Path Length and Trust Quality-based Trust Aggregation with

Journal of Computational Information Systems 10: 23 (2014) 10293–10305
Available at http://www.Jofcis.com
Path Length and Trust Quality-based Trust Aggregation
with Intuitionistic Fuzzy Set
Jun XU 1,2,∗,
1 College
Yuansheng ZHONG 1
of Information Technology, Jiangxi University of Finance and Economics, Nanchang 330013,
China
2 Modern
Economics & Management College of JXUFE, Nanchang 330013, China
Abstract
In social networks, users may ﬁnd increasing challenges in estimating trust scores for users that do not
directly connect each other. In this paper, ﬁrstly, since trust is subjective in nature and intuitionistic
fuzzy sets (IFSs) are most suited for capturing subjective, we represent trust score by IFSs and study
IFSs-based trust aggregated approaches. Then, based on length and trust quality of path, we propose
several hybrid path weight incorporating aggregation strategies. The experiments demonstrate that the
proposed aggregators can eﬀectively improve prediction accuracy.
Keywords: Path Length; Trust Quality; Trust Aggregation; Intuitionistic Fuzzy Sets (IFS); Social
Networks
1
Introduction
With the ever-increasingly web technology, online social networks have become main stream
application, in which people can share information and interact with each other, such as MySpace
and Facebook. However, the success of the online interactions depends on trust that members
have with each other as well as with the service provider. Therefore, trust is an important
element in online social networks [1]. Social networks in which the entities can express their
trust or distrust statements in each other are called trust networks. Trust networks, used to
measure trust relationships between members (called entities in this context) have been studied
in many literatures. In this paper, we pay attention to the trust evaluation between entities which
don’t know each other. Currently, few of trust metrics have been developed in literature for this
problem. Roughly, they may be divided into two categories, which are brieﬂy reviewed in the
following, respectively.
The ﬁrst type is global trust metrics [2]. It computes a trust score that approximates how much
the graph as a whole build a global trust value for the speciﬁc user. Kamvar et al. [3] presented
a distributed and secure method to compute global trust value for a peer by calculating the left
∗
Corresponding author.
Email address: [email protected] (Jun XU).
1553–9105 / Copyright © 2014 Binary Information Press
DOI: 10.12733/jcis12662
December 1, 2014
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principal eigenvector of a matrix of normalized local trust values. Aberer et al. [4] developed
an approach that addresses the problem of reputation-based trust management at both the data
management and semantic level, and assess trust by computing an agent reputation from its
former interactions with other agents. Suryanarayana et al. [5] model a P2P network as a directed
graph. They use the trust inference in the direct graph to infer indirect remote trust. Zhou and
Wang et al. [6] proposed a gossip-based reputation system (Gossip Trust) for fast aggregation of
global reputation scores.
The second type is local trust metrics [2]. It considers the personalized webs of trust, represented
as node to node relationship. It calculates trust score between a source node and a destination
by trust propagation rule. Golbeck [7] combined TidalTrust algorithm to consider the shortest
and strongest trust paths. Though this method decrease the time complexity of the algorithm
but it may negatively impact the coverage of the trust metrics. Avesani et al. [8] implemented
MoleTrust algorithm to be used in Molesking application. The Moletrust trust metric considers
those users who are at a distance not more than the trust propagation horizon. Wierzowiecki
and Wierzbicki [9] presented CloseLook algorithm which is based on the principle of limiting
the amount of computation by selecting the best paths to propagate trust, and by stopping the
trust propagation using scope parameters that can limit the number of considered nodes. Tao et
al. [10] proposed a subjective trust model based on cloud model which can deal with uncertainty
of information in open networks. Chakraborty and karform [11] integrated the path length and
decay of direct trust values along the trust path into trust propagation algorithms based on simple
multiplicative strategy.
The aforementioned two types of methods seem to be eﬀective for modeling trust. However,
they have two main disadvantages that the most of the methods seldom considered vagueness and
uncertainty of trust, and path quality. In the case of vagueness and uncertainty of trust, victor
et al. [12] and Josang et al. [13] who built trust model by using the bi-lattice and probabilistic
approach, respectively. Especially recently, verbiest [14] introduce and evaluate several path
length incorporating aggregation strategies. The simulation experiments demonstrate that using
path length weights improve the aggregation result. Currently, none of them takes into account
the quality of the paths.
