1 Introduction

People live in large social contexts, such as schools, workplaces, neighborhoods, and online communities. In addition, they form smaller groups in which they experience a higher level of communication than the rest of the social context [1]. Uncovering the community structure of a social network and modeling it are important tasks in social network analysis (SNA). SNA studies the relationships between social entities such as the members of a group, corporations or nations [2]. The phenomenon or data reflected by their relationship model are the focus of network analysis. Agents’ interaction in the social environment can be expressed as a pattern or rule based on relationship. The regular pattern based on this relationship reflects the social structure, and the quantitative analysis of this structure is the starting point of SNA. The focus of SNA is relationships and the relationship model, which is conceptually different from traditional statistical analysis and data processing methods.

Trust in general is a multifaceted concept. It is subjective, dynamic and context specific [3]. Trust is defined as an entity behaving in an expected manner, despite the lack of an ability to monitor or control the environment in which it operates [4]. Trust measures have been studied in many disciplines from different perspectives. Businesses use the trust relationship in the social network environment to effectively recommend customers and increase the purchase rate of customers. An important practical application of SNA is trust-enhanced recommender systems (or trust-aware recommender systems).

Trust is a representative relationship in SNs. When the strength of a relationship is related to the concept of “trust,” the social network is referred to as a trust network [5]. For example, WeChat, as a trust network, shows that users accept advice that comes from individuals they trust. A recommendation mechanism induced by both objective and subjective trust will be a more rational approach to conduct measurements. Trust modeling is a meaningful topic for users who have not been exposed to social networks to determine whether a strange user is trustworthy [6]. Levin, Cross [7] discussed implications of trust relationship for theory and practice. A social network is defined by a directed graph. An adjacent matrix can only describe whether trust relationship between each pair of decision makers exists or not in the graph. Dong et al. [8] defined a weighted adjacent matrix to describe trust strength. However, the vagueness of trust strength cannot be reflected completely. Victor et al. [9], Wu et al. [10], Gong et al. [11], Cai et al. [12] and Wu et al. [13] studied trust models that extract some effective social factors from the information in a social network. All these trust models try to interpret trust as a gradual phenomenon. The use of bilattices results for (trust, distrust)-couples is defined as trust score or trust function in [9, 10, 13]. Although the degree of trust and distrust of an agent can reflect his/her uncertainty in some degree, the hypothesis of the coexistence of trust and distrust remains to be discussed.

SNA is becoming an important technology in human behavioral modeling. We can exploit plenty of valuable information from SNA. The provision of a bridge between a social network’s conceptual properties and quantitative model is the premise of future research on SNA. Although existing SNA has been developed, the foundation of the preliminary work is not very solid, which will affect the further application of quantitative models. Existing computational models have developed trust propagation methods for unlinked individuals/organizations via trusted third parties (TTPs) that have direct trust in each other [5]. Wu et al. [10] constructed a uninorm operator that propagates trust and distrust simultaneously. Victor et al. [9] introduced several bilattice-based trust models and their propagation operators. Wu et al. [2] proposed a new dual trust propagator which successfully describes the phenomenon that the distrust value increases during the propagation process. Gong et al. [11] proposed two weighted trust aggregation operators to accomplish a multitrust transitive aggregation mode. However, the difference between direct trust and indirect trust is not taken into account in trust propagation. In another aspect, the objective fact that information attenuation also exists in the process of trust transmission is ignored, which leads to inconsistency with the facts. These trust propagation methods are not effective when dealing with complex trust information, such as interval-valued trust information and linguistic trust information.

How to compute and predict trust between agents more accurately and effectively is still an open problem [14]. In this paper, we discuss how to use fuzzy graph-based approaches to quantify human trust behavior in SNs and give patterns or rules based on trust to reflect the social structure. The remainder of this paper is organized as follows. In Sect. 2, we present a brief review of multigranularity linguistic variables and trust models. In Sect. 3, we propose a fuzzy context-based social network description model. A weighted direct graph can help agents to describe their trust relationships in a visual way which is the basis of trust relationship modeling. The characteristics of SNs are fully described by setting the properties of nodes and edges in the graph. In Sect. 4, a trust relationship model is carried out to compute and predict direct or indirect trust which is a key parameter to support social network group decision making (SN-GDM). In Sect. 5, we provide a framework to trust-based decision model and our proposal is applied to solve an SN-GDM scenario in an incomplete information context. An illustrative example is given. We also compare our proposal with existing methods. In Sect. 5, the advantages and limitations are discussed.

2 Preliminaries

In order to make the paper self-contained, we review some basic concepts and operations of multigranularity linguistic variables and trust models.

2.1 Multigranularity Linguistic Variables

Some activities in the real world cannot be assessed in a quantitative form but rather in a qualitative way. In such a case, a better approach may be the use of linguistic assessments instead of numerical ones. Linguistic variables can be represented as \(({s}_{i},\alpha )\), where \({s}_{i}\) is a linguistic term and \(\alpha\) is a numeric value representing the symbolic translation [15]. This form can be translated to a value \(\beta \in [0,g]\) which is used to represent the value of linguistic 2-tuples. The translation function \(\Delta^{ - 1}\) and retranslation function \(\Delta\) are as follows [15]:

$$\Delta :[0,g] \to S \times [ - 0.5,0.5),$$
(1)
$$\Delta (\beta ) = (s_{i} ,\alpha ),{\text{with}}\;\left\{ {\begin{array}{*{20}l} {s_{i} } & {i = {\text{round}}(\beta )} \\ \alpha & { = \beta - i} \\ \end{array} } \right.,$$
(2)
$$\Delta^{ - 1} :S \times [ - 0.5,0.5) \to [0,g],$$
(3)
$$\Delta^{ - 1} (s_{i} ,\alpha ) = i + \alpha = \beta .$$
(4)

Different experts may have different levels of knowledge about a problem; therefore, multigranularity linguistic information can be used to express their opinions. \(S = \left\{ {s_{0} ,s_{1} , \ldots ,s_{g} } \right\}\) is a linguistic term set characterized by its cardinality or granularity, where \(\#({S}^{g})=g+1\). They use several linguistic term sets with different granularities of uncertainty [16]. If a high precision is needed, then it is possible to select a high granularity value. On the contrary, a low granularity value can be used [17].

Definition 1

[18, 19] Let \(S^{g} = \left\{ {s_{0}^{g} , \cdots ,s_{g}^{g} } \right\}\) be a linguistic term set where \(s_{0}^{g} < s_{1}^{g} < \cdots < s_{g}^{g}\). The linguistic term is represented as \(s_{i}^{g} \in S^{g}\), where superscript \({\text{sup}}(s_{i}^{g} ) = g\) measures the uncertainty; subscript \(sub(s_{i}^{g} ) = i\) is the value of the term to measure order in the set.

Definition 2

[18, 19] Let \({S}^{g}=\left\{{s}_{0}^{g},{s}_{1}^{g},\cdots ,{s}_{g}^{g}\right\}\) be a linguistic term set in hierarchical structure and \(R\) be a real number set. We define the 2-scale numerical function \(2-SNF:{S}^{g}\to R\), which is constituted by two parts: order function and vagueness function \(\left(O,V\right)\).

