust network (b) and the bipartite
network (c).
To improve our LTPN algorithm, we need solve two
problems. First, the personal trust between two individu-
als without a direct connection needs to be inferred. We
will solve this problem by applying an excellent personal
trust in ference metho d to the existen t tru st network. Af ter
these inferred trust relations with maximal evaluated
values are added to LTRN, the expanded listening and
trust relation network (ELTRN for short) will be built.
Second, ELTRN is transformed into the listening and
trust preference network. Obviously, ELTRN is a special
LTRN. For convenience, the process of transforming the
listening and trust relation network into the listening and
trust preference network is called as the LTRN-LTPN
transformation. We use the LTRN-LTPN transformation
method in Ref. [1] to solve the second problem.
Based on the above analysis, we put forward the im-
proved LTPN algorithm via friendships’ expansion
(FELTPN for short).
FELTPN is described as the following four steps:
(1) Expand the user-trusted friend relations by making
use of an excellent trust inference method.
(2) Build the listening and trust preference network by
applying the LTRN-LTPN transformation to ELTRN.
(3) Apply the LTPN algorithm to the listening and
trust preference network for the personal artist recom-
mendation.
Next, our LTRN-LTPN transformation and LTPN al-
gorithm are shortly introduced. Readers are encouraged
to see Ref. [1] for their details.
The u ser-listened artist relations can be expressed by a
listening count matrix, Cn×m, where Cif is the listening
count of ui to af. The user-trusted friend relations can be
expressed by an adjacent matrix Tn×n. If there exists a
mutual trust between ui and uj, Tij = Tji = 1, otherwise Tij
= Tji = 0.
The LTRN-LTPN transformation is as follows:
(1) Transform the user-listened artist relations into the
impartial listening preferences, C'n×m, in which
C'if = Cif／∑gCig. (1)
(2) Take advantage of the cosine similarity between
the listening preferences of users to build trust prefe-
rences, Ťn×n, in which
22
, if1
0 otherwise
ij ij
ij
CC T
CC
′′
⋅
=
′′
∗



(2)
Figure 1. (Color online) A simple listening and trust relation network consisting of four users and eight artists.
Ť
ij
=
artists
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u
3
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users
u
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a
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=+
ab c
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1
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Improving Personal Product Recommendation via Friendships’ Expansion
3
where C'i is the ith row vector of matrix C'n×m.
Now, C'n×m and Ťn×n express together the listening and
trust preference network evolving from the correspond-
ing relation network.
Suppose that ut is a target user, al is an artist to which
he/she has never listened, and {uj} is a set consisting of
all his/her trusted friends having listened to al.
The basic idea of our LTPN algorithm is as follows:
(1) Find all these two-step trust and listening prefe-
rence paths from ut to uj to al, each of which obviously
contains the trust preference and listening one in turn.
Each such path is called path(ut,uj,al).
(2) Acquire the single-path level of preference of ut for
al from path(ut,uj,al), Ptjl, by doing the multiplication on
listening and trust preferences of the path, that is,
Ptjl = ŤtjC' (3)
(3) Obtain the level of listening preference of ut for al,
Ptl, by adding all the above single-path levels of prefe-
rence of ut for al together, that is,
Ptl = ∑jPtjl = ∑jŤtjC'jl (4)
Figure 2 explains the general idea of FELTPN from
the angle of the target user. Now suppose u3 is a target
user.
To avoid interference, plot (a) refines Figure 1(a)
from the angle of u3 and emphasizes these artists not se-
lected by u3 with green bold solid lines. Plot (b), in which
the blue bold dashed curve represents the inferred friend-
ship for u3, is the ELTRN only for u3 after a certain trust
inference algorithm is applied to the trust network in
Figure 1(b). Plot (c), in which the edge labels on the
trust links denote the weights of the trust relations be-
tween u3 and his/her friends and the edge labels on the
Figure 2. (Color online) Illustration of FELTPN.
target
user
artists
friends
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20
a1
a2
a3
a5
a6
a7
a8
230
a
target
user
artists
friends
0.13
0.08
0.45
0.05
0.25
0.66
0.09
0.59
0.41
0.09
0.15
0.11
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0.04
a
1
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8
0.29
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0.32
0.46
ca
3
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8
d
0.29
0.04
0.01
target
user
artists
friends
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a
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b
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target
user