In this paper, we propose a new trust model based on intuitionistic fuzzy set (IFS) [15], which
is a generalization of the concept of fuzzy set [16] whose basic component is only a membership
function. The IFS may describe imprecise information more abundant and ﬂexible than the fuzzy
set. Moreover, as the quality of path gets poor or path gets longer, it may be deemed to be less
trustworthy. Therefore, we introduce path length and quality into trust aggregation strategy.
And test the usefulness of the strategy on the Advogato dataset.
The rest of the paper is organized as follows. Section 2 recalls preliminaries and introduces
necessary concepts on trust metrics. In Section 3, we present several path weight-based trust
aggregation operators. In Section 4, we introduce calculating method of path length weight and
path quality weight, while the trust degree of agents is computed using SNA in Section 5. We
evaluate the performance of our proposed method on the basis of a data set from Advogato in
Section 6. Finally, we conclude in Section 7.
2
Preliminaries
According to the proposal in [17], we treat a trust network as a directed graph with the entities
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as nodes, and the trust relationships as edges.
2.1
Intuitionistic fuzzy set
Definition 1 [15] Let X be a universe of discourse. An IFS over X is an expression given by
A = {< x, uA (x), vA (x) > |x ∈ X}, in which uA : X → [0, 1], vA : X → [0, 1] with the condition
0 ≤ uA (x) + vA (x) ≤ 1 for all x ∈ X. uA (x) and vA (x) denote the membership degree and the
non-membership degree of the element x ∈ A, respectively. We call πA (x) = 1 − uA (x) − vA (x)
the degree of hesitation (or uncertainty) of element x to A.
According to the study in [18], for an IFS A = {< uA (x), vA (x) > |x ∈ X}, the couples
< uA (x), vA (x) > is called an intuitionistic fuzzy number (IFN). For convenience, we denote an
IFN by < u, v >, where 0 ≤ u ≤ 1, 0 ≤ v ≤ 1 and u + v ≤ 1. Let T be the set of all the IFNs.
2.2
Possibility degree
Possibility degree of two intuitionistic fuzzy numbers (IFNs) not only reﬂects the comparison of
two IFNs, but also reﬂects the degree of one is larger than another. We extend the possibility
degree method of interval valued numbers to IFNs and deﬁne a possibility degree expression to
compare two IFNs.
Definition 2 Let α1 =< µ1 , v1 >, α2 =< µ2 , v2 > be two IFNs, the possibility degree of two IFNs
can be denoted as follows.
{
{
} }
1 − v2 − µ1
p(α1 ≻ α2 ) = max 1 − max
,0
(1)
π1 + π2
where π1 and π2 are hesitation degree of α1 and α2 , respectively.
2.3
Lattice-based approach
Definition 3 (Trust network) A trust network is a pair (A, R) in which A is the set of entities
and R is a function A × A → [0, 1]2 = T . For each couple (x, y) of entities in A, trust score
R(x, y) = T .
In this paper, as showed in [17], we represent trust score between entities as an IFN < u, v >→
[0, 1]2 = T , in which u corresponds to a trust degree, v to a distrust degree, and π = 1 − u − v to
hesitation degree. According to the deﬁne in [19], the trust score space is embedded in a lattice
L∗ = {< u, v >∈ [0, 1]2 |u + v ≤ 1}
< u1 , v1 >≤L∗ < u2 , v2 >⇔ u1 ≤ u2 and v1 ≥ v2
.
The trust scores <1, 0> and <0, 1> are respectively the biggest and the smallest value of L∗ ,
corresponding to full trust and full distrust. For additional details, we refer to [17].
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Trust metrics
In a trust network (A, R), assume two connected agents x ∈ A and y ∈ A, but not directly.
However, two such agents may indirectly be connected to each other through a path in the
network. In order to calculate the trust score of x in y, usually including the trust metrics
propagation and aggregation.
2.4.1
Trust propagation
Despite trust is not perfectly transitive, it can be propagated between users [1]. Consider a trust
path p in a trust network as (x, z1 , · · · , zm−1 , y) where z1 , · · · , zm−1 are the agents between agents
x and y, such that α1 =< u1 , v1 > is the direct trust score of x in z1 , αi =< ui , vi > is the direct
trust score of zi−1 in zi for all i ∈ 1, · · · , m − 2 and αm =< um , vm > is the direct trust score
of zm−1 in y. We can then estimate a trust score along the path (x, z1 , · · · , zm−1 , y) from x to y.