\(\left(O,V\right)\) should satisfy these conditions:

  1. (1)

    In order to normalize the values of labels in different levels, we require \(O\in [\mathrm{0,1}]\) and \(V\in [\mathrm{0,1}]\);

  2. (2)

    If the linguistic term \(A\) is vaguer than the term \(B\), then \(V(A)>V(B)\).

If we suppose the term set to be a symmetrical one with uniform distribution, then we can get the functions (5) and (6):

$$O\left({s}_{i}^{g}\right)=i/g,$$
(5)
$$V\left({s}_{i}^{g}\right)=2/g,$$
(6)

Fusion mechanisms need to integrate assessments expressed in multigranularity linguistic variables, accommodating groups of experts with different expertise or uncertainty levels. These two parameters (O,V) to represent a multigranularity linguistic variable can remain necessary information.

Even though the linguistic approaches are appropriate to describe vague concepts associated with natural language, due to the expert’s granules of knowledge, the employment of a single linguistic term might not be enough to express the expert’s assessment. To avoid the situation that a selected linguistic term from a predefined set might not match the expert’s opinion, the use of complex linguistic expressions instead of single linguistic terms is proposed [20]. These related methods [21,22,23] dealt with the problem of expert’s granules of knowledge in another way. Our paper will apply the method [18, 19] to exhibit vagueness and imprecision of trust relationship.

2.2 Trust Model

A binary (crisp) relation is a mapping \(R: Y\times Y\to \{0, 1\}\), i.e., if an agent has a connection with another agent, then there is a link between them. Trust networks based on social relationships are important information sources for choices or decisions based on opinions from people one knows well or with whom one shares common interests. Han et al. [24] indicated that trust and distrust are two distinct but coexisting concepts.

Definition 3

[9]. Trust value \((t,d)\) is an element of \({[\mathrm{0,1}]}^{2}\), where \(t\) is called the degree of trust, and \(d\) is the degree of distrust.

A trust score space \(\mathcal{B}\mathcal{L}=({\left[\mathrm{0,1}\right]}^{2},{\le }_{t},{\le }_{k},\neg )\) consists of the set \({[\mathrm{0,1}]}^{2}\) of trust scores \(({t}_{i},{d}_{i})\), a trust ordering \({\le }_{t}\), a knowledge ordering \({\le }_{k}\), and a negation \(\neg\) defined by

\(({t}_{1},{d}_{1}){\le }_{t}({t}_{2},{d}_{2})\) iff \({t}_{1}\le {t}_{1}\) and \({d}_{1}\ge {d}_{2}\);

\(({t}_{1},{d}_{1}){\le }_{k}({t}_{2},{d}_{2})\) iff \({t}_{1}\le {t}_{1}\) and \({d}_{1}\le {d}_{2}\);

$$\neg \left({t}_{1},{d}_{1}\right)=\left({d}_{1},{t}_{1}\right)$$

Wu et al. [13] provided two functions (7) and (8) to define the trust score and knowledge deficit

$$TS(t,d)=t-d$$
(7)
$$KD(t,d)=\left|1-t-d\right|$$
(8)

Although social and economic networks generally use binary relations, binary networks do not allow us to extract complex knowledge of the relationship intensity between agents. Implicit trust plays a significant role in the overall dynamics of social networks. A fuzzy relation is defined as a mapping \(R: Y\times Y\to [0, 1]\), where \({\mu }_{R}({y}_{i},{y}_{j})\) denotes the degree of membership of the relationship between the pair of actors \(({y}_{i},{y}_{j})\). Zadeh et al. [25] used m-ary fuzzy relations to describe an adjacency matrix. Such relations represent social relationships among m individuals when a group of m individuals is considered:

$$\mu \left({y}_{1},{y}_{2}, \dots , {y}_{m}\right)=\left\{\begin{array}{l}1 if\, {y}_{1},{y}_{2}, \dots , {y}_{m}\,are\,related\,to\,each\,other\\ \left(\mathrm{0,1}\right) \,if\, {y}_{1},{y}_{2}, \dots , {y}_{m}\,are\,related\,to\,each\,other\,to\,some\,extent\\ 0 \,if\, {y}_{1},{y}_{2}, \dots , {y}_{m}\,are\,not\,related\,to\,each,other\end{array}\right.$$
(9)

Genç et al. [26] proposed linguistic summary forms by considering both attributes of social and economic agents and the relations between them. The processes in polyadic quantifiers have been extended to semifuzzy cases.

If trust is used to support decision making, it is important to have an accurate estimate of trust when trust is not directly available. Victor et al. [9] defined the concept of a propagation operator and gave an example of function (10). Trust propagation is often exploited to enable a source user to estimate trust in an unknown target user based on a trust chain of users that links them together.

$$P\left(\left({t}_{0i},{d}_{0i}\right),\left({t}_{im},{d}_{im}\right)\right)=\left(\mathcal{T}\left({t}_{0i},{t}_{im}\right),\mathcal{T}\left({t}_{0i},{d}_{im}\right)\right)$$
(10)

with \(\mathcal{T}\) being a t-norm.

In Fig. 1, there is no direct orthopair of trust/distrust values between experts \({E}_{1}\) and \({E}_{3}\). Through the path via expert \({E}_{2}\), we can calculate the trust/distrust values between experts \({E}_{1}\) and \({E}_{3}\).

Fig. 1
figure 1

An example of trust propagation

Full size image

Kuter,Golbeck [27] described a trust inference algorithm that uses a probabilistic sampling technique to estimate our confidence in the trust information from some designated sources. The confidence of \(n\) for \(n{^{\prime}}\) as the conditional probability \(P(n|n{^{\prime}})\) is defined as follows: Given that \(n\) conveys some information to \(n{^{\prime}}\), the probability that \(n\) believes in the correctness of that information is \(P(n|n{^{\prime}})\).

3 Representation of a Social Network

SNA enables us to examine the structural and locational properties including prestige, centrality, trust relationship, etc. Firstly, we explain why current models are not fully suitable for the measurement of trust in a social network. Then, we construct a fuzzy context-based social network where multigranularity linguistic variables are used to describe the trust relationships among agents.

3.1 New Challenges to Measure Trust

Although people’s understanding of trust relationships has been studied for a long time, the knowledge is not unified. The pattern or rule based on trust is developed based on people’s understanding of trust. We face some challenges when we attempt to quantify it. We discuss them in detail.

1. Context Specific (or Context Dependence)

Trust is context specific in its scope. Sherchan et al. [3] gave an example. John, as a professional doctor, receives Mike’s trust. Mike will ask John about his health. However, he does not ask John about vehicle maintenance and repair because he does not trust John as an expert in vehicle maintenance and repair. Because of this property, the trust between a pair of agents should be multidimensional. We focus on the relationship composed of multidimensional factors and how such relationship affects the behavior of network members. A single-dimensional network is not enough to describe trust relationships. Our trust model should represent the multidimension of trust between a pair of agents.

2. Asymmetry

The trust relationship between A and B is not equal. It is common for one side to trust the other side slightly more or slightly less. Hence, trust is directed and asymmetric. Yager [14] discussed the relationship in two situations, which are symmetry and asymmetry, and primarily constructed an undirected graph to represent an SN. Our model assumes that the trust network should be a directed graph.