Improving Personal Product Recommendation via Friendships’ Expansion
4
user-listening artist links of u3 represent the levels of
listening preference of u3 for the artists, presents the lis-
tening and trust preference network only for u3 after the
LTRN_LTPN transformation is applied to the ELTRN in
plot (b). Eventually, t he pr edict ed sco res for u3 are shown
in plot (d) after our LTPN algorithm is applied to the
listening and trust preference network in plot (c).
3. Experiment Data and Evaluation Metrics
We use a benchmark data set, lastfm, to evaluate the ef-
fectiveness of the personal trust inference method and
artist recommendation algorithm. The lastfm data, re-
leased in the framework of HetRec 2011
(http://ir.ii.uam.es/hetrec2011), consists of 1892 users
and 17632 artists from Last.fm online music system
(http://www.last.fm). It records user-trusted friend and
user-listened artist relations. For the purpose of effec-
tively testing our algorithm, we remove those users hav-
ing listened to less than ten artists. As a result, th e listen-
ing and trust relation network, originating from a random
sampling of 990 users, contains 49191 user-listened artist
relations and 3878 mutual trust relations. Therefore, the
average number of friend relations per user is 7.8 and the
average number of user-listened artist relations per user
is 49.69.
Figure 3, where Nd is the number of users with d
friends, exhibits the degree distribution of trust network
from the lastfm data. Obviously, it satisfies heavy-tailed
distribution, which means most users own sparse friend-
ships.
The personal trust inference method need calculate the
likelihood of existence of all non-observed tru st relations
of the target user, then sorts them in a descending order
according to their likelihoods and finally recommend the
target user these top-L en friends. To determine how ac-
curately a personal trust inference method is able to mine
Figure 3. The degree di stribution of trust network from the
lastfm data.
friendships, the observed user-trusted friend relations E1
need to be randomly divided into two parts: The training
set Figure 3.
E1Tr contains 90% of user-trusted friend relations from
each user, and the remaining 10% constitutes the probe
set E1Pr. E1Tr is treated as known information, while no
information in E1Pr is used for prediction.
Similarly, the personal artist recommendation algo-
rithm need calculate the listened likelihood of all non-
listened artists of the target user in the future, then sorts
them in a descending order according to their likelihoods
and finally recommend the target user these top-Len art-
ists. To determine the effectiveness of our artist recom-
mendation algorithm, user-listened artist relations E2
also need to be randomly divided into two parts: The
training set E2Tr contains 90% of user-listened artist rela-
tions from each user, and the remaining 10% constitutes
the probe set E2Pr. E2Tr and (observed and expanded)
mutual trust relations are treated as known information,
while no information in E2Pr is used for prediction.
We use three standard metrics, area under the receiver
operating characteristic curve (AUC) [8], Precision [9]
and Recall [9], to quantify the accuracy of a personal
trust inference method. Note that the AUC evaluates the
whole ranking list of all candidate friends while both
Precision and Recall only focus on the top-Len candidate
friends of the list.
We use precision, recall, inter-diversity [10], intra-
diversity [11] and popu larity [10] to evaluate the quality
of an artist recommendation algorithm. The first two are
different accuracy metrics. The inter-diversity (De) mea-
sures the personalization of recommendations for differ -
ent users with diff erent habits and tas tes. The intradiver-
sity (Da) surveys the diversity of topics in a recommen-
dation list. The last one measures the capability of recom-
mendin g da rk (less pop ular) arti s t s .
Before introducing concrete definitions of the above
metrics, we must interpret some symbols to be used in
these definitions.
Denote by Ωi the set containing all n − 1 possible trust
relations of ui. Then the set of all evaluated trust relatio ns
of ui is
Tr
1
ii
EΩ−
where
Tr
1
i
E
denotes the training set
of trust relations of ui. Accordingly,
Pr
1
i
E
is the probe
set of trust relations of ui. Here, trust relations of ui can
be interpreted as his/her trusted friends.
As a matter of convenience, the (friend or artist) rec-
ommendation list for ui is consistently expressed as Li
and the (friend or artists) probe set for ui is consistently
expressed as Pi.
| · | denotes the number of elements in a set or list. The
number of users having listened to af is expressed as df.
Bn×m is an adjacency matrix. If ui has listened to af, Bif =
1, otherwise Bif = 0.
In the present case, the AUC statistic can be inter-
0
20
40
010 2030 40 50
Degree d
Nd
60
60 70
80
100
120
140
160
180