The iterative multiplication strategy for trust propagation is discussed in [20]. In this section, we
deﬁne the propagation operator by
P rop(x, z1 , · · · , zm−1 , y) = α1 ⊗ α2 ⊗ · · · ⊗ αm−1 ⊗ αm
(2)
for all < u1 , v1 > and < um , vm > in L∗ .
According to multiplication algorithm of IFS [21], Eq. (2) can be rewritten as follows:
Pr op(x, z1 , · · · , zm−1 , y) =<
m
∏
ui , 1 −
i=1
2.4.2
m
∏
(1 − vi ) >
(3)
i=1
Trust aggregation
When there are several paths linking two agents, each path has its own trust score. We need to
integrate the information provided by each of those paths. Consider n paths linking agents x and
y, qi =< ui , vi > (i = 1, 2, · · · , n) be the trust score, created by the i-th path. Victor et al. [12]
deﬁned basic trust aggregators motivated by a set of boundary conditions. Inspired by this work,
the trust aggregators are deﬁned as follows:
Definition 4 (TMAX). The trust degree maximizing trust aggregator TMAX is defined as
T M AX(x, y) =< max(u1 , · · · un ), max(u1 + v1 , · · · un + vn ) − max(u1 , · · · un ) >
(4)
Definition 5 (DMAX). The distrust degree maximizing trust aggregator DMAX is defined as
DM AX(x, y) =< max(u1 + v1 , · · · un + vn ) − max(v1 , · · · vn ), max(v1 , · · · vn ) >
(5)
Definition 6 (HMAX). The hesitation degree maximizing trust aggregator HMAX is defined as
HM AX(x, y) =< min(u1 , · · · un ), min(v1 , · · · vn ) >
(6)
Definition 7 (TSMAX). The trust score maximizing trust aggregator TSMAX is defined as
T SM AX(x, y) =< max(u1 , · · · un ), min(v1 , · · · vn ) >
Obviously, these results of trust aggregators are also IFNs.
(7)
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3
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Advanced Trust Aggregation Operators
Since the above trust aggregators are too extreme, they are not very comprehensive. In some
cases, users might care more about some reliable trust scores because they aggregated along more
trusted or shorter paths. In this section, we propose three families of aggregation operators such
as the path weighted averaging (PWA) operator, the ordered weighted averaging (OWA) operator
and the induced OWA (IOWA) operator.
3.1
PWA operators for trust scores
Definition 8 Let qi =< ui , vi > (i = 1, 2, · · · , n) be a collection of trust scores, obtained by
propagating n paths linking agents x and y, and let P W A : (L∗ )n → L∗ . If
P W A(x, y) = w1 q1 ⊕ w2 q2 ⊕ · · · ⊕ wn qn
(8)
then the function PWA is called path weighted averaging (PWA) operator, where Wp = (w1 ,
w2 , · · · , wn ) is the path
∑ weight vector based on length and quality of path, such that ∀i ∈ {1, 2, · · · ,
n}, wi ∈ [0, 1] and ni=1 wi = 1. The computation of weight can refer to the expression (12).
According to addition algorithm of IFS [21], Eq. (8) can be rewritten as follows:
P W A(x, y) =< 1 −
n
∏
(1 − ui ) ,
i=1
3.2
wi
n
∏
viwi >
(9)
i=1
OWA operators for trust scores
Definition 9 Let qi =< ui , vi > (i = 1, 2, · · · , n) be a collection of trust scores, obtained by
propagating n paths linking agents x and y. An ordered weighted averaging operator of dimension
n is a mapping ∑
OW A : (L∗ )n → L∗ , such that w = (w1 , w2 , · · · , wn )T , with wi ∈ [0, 1](i =
1, 2, · · · , n) and ni=1 wi = 1, and
OW A(x, y) = w1 qσ(1) ⊕ w2 qσ(2) ⊕ · · · ⊕ wn qσ(n)
(10)
where qσ(i) is the i-th largest of the qi . w = (w1 , w2 , · · · , wn )T can be determined by the normal
distribution based method [23]. It can relieve the influence of unfair arguments on the decision
results by weighting these arguments with small values.