3. Transitivity/Nontransitivity

Social networks consist of direct and indirect trust relationships (recommended trust relationships) between nodes [11]. Transitivity captures the property “friend of a friend is my friend.” Therefore, most computational models of trust prediction [9, 11, 13, 14] include the property of transitivity. However, Sherchan et al. [3] stated that trust is not transitive. If Mary trusts Jack and Jack trusts Jim, we cannot conclude that Mary trusts Jim. Whether the trust model is based on transitivity or nontransitivity is a key problem. However, there is no consensus yet.3

4. Propagation

Although there is no consensus on the issue of transitivity/nontransitivity, people widely admit that through an indirect chain of TTPs, trust can be propagated to an unknown person. A trust propagation chain may involve more than three agents. There are often more than two trust propagation chains from A to B. Designing a reasonable operator to calculate the degree of trust from A to B and giving an explanation of the real environment will be difficult.

5. Subjective

Trust is not a crisp and complete relation. Gong et al. [11] pointed out that trust relationships are characterized by subjective uncertainty and are difficult to quantify accurately. This property makes it difficult to quantify the trust network. Fuzzy set theory has the ability to model and analyze imprecise relations and connections between individuals or groups [25]. Sherchan et al. [3] illustrated the importance of combining computing with words to model SNs. A linguistic term set may not be sufficient to represent such subjectivity because a trust network is a multiagent network. The information that the agents provide is more related to their own opinions and feelings than to specific and measurable facts and objects. Modeling social relationships in GDM by integrating and exploiting relationship information, e.g., trust between agents, is facing major challenges inherent to GDM problems [20]. Multigranular fuzzy linguistic modeling methods make information transformed and presented in an organized way [17]. Therefore, we combine multigranular fuzzy linguistic modeling methods to complete the representation and measure of trust.

In the following sections, we construct a representation model that is a fuzzy direct graph, where linguistic variables such as strong or very strong are applied to quantify the degree of trust of the arc in the graph. Context specificity can be reflected by the multidimensions of trust of an arc.

3.2 Fuzzy Context-Based Social Network Description Method

Given a weighted direct graph \(G =\left\{V,E\right\}\), let \(V=\left\{{v}_{1},{v}_{2},\cdots ,{v}_{n}\right\}\) be a set of nodes and each of the nodes has an associated vector of attribute (feature) values. Node \({v}_{i}\in V\) represents an agent in an SN. \({c}_{ik}\) is the value of attribute \({C}_{k}\) according to agent \({v}_{i}\). The vector of attributes \({C}_{i}=\{{c}_{i1},{c}_{i2},\cdots ,{c}_{im}\}\) shows the diversity of a person’s characters. Therefore, an SN is also diverse. Attribute matrix \(C={\left[{c}_{ik}\right]}_{n\times m}\) is important to help understand the multidimensional property of social networks. SNA is based on a source of multidimensional information, which has the property of being context specific. This property is reflected in the structure of SNs. In fact, as the attribute space varies, the trust relationship also varies. We name \(G\) a fuzzy context-based social network.

E is a set of arcs. \(R({v}_{i},{v}_{j})\) can be seen as defining the weight on arcs \(({v}_{i},{v}_{j})\). A fuzzy relationship on \(V\times V\) is in the form of a fuzzy multigranularity linguistic subset, where \(R({v}_{i},{v}_{j})\) indicates the degree of trust from \({v}_{i}\) to \({v}_{j}\). Different agents may have different confidence or preferences to describe their degrees of trust. Therefore, it is reasonable to map the relationship on \(V\times V\) to a fuzzy multigranularity linguistic subset.

Social networks consist of direct and indirect relationships between nodes. A direct relationship from \({v}_{i}\) to \({v}_{j}\) means there is an arc \(({v}_{i},{v}_{j})\in E\) in \(G\). For each node \({v}_{i}\), let \({NG}_{j}=\left\{{v}_{j}:<{v}_{i},{v}_{j}>\right\}\) represent the set of nodes neighboring \({v}_{j}\), which has an arc \(({v}_{i},{v}_{j})\).

An indirect relationship from \({v}_{i}\) to \({v}_{j}\) means there is no arc \(({v}_{i},{v}_{j})\in E\), but we can find a chain from \({v}_{i}\) to \({v}_{j}\). The definition of a relationship chain is as follows.

Definition 4

For two agents \({v}_{i}\) and \({v}_{j}\), if there is a path \({v}_{i}\to {v}_{\sigma (1)}\to {v}_{\sigma (2)}\cdots \to {v}_{\sigma (q)}\to {v}_{j}\), where \({v}_{\sigma (k)}\in V (k=\mathrm{1,2},\cdots ,q)\), then the path from individual \({v}_{i}\) to \({v}_{j}\) is reachable. It is denoted as a relationship chain \(v_{i} \Rightarrow v_{j}\)

If two nonadjacent nodes do not have direct interaction experience, then there is no arc among them. Some research assumes that trust is transitive and supposes there is recommended trust between two nonadjacent nodes if there is a trust propagation chain between them. Recommended trust means that the trust relationship between the two nonadjacent nodes can be obtained through a chain connecting them. Regardless of whether trust is transitive or nontransitive, the property of propagation is accepted anyway. Therefore, the definition of a relationship chain is in fact an interaction chain that shows the interactions among the members in the chain. Whether it is a trust propagation chain depends on another property of SNs, which is context specific.

We believe the trust between a pair of agents should be multidimensional. In addition, we use the attribute matrix \(C={\left[{c}_{ik}\right]}_{n\times m}\) to help us understand the multidimensional property of the trust network. In path \({v}_{i}\to {v}_{\sigma (1)}\to {v}_{\sigma (2)}\cdots \to {v}_{\sigma (q)}\to {v}_{j}\), if the trust from \({v}_{i}\) to \({v}_{\sigma (1)}\) and the trust from \({v}_{\sigma (1)}\) to \({v}_{\sigma (2)}\) are in different dimensions, the trust relationship cannot be transitive. Some existing research assumes that these trust relationships are in the same dimension. This assumption makes modeling easier but also deviates from the understanding of the essence of social networks. Take the example in Sect. 3.1 again. Mike and John have interactions in the dimension of health consulting, so there is an arc from Mike to John in this dimension. However, Mike and John have no interactions in the dimension of mechanical consulting, so there is no arc from Mike to John in that dimension. One day John introduced his colleague Alice to Mike. Mike cannot transmit trust to Alice in the dimension of mechanical consulting. Therefore, the fuzzy relationship \(R({v}_{i},{v}_{j})\) on \(V\times V\) is extended to \({R}^{{C}_{k}}({v}_{i},{v}_{j})\), where \({v}_{i}\)’s trust to \({v}_{j}\) is based on the same attribute \({C}_{k}\). In the above example, the same occupation makes the trust transitive from Mike to Alice. The attribute \({C}_{k}=occupation\) and \({c}_{Alice,k}={c}_{John,k}=doctor\). In other words, Mike trusts Alice in the dimension of health consulting.