Improving Personal Product Recommendation via Friendships’ Expansion
5
preted as the probability that a randomly chosen trust
relation in E1Pr is given a higher trust score than a ran-
domly chosen nonexistent relation. Given the whole
ranking list of all evaluated trust relations of ui, among w
independent comparisons, if there are w′ times the rela-
tion in
Pr
1
i
E
having a higher score and w′′ times being
of the same score, the AUC value for ui is computed as
follows:
AUCi = (w′ + 0.5w′′)/w. (5)
The AUC for the whole system is the average over all
users, that is,
AUC = ∑AUCi/n. (6)
The degree to which the AUC value exceeds 0.5 indi-
cates how much better predictions are than pure chance.
Precision is defined as the ratio of the number of rele-
vant items to the total number of recommended items.
Given the top-Len items of a ranking list for ui, the preci-
sion for ui obviously equals |Li∩Pi|/Len. Therefore the
precision of the whole system is
1
ii
i
PrecisionL P
n Len
= ∩
×
∑
. (7)
Clearly, a higher value of precision means a better
performance.
Recall is defined as the ratio of the number of relevant
items to the total number of items in the probe set. Given
the top-Len items of a ranking list for ui, the recall for ui
obviously equals |Li∩Pi|/|Pi|. Therefore the recall of the
whole system is
1
ii
ii
LP
Recall nP
∩
=
∑
. (8)
A higher value of recall means a better performance.
A basic De measures the diversity of artists between
two recommendation lists. De between two recommenda-
tion lists can be quantified by the Hamming distance
between the two lists. Therefore th e inter-diversity of the
whole system is
( )
21
1
ij
e
ij
LL
Dn nLen
≠

∩

= −

−
∑
. (9)
Generally speaking, the greater value of De means the
more personalized recommendations for different users.
A basic Da measures the diversity of artists within a
recommendation list. Therefore, the intra-diversity of the
whole system is
( )
12
11
afg
i fg
ii
DS
nLL ≠

= − 

−

∑∑
, (10)
where Sfg denote s the cosine similarity betw een af and ag,
namely,
1
fgif ig
i
fg
S BB
kk
=
∑
. (11)
Generally speaking, the higher value of Da means the
more diversified topics within recommendatio n lists.
The popularity of a recommendation list is defined as
the average degree of artists in the list. Therefore the
popularity of the whole system is
11
fif
i aL
i
Popularity d
nL ∈
=
∑∑
. (12)
The smaller Popularity, corresponding to the stronger
capability of mining dark artists, is preferred since those
lower-degree artists are hard to be found by users them-
selves.
4. The Inference of User-Trusted Friend
Relations
In this section, we seek the most excellent (accurate and
fast) trust inference method among the most influential
and representative sixteen trust inference indices. These
indices can be clas sified into three catego ries: local , quasi-
local and global in accordance with the scope of topo-
logical information used by the inference measure.
Clearly, a local measure only requires the information
about node degree and the nearest neighborhood, and a
global measure asks for the global knowledge of network
topology. Note that the quasi-local measure [12] consid-
ers more topological information than the local but does
not require global topological information. The refore , the
global is the most time-consuming, the quasi-local takes
second place, and the local needs the minimum time.
We compare the accuracies of sixteen indices, which
are CN [13,14], Jacard [15], Salton [16], LHN [17], HPI
[18], HDI [10], Sørensen [19], PA [20], AA [21], RA
[22], LP [22], LRW [23], SRW [23], Katz [24], MF I [25]
and RW R [26], about predicting friends in Table 1. CN,
Table 1. Accuracies of the sixteen trust inference indices
when Len = 10.
friend inference indices AUC Precision Recall
CN 0.8353 0.014 0.6797
Jacard
0.8292
0.0126
0.67
Salton
0.8277
0.0115
0.6646
LHN
0.8096
0.0014
0.6271
HPI
0.819
0.0028
0.6372
HDI 0.8288 0.0126 0.668
Srensen 0.8292 0.0126 0.67
PA 0.7919 0.0029 0.6289
AA 0.8375 0.0147 0.6848
RA 0.8362 0.0137 0.6794
LP (
α
= 0.001) 0.6662 0.0021 0.6314
LRW (t = 3)
0.9635
0.0226
0.7723
SRW (t = 3)
0.9875
0.0776
0.9565
Katz (α = 0.001)
0.6662
0.0021
0.6314
MFI
0.429
0.0019
0.6305
RWR (c = 0.9) 0.9702 0.049 0.8318