3.3
IOWA operators for trust scores
Another strategy to incorporate path weight is use IOWA operators [24], which is an extension of
the OWA operator. The main diﬀerence is that the reordering step of the IOWA is not according
to their values but rather using order-inducing variables. Inspired by this work, we can deﬁne the
path weight incorporating aggregator P IOW A as follows:
Definition 10 Let qi =< ui , vi > (i = 1, 2, · · · , n) be a collection of trust scores, obtained by
propagating n paths linking agents x and y. An induced ordered weighted averaging operator
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of dimension n is a mapping ∑
P IOW A : (L∗ )n → L∗ , such that w = (w1 , w2 , · · · , wn )T , with
wi ∈ [0, 1](i = 1, 2, · · · , n) and ni=1 wi = 1, and
P IOW A(x, y) = w1 b1 ⊕ w2 b2 ⊕ · · · ⊕ wn bn
(11)
where bi is the qi value of the OWA pair < ui , qi > having the ith largest ui , and ui in < ui , qi > is
referred to as the order inducing variable according to path weight and qi as the argument variable.
4
Path Weight
In order to address the reﬁnement of trust aggregator, we propose the hybrid path weights, which
combine path length weights and path quality weights between two agents. Path length weight
is derived from the inverse of length by adopting Verbiest and Victor’s [14] computation method.
Path quality weight is measured according to the trust degree of each path between two agents.
Generally speaking, it is diﬃcult to decide automatically the weights of path length and quality.
Some special situations should be considered, for example, there is only path length or path
quality, so the ﬁnal weight is deﬁned in as follow:

W l i × W qi


W l ̸= ∅, W q ̸= ∅



 W li + W q i
W l ̸= ∅, W q = ∅
W p i = W li
(12)


W qi
W l = ∅, W q ̸= ∅



1/n
W l = ∅, W q = ∅
in which W li ∈ [0, 1] is the path length weight. W qi ∈ [0, 1] returns the quality of a path. These
details will be discussed in the next section. In Eq. (12), four cases are discussed.
From both path length and path quality perspectives, the trust aggregation is derived not only
by the opinion of length decay, but also by those of the path reliability. Therefore, the PWA
operator reduces to a path hybrid weighted averaging (PHWA) operator:
P HW A(x, y) = wp1 q1 ⊕ wp2 q2 ⊕ · · · ⊕ wpn qn
where wpi =
(13)
W li ×W qi
.
W li +W qi
In the same way, the PIOWA operator reduces to a path hybrid ordered weighted averaging
operator (PH-IOWA).
When only path length weight exists, the ﬁnal weight is equal path length weight. The PWA
operator reduces to a path length weighted averaging (PLWA) operator:
P LW A(x, y) = wl1 q1 ⊕ wl2 q2 ⊕ · · · ⊕ wln qn .
(14)
When only path quality weight exists, the ﬁnal weight is equal path quality weight. The PWA
operator reduces to a path quality weighted averaging (PQWA) operator:
P QW A(x, y) = wq1 q1 ⊕ wq2 q2 ⊕ · · · ⊕ wqn qn .
(15)
Similarly, the PIOWA operator reduces to a path quality ordered weighted averaging operator
(PQ-IOWA).
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When both path length and path quality weight do not exist, which means each path linking
agents x and y is equally important. The PWA operator reduces to an average (AV) operator:
1
AV (x, y) = (q1 ⊕ q2 ⊕ · · · ⊕ qn ).
(16)
n
4.1
Factor based on path length
In social network, people tend to trust others with shorter paths. As shown in [14], there was
a negative correlation between trust score and path length; the more close two agents were,
the greater the trust score between them. Consider n paths linking agents x and y, we call
pi (i = 1, 2, · · · , n) the path length of the i-th path. We deﬁne path length weight as:
λ
λ + pi − 1
wli = n
(i = 1, 2, · · · , n)
∑
λ
λ + pi − 1
i=1
(17)
where λ denotes the decay factors and is used to control the decay speed.