3.3 Trust Score

For a pair of agents \({v}_{i}\) and \({v}_{j}\), \(R({v}_{i},{v}_{j})\) is quantified by the trust score \({\lambda }_{ij}^{c}\). \({\lambda }_{ij}^{c}\) is defined as a trust score from \({v}_{i}\) to \({v}_{j}\) according to context \(c\). Because humans do not merely reason in terms of “trusting” and “not trusting”, but rather trusting someone “very much” or “more or less” [9]. We apply linguistic variables in trust score. However, an SN is a multiagent network, and different preferences of agents make multigranularity linguistic variables better. We adopt the computational model of [19, 28]. \(\wedge ({\lambda }_{ij}^{c})\to \left(O,V\right)\), in which \(O\) is the degree of trust, and \(V\) is the uncertainty of trust (the confidence of the agent). The trust score will be denoted by

$$\wedge ({\lambda }_{ij}^{c})=\{(O({\lambda }_{ij}^{c}),V({\lambda }_{ij}^{c}))| O({\lambda }_{ij}^{c}),V({\lambda }_{ij}^{c})\in [\mathrm{0,1}]\}\equiv {\left[\mathrm{0,1}\right]}^{2}$$
(11)

Given two trust scores, \({\lambda }_{1}\) and \({\lambda }_{2}\), the comparison rules are as follows:

  1. 1.

    If \(O\left({\lambda }_{1}\right)<O\left({\lambda }_{2}\right)\), then we obtain \({\lambda }_{1}\prec {\lambda }_{2}\);

  2. 2.

    If \(O\left({\lambda }_{1}\right)=O\left({\lambda }_{2}\right)\) and \(V\left({\lambda }_{1}\right)>V\left({\lambda }_{2}\right)\), then we obtain \({\lambda }_{1}\prec {\lambda }_{2}\);

  3. 3.

    If \(O\left({\lambda }_{1}\right)=O\left({\lambda }_{2}\right) \mathrm{and} V\left({\lambda }_{1}\right)=V\left({\lambda }_{2}\right)\) then we obtain \({\lambda }_{1}\sim {\lambda }_{2}\).

Comparison rules are based on the assumption that the value of the degree of trust is the primary factor determining the rank of the trust score. When the value of the degree of trust is the same, we prefer the trust score with a smaller value of \(V\) because a smaller value of \(V\) means more confidence in the value of the degree of trust.

Here, we set two special values of \({\lambda }_{ij}^{c}\), which are named absolute trust and absolute distrust. Absolute trust is represented as \(\Lambda ({\lambda }_{ij}^{c})=\left(\mathrm{1,0}\right)\), which means that the degree of trust is the largest value and the confidence is complete. Absolute distrust is represented as \(\Lambda ({\lambda }_{ij}^{c})=\left(\mathrm{0,0}\right)\), which means that these two agents have no interactions, and the value reflecting uncertainty is the largest.

We define \({({\lambda }_{ij}^{c})}_{\alpha }\) as the \(\alpha\)—cut set trust score, which means \(V\left({\lambda }_{ij}^{c}\right)\le \alpha\). \({({\lambda }_{ij}^{c})}_{\alpha }\) can help us eliminate trust relationships that do not have much credibility. By setting the value of \(\alpha\), we can modify an SN’s strength. The lower the value of \(\alpha\) is, the stronger the SN’s tie. We can cut the arcs whose \(V\left({\lambda }_{ij}^{c}\right)>\alpha\) to ensure that the remaining arcs are reliable to a certain extent.

4 Trust Relationship Modeling

In this section, we design a trust propagation operator based on the trust score defined in Sect. 3.3 firstly. Then, we propose a weight identification method that is useful for decisions of social network environment.

4.1 Trust Propagation Operator

Definition 4 defines the concept of a relationship chain. However, the relationship chain cannot improve the transitivity of trust. Because we consider trust to be context specific. Only when the interactions among the three agents are in the same attribute dimension, trust can be transmitted. A relationship chain is transformed to a trust propagation chain. The definition is as follows.

Definition 5

For two individuals \({v}_{i}\) and \({v}_{j}\), according to a specific context \(c\), if there is a path \({v}_{i}\stackrel{c}{\to }{v}_{\sigma (1)}\stackrel{c}{\to }{v}_{\sigma (2)}\cdots \stackrel{c}{\to }{v}_{\sigma (q)}\stackrel{c}{\to }{v}_{j}\), where \({v}_{\sigma (k)}\in V (k=\mathrm{1,2},\cdots ,q)\) and \(\stackrel{c}{\to }\) represents an arc according to context \(c\), then the path from individual \({v}_{i}\) to \({v}_{j}\) is reachable. It is denoted as a trust propagation chain according to context \(c\): \({v}_{i}\stackrel{c}{\Rightarrow }{v}_{j}\).

Trust propagation operator \(P\left({\lambda }_{ij}^{c},{\lambda }_{jk}^{c}\right)\) is used to obtain trust score \({\lambda }_{ik}^{c}\), where there is a path \({v}_{i}\to {v}_{j}\to {v}_{k}\). First, we introduce the properties of a trust propagation operator \(P\left({\lambda }_{ij}^{c},{\lambda }_{jk}^{c}\right)\).

  1. 1.

    Completely transitive: If agent \({v}_{j}\) fully trusts agent \({v}_{k}\), then the trust relationship from \({v}_{i}\) to \({v}_{j}\) is completely transmitted to \({v}_{k}\). In real life, if a friend whom you fully trust tells you to trust someone and you have no other information about this person, you will likely choose to trust him. In other words, \(P\) is used to denote an operator for trust score propagation in this situation: \(P\left({\lambda }_{ij}^{c},{\lambda }_{jk}^{c}\right)={\lambda }_{jk}^{c}\).

  2. 2.

    Completely nontransitive (or trust block): If agent \({v}_{i}\) fully distrusts agent \({v}_{j}\), then the trust relationship from \({v}_{j}\) to \({v}_{k}\) is blocked. In real life, if a friend whom you fully trust tells you to distrust someone and you have no other information about this person, you likely will choose to distrust him. \(P\) is used to denote an operator for trust score propagation in this situation: \(\Lambda (P\left({\lambda }_{ij}^{c},{\lambda }_{jk}^{c}\right))=(0,V\left({\lambda }_{jk}^{c}\right))\)

  3. 3.

    Associativity: \(P\left( {\lambda _{{ij}}^{c} ,P\left( {\lambda _{{jk}}^{c} ,\lambda _{{kl}}^{c} } \right)} \right)\sim P\left( {P\left( {\lambda _{{ij}}^{c} ,\lambda _{{jk}}^{c} } \right),\lambda _{{kl}}^{c} } \right)\). In path \({v}_{i}\to {v}_{j}\to {v}_{k}\to {v}_{l}\), the subsequence of propagation will not affect the final trust value from \({v}_{i}\) to \({v}_{l}\).

  4. 4.

    Monotonicity: If the members in a chain trust each other more than those in another chain, the result of propagation will be larger. \(P\left({\lambda }_{ij}^{c},{\lambda }_{jk}^{c}\right)\succ P\left({\lambda }_{i{^{\prime}}j{^{\prime}}}^{c},{\lambda }_{j{^{\prime}}k{^{\prime}}}^{c}\right)\) if \({\lambda }_{ij}^{c}\succ {\lambda }_{i{^{\prime}}j{^{\prime}}}^{c}\) and \({\lambda }_{jk}^{c}\succ {\lambda }_{j{^{\prime}}k{^{\prime}}}^{c}\).