Improving Personal Product Recommendation via Friendships’ Expansion
6
Jacard, Salton, LHN, HPI, HDI, Sørensen, PA, AA and
RA are local indices. LP, LRW and SRW are quasi-local
indices. Katz, MFI and RWR are global indices.
The results reported here are averaged over 30 inde-
pendent runs, each of which corresponds to a random
division of training and testing sets of user -trusted friend
relations. The entries corresponding to the highest accu-
racies are emphasized in bold. The entries of LP, LRW,
SRW, Katz, and RWR are the accuracies corresponding
to their optimal parameters. The numbers inside the
brackets denote the optimal values of their parameters.
We can see that the highest accuracies are all acquired
by SRW with t = 3. Cleary, this optimal walking step is
very small, w hich unfolds two points: SRW with the op-
timal step owns the low computational complexity al-
lowing high scalability and it abandons the superfluous
information that makes no contribution or negative con-
tribution to the prediction accuracy. Therefore, SRW can
be expected to improve the artist recommendation of
LTPN algorithm by expanding users’ friendships.
5. The Personal Artist Re com mendation
via Friendships’ Expansion
These results in Figure 4 report the comparisons be-
tween FELTPN and three other typical recommendation
algorithms, LTPN, the collaborative filtering algorithm
(CF) and the standard network-based inference (NBI)
[27], for different moderate list lengths.
The basic CF recommends artists for the target user ut
by the following three steps:
1) Co mpute the affinity Qtj between ut and every other
user uj as follows:
( )
( )
1
=∑
tjtf jf
f
tj
Q BB
du du
, (13)
where d(uj) denotes the number of artists uj has listened
to.
2) Compute the level of listening preference Ptl of ut
for his/her unheard artist al as follows:
≠
≠
=
∑
∑
tj jl
jt
tl
tj
jt
QB
P
Q
. (14)
3) Sort these artists, which ut has never heard before,
in decreasing order according to their predicted scores,
and finally recommend the top Len artists to ut.
The results are averaged over 30 independent runs,
each of which corresponds to a random division of train-
Figure 4. (Color online) The precision (a), recall (b), inter-diversity (c), intra-diversity (d), popularity (e) vs. Len for four dif-
ferent algorithms.
LTPN
NBI
CF
FELTPN
0.001
0.003
0.005
0.007
0.009
0.011
02040 60 80 100
Len 0.00
0.02
0.04
0.06
0.08
0.10
02040 60 80 100
Len
Preccision
Recall
0.12
58
98
138
02040 60 80 100
Len
Popularity
178
0.5
0.7
0.9
1.0
020 40 60 80 100
Len
D
e
0.6
0.8
1.0
020 40 6080 100
Len
D
a
0.13
(a) (b)
(c) (d)
(e)

Improving Personal Product Recommendation via Friendships’ Expansion
7
ing and testing sets of user-listened artist r elations. Here,
ELTRN are produced by the following friendship-ex-
panding method: only when the number of friends of a
user is less than 10, his friendships are expanded via
SRW and the number of expanded friends is set to 1. We
can see from Figure 4 that the expanded friendships via
SRW can improve both precision and recall of our LTPN
algorithm while both outstanding diversity and excellent
popularity are maintained.
6. Conclusion and Discussion
In this paper, we exploit SRW to expand the personal
friendships and then make use of our LTPN algorithm to
recommend personal artists. The experimental results
show that the expanded friendships via SRW can im-
prove the recommendation accuracy of our LTPN algo-
rithm. However, we can also see that the improvement is
not large. External information, such as tag information
and rating information, may provide more useful infor-
mation and insights for personal recommendation. Fur-
thermore, the sparsity of existent data and the huge size
of real systems are two most main difficulties for the
studies of personal recommendation. The former may
result in large difficulties in building statistical models.
The latter requires highly efficient algorithms because
any highly accurate algorithm will become meaningless
if the consuming time is unacceptable. Therefore, de-
signing a better algorithm in both accuracy and speed is
our future challenge, especially for sparse and huge net-
works.
7. Acknowledgements
The authors thank the anonymous referees for helpful
suggestions and comments and acknowledge HetRec
2011 for providi n g us the lastfm data.
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