Intermediate agents
A
S
B
T
C
Source agent
Target agent
D
Fig. 1: Social network
4.2
Factor based on path quality
Since several paths may exist between two nonadjacent agents, such as the path S → C → T and
S → A → B → T in Fig. 1. The source agent can evaluate the trustworthiness of the target agent
based on the trust information between the intermediate agents along the path [1]. In general,
intermediate agents along the path which have higher trust degree can give more trustworthiness
to the target agent. Therefore, we propose a new concept, Path Trust Quality (PTQ) as follows.
Definition 11 Path Trust Quality (PTQ) is the ability to guarantee a certain level of trustworthiness in trust propagation along a path, taking average trust degree of intermediate agents along
a path as attribute. Consider a trust path pi (i = 1, · · · , n) in a trust network have t intermediate
agents z1 , · · · , zt (zj ∈ E) between agents x and y, the PTQ weight is calculated as in Eq. (18),
1∑
T D zj
t j=1
t
wqi =
t
n
∑
1∑
i=1
t
j=1
TD
(i = 1, 2, · · · , n)
zj
(18)
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in which T Dzj is the trust degree of the intermediate agent zj , illustrated in Section 5.
5
Trust Degree Based on Social Network Analysis
Before aggregating trust scores of paths, it is necessarily to determine the weights associate to each
path. In most trust aggregation problems, they counted the length of path, but not its quality.
However, according to Deﬁnition 2, PTQ is associated with average trust degree of intermediate
agents. Therefore, trust degree should be regarded as a reliable source to be used in deriving
aggregation weights for paths. Social Network Analysis (SNA) is the study of social relationships
between entities, and entities can be individuals, group, or communities [22]. It makes us able to
research properties of networks including centrality, prestige and trust relationship. Accordingly,
a method is given below for measuring the trust degree of an agent with the concepts of SNA.
5.1
In-degree centrality value
In a directed graph, in-degree is the extent to which an actor receives information about another
actor from the other actors. According to social network theory, an actor with high in-degree has
high prestige, since many other actors seek to direct him ties. He also has more power, because he
can inﬂuence other actors as far as his opinions are taken into account. Therefore, the in-degree
of centrality can be applied in our study.
Definition 12 Let G = (E, L) be a directed graph composed of nodes E = {e1 , · · · , em } and
edges L = {l1 , · · · , lq } representing directed lines between nodes, ω = {ω1L , · · · , ωqL } be the set of
intuitionistic fuzzy assessments of the corresponding lines, SL = (Skh )m×m be the sociomatrix of
the graph G = (E, L), then the in-degree centrality value of node is defined as follow:
1 ∑
Skh
=
m − 1 k=1
m
L
DD
(eh )
5.2
(19)
Computing trust degree
According to the Deﬁnition 3, the trust degree of each node is computed as follows:
Definition 13 Let G = (E, L) be a directed graph representing the trust relationship between
L
L
agents E = {e1 , · · · , em } and {DD
(e1 ), · · · , DD
(eh )} be the set of in-degree centrality values. The
trust degree of agents eh can be computed as:
1 ∑
L
L
TD =
p(CD
(eh ) ≻ CD
(ek ))
m − 1 k=1
m
h
L
L
L
L
in which p(CD
(eh ) ≻ CD
(ek )) is possibility degree of CD
(eh ) ≥ CD
(ek ).
(20)
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6
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Experimental Evaluation
In this chapter, we conduct experiments on our proposed hybrid path weight aggregation (PHWA)
strategies, and compare them to other classical counterparts presented in the literatures [14], for
evaluating their performance. All aggregators compared in our experiments are described in
Table 1. Besides conventional operators we used the average AV, which serves as a baseline for
the HPWA.
Table 1: Overview of trust aggregators used in experiments
TMAX
Trust degree maximizing
DMAX
Distrust degree maximizing
HMAX
Hesitation degree maximizing
TSMAX
Trust score maximizing
AV
Average of trust and distrust degree
PLWA
Path length weight average
PQWA
Path quality weight average
PHWA
Hybrid path weight average based on length and quality
PQ-IOWA Uses path quality dependent order to rank trust scores before applying OWA
weight vector to them
PH-IOWA
6.1
Uses both path length and path quality dependent order to rank trust scores
before applying OWA weight vector to them
Data set
In our experiment, we collected the Advogato data set downloaded from Trustlet.org on May 19th
2012. Advogato.org is an online community site of free software development. On Advogato users
can rate each other on 4 diﬀerent levels: Observer, Apprentice, Journeyer, and Master and the
Advogato trust metric uses this information in order to assign a trust score to every user. These
ratings among members constitute a rich trust network of 7431 users and 56615 trust ratings.