Then, we design a propagation operator \(P\) that satisfies the above properties.

Definition 6

The trust propagation operator \(P\left( {\lambda _{{ij}}^{c} ,\lambda _{{jk}}^{c} } \right)\) associates two trust scores \(\Lambda ({\lambda }_{ij}^{c})=\left(O({\lambda }_{ij}^{c}),V({\lambda }_{ij}^{c})\right)\), \(\Lambda ({\lambda }_{jk}^{c})=\left(O({\lambda }_{jk}^{c}),V({\lambda }_{jk}^{c})\right)\) with the following trust score output:

$$\Lambda (P\left({\lambda }_{ij}^{c},{\lambda }_{jk}^{c}\right))=\left({\mathcal{T}}_{p}\left(O({\lambda }_{ij}^{c}),O({\lambda }_{jk}^{c})\right), \mathcal{S}(V\left({\lambda }_{ij}^{c}\right),V\left({\lambda }_{jk}^{c}\right))\right)$$
(12)

where \({\mathcal{T}}_{p}\) is the product t-norm function \({\mathcal{T}}_{p}\left(O({\lambda }_{ij}^{c}),O({\lambda }_{jk}^{c})\right)=O({\lambda }_{ij}^{c})\times O({\lambda }_{jk}^{c})\), and \(\mathcal{S}\) is the t-conorm function \(\mathcal{S}(V\left({\lambda }_{ij}^{c}\right),V\left({\lambda }_{jk}^{c}\right))=Max(V\left({\lambda }_{ij}^{c}\right),V\left({\lambda }_{jk}^{c}\right))\). \(P\) has two neutral elements \(\left(1,0\right)\) and \(\left(0,1\right)\)

Functions \({\mathcal{T}}_{P} :\left[ {0,1} \right]^{2} \to \left[ {0,1} \right]\) and \(\mathcal{S}:\left[ {0,1} \right]^{2} \to \left[ {0,1} \right]\) are arbitrary associative, commutative functions having a neutral element \(e=1(e=0)\), which is increasing in each of its arguments [29]. Now, we prove that \(P\) can satisfy the properties of being completely transitive, trust block, associative, and monotonic.

  1. 1.

    Completely transitive:

    Proof

    Agent \({v}_{i}\) fully trusts agent \({v}_{j}\) means absolute trust and \({\lambda }_{ij}^{c}=\left(\mathrm{1,0}\right)\)

    $$\Lambda (P\left({\lambda }_{ij}^{c},{\lambda }_{jk}^{c}\right))=\left({\mathcal{T}}_{p}\left(1,O({\lambda }_{jk}^{c})\right), \mathcal{S}(0,V\left({\lambda }_{jk}^{c}\right))\right)=\left(O({\lambda }_{jk}^{c}),V\left({\lambda }_{jk}^{c}\right)\right)={\lambda }_{jk}^{c}$$
  2. 2.

    Completely nontransitive trust block

    Proof

    Agent \({v}_{i}\) fully distrusts agent \({v}_{j}\) means absolute distrust and \({\lambda }_{ij}^{c}=\left(\mathrm{0,0}\right)\)

    $$\Lambda (P\left({\lambda }_{ij}^{c},{\lambda }_{jk}^{c}\right))=\left({\mathcal{T}}_{p}\left(0,O({\lambda }_{jk}^{c})\right), \mathcal{S}(0,V\left({\lambda }_{jk}^{c}\right))\right)=(0,V\left({\lambda }_{jk}^{c}\right))$$
  3. 3.

    Associativity:

    Proof

    \(\begin{aligned} \Lambda \left( {P\left( {\lambda _{{ij}}^{c} ,P\left( {\lambda _{{jk}}^{c} ,\lambda _{{kl}}^{c} } \right)} \right)} \right) = & \left( {O\left( {\lambda _{{ij}}^{c} } \right) \times O\left( {\lambda _{{jk}}^{c} } \right) \times O\left( {\lambda _{{kl}}^{c} } \right),{\text{Max}}\left( {V\left( {\lambda _{{ij}}^{c} } \right),{\text{Max}}\left( {V\left( {\lambda _{{jk}}^{c} } \right),V\left( {\lambda _{{kl}}^{c} } \right)} \right)} \right)} \right) \\ = & \Lambda \left( {P\left( {P\left( {\lambda _{{ij}}^{c} ,\lambda _{{jk}}^{c} } \right),\lambda _{{kl}}^{c} } \right)} \right) \\ \end{aligned}\)

  4. 4.

    Monotonicity:

    Proof

    \(\begin{aligned} \lambda _{{ij}}^{c} > \lambda _{{i^{\prime}j^{\prime}}}^{c} \Rightarrow & \left\{ {\begin{array}{*{20}l} {O\left( {\lambda _{{ij}}^{c} } \right) > O\left( {\lambda _{{i^{\prime}j^{\prime}}}^{c} } \right)} \\ {O\left( {\lambda _{{ij}}^{c} } \right) = O\left( {\lambda _{{i^{\prime}j^{\prime}}}^{c} } \right)~{\text{and}}~V\left( {\lambda _{{ij}}^{c} } \right) < V\left( {\lambda _{{i^{\prime}j^{\prime}}}^{c} } \right)} \\ \end{array} } \right\} \\ \lambda _{{jk}}^{c} > \lambda _{{j^{\prime}k^{\prime}}}^{c} \Rightarrow & \left\{ {\begin{array}{*{20}l} {O\left( {\lambda _{{jk}}^{c} } \right) > O\left( {\lambda _{{j^{\prime}k^{\prime}}}^{c} } \right)} \\ {O\left( {\lambda _{{jk}}^{c} } \right) = O\left( {\lambda _{{j^{\prime}k^{\prime}}}^{c} } \right)~{\text{and}}~V\left( {\lambda _{{jk}}^{c} } \right) < V\left( {\lambda _{{j^{\prime}k^{\prime}}}^{c} } \right)} \\ \end{array} } \right\} \\ \Rightarrow & \left. {\begin{array}{*{20}l} {O\left( {\lambda _{{ij}}^{c} } \right) > O\left( {\lambda _{{i^{\prime}j^{\prime}}}^{c} } \right)} \\ {O\left( {\lambda _{{ij}}^{c} } \right) \times O\left( {\lambda _{{jk}}^{c} } \right) = O\left( {\lambda _{{i^{\prime}j^{\prime}}}^{c} } \right) \times O\left( {\lambda _{{j^{\prime}k^{\prime}}}^{c} } \right){\text{~~and~~~}}V\left( {\lambda _{{ij}}^{c} } \right) \times V\left( {\lambda _{{jk}}^{c} } \right) < V\left( {\lambda _{{i^{\prime}j^{\prime}}}^{c} } \right) \times V\left( {\lambda _{{j^{\prime}k^{\prime}}}^{c} } \right)} \\ \end{array} } \right\} \\ \Rightarrow & P\left( {\lambda _{{ij}}^{c} ,\lambda _{{jk}}^{c} } \right) > P\left( {\lambda _{{i{\text{'}}j{\text{'}}}}^{c} ,\lambda _{{j{\text{'}}k{\text{'}}}}^{c} } \right) \\ \end{aligned}\)

Now, let us discuss the problem that there is more than one path from \({v}_{i}\) to \({v}_{j}\). Let \(p\) be the number of trust propagation chains from agent \({v}_{i}\) to \({v}_{j}\). We denote \({\rho }_{l}({v}_{i}\rightarrow {v}_{j})\) as the \(l\)th chain in \(G\), from agent \({v}_{i}\) toward \({v}_{j}\). Figure 2 shows a parallel network of \(p\) trust propagation chains between two agents, where each chain consists of at least one node. These chains are parallel, and we can calculate \({\lambda }_{ij}^{c}({\rho }_{l})\) according to path \({\rho }_{l}({v}_{i}\rightarrow {v}_{j})\).