The distribution of trust values is listed in Table 2. The average degree is 7.62. The average path
length is 3.78. The average clustering coeﬃcient is 0.13.
Since Advogato data is not exactly equal to our trust score space setting, we translate the trust
statements to IFNs can be found in Table 2.
6.2
Setup
To evaluate the performance of our proposed method, we use the 1000 leave-one-out method, in
which a direct trust score is taken out of the trust network and then estimate it by applying our
proposed aggregator. In order to study the impact of aggregators, we use the same propagation
operator. Moreover, we consider only paths of maximal length h = 3 and the coverage is 64.32%.
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Table 2: Distribution and translation of trust in the data set
Trust statements
IFNs
#
%
Master
< 0.95, 0.05>
12370 29.20
Journeyer
< 0.70, 0.25>
18580 43.86
Apprentice
< 0.40, 0.50>
6300
14.87
Observer
< 0.25, 0.70>
5110
12.07
After obtaining all pairs of the real and predict trust score, we can calculate the mean absolute
error (MAE) between the two variables for evaluating the accuracy of trust estimation. The trust
score MAE is given as follows:
N
1 ∑
M AE =
|ui − u′i | + |vi − vi′ |
N i=1
(21)
where < ui , vi > is real trust score and < u′i , vi′ > is the predicted trust score.
6.3
Results
MAE
First, we compare the performance of the basic trust aggregators introduced in Section 4.2. The
experiment result is shown in Fig. 2. There is small diﬀerence among TSMAX, TMAX and
HMAX, which performs better than DMAX.
Fig. 2: MAE for basic trust aggregators
Next, we study the eﬀect of the path length and path quality dependent trust aggregators.
Fig. 3 shows that PLWA and PQWA perform better than AV. Both path length factor and path
quality factor can improve the simple AV. In general, the hybrid of path length factor and path
quality factor, PHWA, performs slightly better than PLWA. Note that when λ < 0.8, PHWA
also performs better than PQWA. These implies that considering both the path length factor and
path quality factor in aggregating the trust score can have better improvement of the aggregation
result than the method considering only path length or path quality. Similar study can be made
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10303
0$(
for the MAE of OWA, PQ-IOWA and PH-IOWA in Fig. 4. The accuracy of PH-IOWA is better
than that of PQ-IOWA and OWA. These results conﬁrm that it is necessary to give the greater
weight to the shortest paths and the best quality paths. We have also tried to examine that
the hybrid path weight incorporating to our aggregator can improve the accuracy of trust score
predictions.
λ
Fig. 3: MAE for weighted average operators PLWA, PQWA and PHWA compared to the classical av-
MAE
erage AV
Fig. 4: MAE for ordered weighted averaging operators PH-IOWA, PQ-IOWA and OWA
7
Conclusion
In this paper, we built upon previous work in which trust scores are represented by an intuitionistic
fuzzy relation. It is characterized by trust degree, distrust degree and hesitancy degree. Based on
these element boundary conditions, we proposed some basic trust aggregators, such as TMAX,
DMAX, HMAX and TSMAX. However, since trust scores propagated along longer or poor paths
may lead to big error, they should be considered less important than other trust scores during
the trust aggregation process. From this perspective, we introduced other families of aggregation
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operator with path length weight and path quality weight. The path length weight is deﬁned by
the inverses of the path lengths of the trust scores. The path quality weight is obtained by using
trust degree of intermediate agents, via its representation using an SNA methodology approach.
Through an experiment on the Advogato trust network, we showed that trust scores propagated
along longer paths or poor quality paths indeed contain more errors than others. Furthermore,
we learned that the accuracy of aggregation is improved using the PHWA strategy.
Acknowledgement
The research developed in this paper is supported by the National Natural Science Foundation
of China (Nos. 71361012, 71061006), the Natural Science Foundation of Jiangxi Province of
China (No. 20132BAB201050), the Colleges Humanities Social Science Research Project of Jiangxi
Province of China (No. JC1338).
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