Fig. 2
figure 2

Illustration of parallel paths from agent \({v}_{i}\) to \({v}_{j}\)

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However, when there is more than one path from \({v}_{i}\) to \({v}_{j}\), we can obtain different trust scores via the trust propagation chain. How should the trust score be calculated? Mui et al. [30] gave two possible methods: additive and multiplicative. The form of an additive estimate for \({\lambda }_{ij}^{c}\) is

$${\lambda }_{ij}^{c}=\sum_{l=1}^{p}{\lambda }_{ij}^{c}\left({\rho }_{l}\right)/p$$
(13)

Function (13) combines the parallel information about \({\lambda }_{ij}^{c}\) and may be one option. However, we propose a new method to estimate the trust score considering the propagation chain’s “reliability.” In the parallel network of p chains from \({v}_{i}\) to \({v}_{j}\), some chains are tied with strong connections, and some chains are tied with weak connections. A stronger tie means this chain is more reliable.

An important concept in trust propagation chain analysis is the strength of the path \({v}_{i}\stackrel{c}{\to }{v}_{\sigma (1)}\stackrel{c}{\to }{v}_{\sigma (2)}\cdots \stackrel{c}{\to }{v}_{\sigma (q)}\stackrel{c}{\to }{v}_{j}\). The function to calculate the strength of the path is proposed by [31]. We modified the strength function in [31] to suit our trust score function. The strength \(ST({\rho }_{l})\) of path \({\rho }_{l}\) is defined as

$$ST({\rho }_{l})=O({\lambda }_{i\sigma \left(1\right)}^{c})\times O({\lambda }_{\sigma \left(1\right)\sigma \left(2\right)}^{c})\times \cdots \times O\left({\lambda }_{\sigma \left(q\right)j}^{c}\right)$$
(14)

We can say that the strength of path \({\rho }_{l}\) is determined by the lowest degree of trust of the pair in the chain. Two nodes for which there is a path \(\rho\) with \(ST(\rho )>0\) between them are called connected. In other words,\({v}_{i}\) and \({v}_{j}\) are connected according to the lowest degree of trust.

We select the path \({\rho }^{*}\) whose strength is \(\underset{l=1 to p}{\mathrm{max}}ST\left({\rho }_{l}\right)\) as the strongest path that provides the strongest connection between two nodes. The propagation trust value of path \({\rho }^{*}\) is most reliable as the trust score from \({v}_{i}\) to \({v}_{j}\).

However, when there is a direct arc from \({v}_{i}\) to \({v}_{j}\), we prioritize the direct arc. After all, the more people that participate in this trust propagation chain, the greater the uncertainty of this chain, and the value of the degree of trust is not as reliable. The transmission of uncertainty in a group is more complex and needs further research. Therefore, we prefer a direct arc.

We conclude the following situations.

  1. 1.

    If \({v}_{i}\) has no knowledge about \({v}_{j}\), there is no path in \(G\) and \({\lambda }_{ij}^{c}=(\mathrm{0,1})\).

  2. 2.

    If \({v}_{i}\) has no direct arc with \({v}_{j}\) but it has at least a propagation path to \({v}_{j}\), we set \({\lambda }_{ij}^{c}={\lambda }_{ij}^{c}({\rho }^{*})\).

  3. 3.

    If \({v}_{i}\) connects directly with \({v}_{j}\) and there are other indirect paths to \({v}_{j}\), we prefer the trust score of the direct arc, not the propagation trust score from an indirect path.

4.2 Weight Identification in Social Network Group Decision Making

As a weighted directed graph, a node’s centrality is an important parameter. The idea of what kind of power an individual or an organization has in its social network, or what kind of central position it occupies, is one of the earliest contents discussed by network analysis. The centrality of an individual measures the degree to which the individual is in the center of the network, reflecting the importance of the node in the network. In a trust-enhanced social network, it represents how much trust he/she received from other agents in the SN. More important he/she is, much trust he/she receives. The centrality of a node is closely related to its importance in the network [14]. The measure of the centrality of node \({v}_{j}\) is the aggregated trust of the nodes connected to it by arcs. The definition of the weight of node \({v}_{j}\), which is also the centrality.

$$w\left({v}_{j}\right)=\sum_{{v}_{i}\in {NG}_{j}}O\left({\lambda }_{ij}^{c}\right)$$
(15)

The previous t-norm and uninorm-based trust propagation operators treat trust values with different knowledge deficits equally [5]. We extend functions (14) to (15) by combining the uncertainty of the trust to increase the reliability of the degree of trust. In other words, one agent receiving a high degree of trust from another agent may not increase the weight because this knowledge is not reliable. We use a level-set representation and obtain an \(\alpha -\) cut weight measure considering the uncertainty of agents. Here, \(\alpha -\) cut weight can be expressed as

$${w}_{\alpha }\left({v}_{j}\right)=\sum_{{v}_{i}\in {NG}_{j}}O({\left({\lambda }_{ij}^{c}\right)}_{\alpha })$$
(16)

\({w}_{\alpha }\left({v}_{j}\right)\) is the aggregation trust of nodes connected to \({v}_{j}\) with an uncertainty of at least \(\alpha\).

Currently, information is produced and processed by many people, such as trust-enhanced recommender systems, and trust is used to support decision making. SN-GDM is very popular and is particularly relevant to decision contexts involving historical interconnections between individuals within a group. A key issue in SN-GDM problems is how to aggregate individual preferences into a collective one to derive a final solution. Let an SN contain a set of agents \(V=\{{v}_{1},{v}_{2},\cdots ,{v}_{n}\}\) and the opinion set \(A=\{{a}_{1},{a}_{2},\cdots ,{a}_{n}\}\). Based on their historical interactions, we construct an SN. When aggregating individual opinions, the weight of a node is closely related to its centrality.

Then, the collective opinion \({A}^{c}\) is \(F\left({a}_{j}, {w}_{\alpha }\left({v}_{j}\right)\right)\), where \(F\left(\cdot \right)\) is an aggregation operator. Here, we utilize a weighted average operator, and the form of collective opinion is as follows.

$${A}^{c}=\sum_{j=1}^{n}{w}_{\alpha }\left({v}_{j}\right)\cdot {a}_{j}$$
(17)

4.3 Discussion

This paper constructs a novel trust relationship model and applies the model to SN-GDM based on properties of trusted relationship which is closer to human cognition. People’s cognition of trust relationship is the basis of decision making model. Therefore, the assumptions of the properties of trust relationship are very important. Therefore, we give the comparisons of the proposal and the existing references from two aspects. One is from the understanding of trust relationship, and the other is from the measuring and computing of trust relationship in an SN. The detailed comparisons are provided in Tables 1 and 2.

Table 1 Comparisons of the proposal and the existing trust relationship modeling
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Table 2 Comparisons of the proposal and the existing SN-GDMs
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The explanations of the differences of our trust-enhanced SN are as follows.

1. Multidimensional Feature of a Social Network

Trust can increase or decrease with time. This is the property of being dynamic which is ignored by existing references. However, our model seeks to describe the change not with time but with context. Few studies have been carried out in this direction. This improvement is very important. The multidimensional feature of social networks can explain the contradiction of transitive and nontransitive networks.

We assume that agent A can trust agent B to some degree in one context and can distrust agent B to some degree in another context in an SN. However, these two kinds of relationships cannot exist at the same time. This feature strongly illustrates the multidimensional feature of social networks. Because of the multidimensional features of social networks, people simply see that one person shows trust and distrust in another at the same time. However, they do not realize that trust and distrust in fact exist in different dimensions.

2. Uncertainty in a Trust Network

Victor et al. [9] defined a knowledge deficit \(KD(t,d)\) to evaluate the degree of uncertainty of the trust function. \(KD(t,d)\) is similar to our function \(O\). Victor et al. [9] thought \(KD(\lambda )=0\) (i.e., \(t+d=1\)) means perfect knowledge; otherwise, there is uncertainty in the knowledge of trust. However, in our model, we think that trust and distrust cannot exist at the same time. Therefore, we do not agree with the definition of the trust function, which uses a tuple \(\lambda =(t, d)\).

The ability of fuzzy sets to represent the degree of relation between individuals changes the depth of the analysis and provides new more realistic results [25]. In such a case, a binary fuzzy relation can be perceived as a generalization of a binary relation \({R}_{b}\). Its membership function is:

$$\mu ({R}_{b})={\mu }_{b}:A\times A\to \left[\mathrm{0,1}\right]$$
(18)

Compared with crisp values used to describe absolute trust or distrust, fuzzy sets provide great improvements.

$${R}_{b}:A\times A\to \left\{\mathrm{0,1}\right\}$$
(19)

Humans usually employ words in most of their computing and reasoning processes without the necessity of any precise number [34]. We apply the words and propositions drawn from natural language to emulate human trust relationships and describe uncertainty in trust networks.

$$\mu ({R}_{b})={\mu }_{b}:A\times A\to {S}^{g}.$$
(20)

5 An Application of Trust Relationship Model

In this section, we provide a framework of a trust-based decision model to SN-GDM scenario in an incomplete information context. An illustrative example is given to illustrate the proposed method. We discuss and compare several trust-based decision methodologies with the proposal.

5.1 A Framework to SN-GDM with Incomplete Preferences

In a real group decision process, there is often the problem of missing preference values. We develop a trust-enhanced social network for SN-GDM with incomplete preferences. A key issue that needs to be addressed in this type of decision making environment is to estimate unknown preference values [13]. One agent can use other agents’ knowledge to estimate the unknown preference values in his/her personal decision matrix. We can use the trust relationship model to estimate agents’ unknown preferences in this problem. We need to complete two tasks: (1) to estimate the unknown preference values and (2) to aggregate agents’ preferences.

The application of trust relationship model for incomplete SN-GDM consists of the following five steps: (1) computing trust degrees; (2) collecting preference; (3) estimating unknown preference values. (4) aggregating preferences; and (5) ranking alternatives. A framework is shown in Fig. 3.

Fig. 3
figure 3

A framework to SN-GDM problem with incomplete preferences

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5.2 An Illustrative Example

In this subsection, we give an illustrative example of decision making with incomplete preference information. Let \(I=\{{I}_{1},{I}_{2},{I}_{3}\}\) be a set of items needed to be recommended. There are a set of agents \(V=\{{v}_{1},{v}_{2},\cdots ,{v}_{6}\}\) and a set of criteria to be considered \(C=\{{c}_{1},{c}_{2},{c}_{3}\}\). \({c}_{1}\) is outward appearance. \({c}_{2}\) is intrinsic performance. \({c}_{3}\) is product upgrade in the future. The evaluation information is \({a}_{ijk}\in [\mathrm{0,10}]\), which is the evaluation of \({I}_{i}\) according to criteria \({c}_{k}\) by\({v}_{j}\). The higher the value is, the higher the evaluation. These agents construct an SN. We obtain the trust relationship matrix \(R={\left[{r}_{ij}\right]}_{6\times 6}\), where \({r}_{ij}=R({v}_{i},{v}_{j})\). We select one label from a constructed ordered linguistic term set \(S^{g} = \left\{ {s_{0}^{g} , \cdots ,s_{g}^{g} } \right\}\) to describe trust relationship.

Three agents supply evaluation matrix in Table 3.

Table 3 Evaluation matrices for items
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We assume that the trust relationship \(R({v}_{i},{v}_{j})\) on \(V\times V\) is context specific. In other words, their trusts vary according to criteria. For example, someone is the expert in designing products and he may be trustable in the criterion of intrinsic performance, while another one is the expert in fashion and he may be trustable in the criterion of outward appearance. \({R}^{{C}_{k}}({v}_{i},{v}_{j})\) is the trust from \({v}_{i}\) to \({v}_{j}\) according to criterion \({C}_{k}\). \({R}^{{C}_{k}}({v}_{i},{v}_{j})\) is in the form of multigranularity linguistic variables. We can obtain three trust relationship matrices and construct SNs accordingly. We take the trust relationship based on attribute \({C}_{1}\) as an example (see Table 4). We present the process of using the trust relationship to estimate the unknown preference values in Table 3.

Table 4 Trust relationship matrices \({R}^{{C}_{1}}\)
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We construct an SN (see Fig. 4.) according to the information in Table 4.

Fig. 4
figure 4

An SN based on attribute \({C}_{1}\)

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  1. Step 1.

    Computing the trust score.

    We adopt the computational model of multigranularity linguistic variables to obtain \(\Lambda ({\lambda }_{ij}^{c})\to \left(O,V\right)\) (Table 5).

  2. Step 2

    Estimating unknown preference values.

    The trust score of a particular agent can be used to predict the trust-enhanced evaluation matrices of the other agents in a trust propagation chain when the evaluation matrix is incomplete. In Table 3, \({a}_{111}\) and \({a}_{311}\) are unknown. Therefore, we need to use related agents’ trust to predict these two values.\({v}_{1}\)’s evaluations are unknown.\({v}_{1}\) has arcs or trust propagation chains from \({v}_{1}\) to \({v}_{3}\),\({v}_{4}\), \({v}_{5}\), and \({v}_{6}\) that show that there are trust relationships from \({v}_{1}\) to them. We predict \({p}_{111}\) and \({p}_{311}\) by aggregating their evaluations. We assume that the agent with a high trust degree from \({v}_{1}\) will be assigned a high weight. Therefore, we set weight of \({v}_{j}\) as

    $${w}_{j}=O((\Lambda \left({\lambda }_{1j}^{c}\right))/\sum_{j=3}^{6}O((\Lambda \left({\lambda }_{1j}^{c}\right))$$
    (21)

    First, we calculate the trust scores through the trust propagation chains from \({v}_{1}\). There are two trust scores (\({\lambda }_{14}^{{C}_{1}}\) and \({\lambda }_{15}^{{C}_{1}}\)) that are calculated by trust propagation chains.

    Table 5 Trust score matrices
    Full size table

    There is one trust propagation chain \({\rho }_{1}:{v}_{1}\to {v}_{3}\to {v}_{4}\).

    Therefore, \({\lambda }_{14}^{{C}_{1}}=(\frac{1}{2},\frac{1}{2})\).

    There are two trust propagation chains \({\rho }_{1}:{v}_{1}\to {v}_{3}\to {v}_{4}\to {v}_{5}\) and \({\rho }_{2}:{v}_{1}\to {v}_{6}\to {v}_{5}\).

    $$ST\left({\rho }_{1}\right)=0.4167$$
    $$ST({\rho }_{2})=0.4375$$

    \({\rho }_{2}\) is stronger, and we select \({\rho }_{2}\) to calculate \({\lambda }_{15}^{{C}_{1}}=(\frac{7}{16},\frac{1}{2})\).

    Then, we obtain the trust scores from \({v}_{1}\) to \({v}_{3}\),\({v}_{4}\), \({v}_{5}\), and \({v}_{6}\) (see Table 6).

    Table 6 Trust score from \({v}_{1}\) to \({v}_{3}\),\({v}_{4}\), \({v}_{5}\), and \({v}_{6}\)
    Full size table

    We predict the evaluation of \({v}_{1}\).

    $${a}_{111}=\sum_{j=3}^{6}O\left({\lambda }_{1j}^{c}\right)\times {a}_{1j1}/\sum_{j=3}^{6}O\left({\lambda }_{1j}^{c}\right)=7.2$$
    $${a}_{311}=\sum_{j=3}^{6}O\left({\lambda }_{1j}^{c}\right)\times {a}_{2j1}/\sum_{j=3}^{6}O\left({\lambda }_{1j}^{c}\right)=5.95$$
  3. Step 3

    Determining the weights of agents.

    According to expression (16), the \(\alpha -\) cut centrality values of agents are given in Table 7.

    Table 7 The centralities of agents (\(\alpha =1/2\))
    Full size table
  4. Step 4

    Aggregation Process.

We compute the collective overall evaluation values \({a}_{i}^{{c}_{1}},(i=\mathrm{1,2},3)\) of the three items:\({a}_{1}^{{c}_{1}}=\) 7.1

$${a}_{2}^{{c}_{1}}=6.1$$

\({a}_{3}^{{c}_{1}}=\) 6.6

We repeat the above steps to complete the evaluation matrix.

We construct the SNs (see Figs. 5 and 6) according to trust relationship matrices \({R}^{{C}_{2}}\) and \({R}^{{C}_{3}}\).

Fig. 5
figure 5

An SN based on attribute \({C}_{2}\)

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Fig. 6
figure 6

An SN based on attribute \({C}_{3}\)

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According to expression (16), the \(\alpha -\) cut centrality values of agents in context of \({c}_{2}\) and \({c}_{3}\) are given in Table 8.

Table 8 The centralities of agents in the context of \({c}_{2}\) and \({c}_{3}\) (\(\alpha =1/3\))
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We obtain \({a}_{212}=7.1\) by analyzing the SN in context \({c}_{2}\) and \({a}_{313}=6.3\) by analyzing the SN in context \({c}_{3}\).

We obtain the global evaluation of the three items in three SNs decision environments (see Table 9).

Table 9 Global evaluation matrices for items
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Using the average operator, we obtain

$${a}_{1}=\frac{7.2+7.0+6.9}{3}=7.0$$
$${a}_{2}=6.7$$
$${a}_{3}=6.5$$

The final ranking of items is:

$${I}_{1}\succ {I}_{2}\succ {I}_{3}$$

5.3 Comparisons

In the following, we give a comparative example using the method in [8] to solve this problem.

Firstly, we obtain the trust scores from \({v}_{1}\) to \({v}_{3}\),\({v}_{4}\), \({v}_{5}\), and \({v}_{6}\) (see Table 10).

Table 10 Trust score from \({v}_{1}\) to \({v}_{3}\),\({v}_{4}\), \({v}_{5}\), and \({{\varvec{v}}}_{6}\)
Full size table

Then, the complete evaluation of \({v}_{1}\) is obtained according to the weighted average (see Table 11).

Table 11 Evaluation matrix for items
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Calculate the weight of each agent (see Table 12). Different from this paper, this method has equal weight on different criteria.

Table 12 The weight of each agent
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Finally, the global evaluation of the three items in three SN decision environments is obtained (see Table 13).

Table 13 Global evaluation matrices for items
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Using the average operator, we obtain

$${a}_{1}=6.92$$
$${a}_{2}=6.58$$
$${a}_{3}=6.42$$

The final ranking of items is:

$${I}_{1}\succ {I}_{2}\succ {I}_{3}$$

Remark:

Through comparison, we find that although the ranking of the items obtained by these two methods is the same. It is worth noting that the values of the incomplete evaluation matrix and the collective evaluation matrix are inconsistent. This shows that in different decision making environments, the ranking of final items will change accordingly.

In this example, we illustrate one key point that social networks are context specific. The same group of people constructs three different SNs according to attributes. We construct three SNs according to three criteria. We only trust the judgment of the areas where experts are good at. Therefore, we predict unknown evaluations based on different weights of agents, while other methods, like the method in [8], can also estimate the missing information, but do not distinguish multidimensional degree of trust between a pair of agents. Our computational model can be applied in more areas.

6 Conclusion

This paper proposes a fuzzy context-based social network description method after analyzing the properties of trust. The weighted direct graph allows us to design a trust propagation model to reflect the new challenges of measuring trust. In this section, we point out the contributions and limitations of our proposal. The contributions of our paper are in two aspects:

  1. 1.

    Our proposal reveals multidimensional feature of a social network which can explain the property of transitivity or nontransitivity.

  2. 2.

    Our proposal allows to address complex real-world trust relationships where humans exhibit vagueness and imprecision.

The significant opportunities also exist for future research:

  1. 1.

    Consistency of incomplete preferences

    Cabrerizo et al. [35] pointed out that the missing values of incomplete fuzzy preference relations should be consistent with the complete fuzzy preference relations. So Cabrerizo et al. [35] proposed a process to adjust the established value to maximize the consistency level. However, our proposal lacks this step. As a future work, we would add a consistency test to perfect the process of estimating incomplete information.

  2. 2.

    Dynamic change of trust relationship

    Many facets of social networks—including their agents, interconnections, discussed topics, interests and trends—are dynamic [25]. When trust increases or decreases with new experiences, our fuzzy graph needs to adjust to suit this situation. We consider the trust prediction in the context space more and in time series less. Further research should develop in this aspect. Perhaps a dynamic network is better at describing trust relationships.

  3. 3.

    Uncertainty in a trust-enhanced social network

    The mathematical decision models should be closer to human common sense in the representation of uncertainty and in the process of human reasoning in decision making [36]. We must rethink an SN as a fuzzy system and extend fuzzy models to represent uncertainty not only in trust relationship, but also in centrality of an SN. This expansion will have greater managerial and academic impact.