Community, also called a cluster or module, is a group of vertices which probably share common properties. Community detection refers to the recognition of modules and their boundaries based on the structural positions of vertices and the classification of vertices [48]. The community detection method is the key based on a trust network for DMs to reduce dimensionality clustering. Some methods have been developed, such as the GN method [49], the spin-glass method [50], the random walk community detection [51], the label propagation method [52] and so on. The Louvain method is a commonly used community detection method, which is based on the modularity theory proposed by Blondel et al. [46]. This method is an agglomerative clustering algorithm, which reveals the complete hierarchical community structure of the network and can cluster subgroups automatically without setting the initial number of subgroups. Therefore, this study uses the Louvain method for DMs in LGDM problems.

Assuming that a network has t nodes, the algorithm of the Louvain method can be expressed as follows [39]:

Step 1: Each node in the network is treated as a separate community. In the beginning, the number of communities is the same as the number of nodes.

Step 2: Assign each node to the community according to the node near each node, calculate the value of ΔQ before and after the module allocation, and record the maximum ΔQ value of the node. If maxΔQ>0, then the node d^α is assigned to the community where the node with the maximum ΔQ value is located, otherwise, it will remain unchanged.

The definition of ΔQ is as follows:

ΔQ=[∑in+2e^r,in2m−(∑tot+e^r2m)]−[∑in2m−(∑tot2m)2−(e^r2m)2]

where ∑in is the total degree of all edges in the community c, ∑tot is the total degree of all edge associated to the node in the community c, e^r,in is the number of edges of the node from node er(r=1,2,…,m) to all nodes in the community c, m is the number of all edges in the network.

Step 3: Repeat Step 2 until the network in the community no longer changes.

Step 4: Streamline the entire network and treat the nodes in a community as a new node. The weights between nodes within the community are converted to the weights of the new nodes. The weights between the edges of the community are converted to weights between the edges of the new nodes. The boundary value of the weight is 1.

Step 5: Repeat Steps 1–4 until the modules of the entire network no longer change.

(1) Network weights of each DM

Set V nodes in the network. Firstly, the degree centrality CD(er) and the eigenvector centrality CE(er) of each node er(r=1,2,…,m) are calculated by using Pajek software, and then they are standardized [5]:

(11)CD′(er)=CD(er)∑r=1mCD(er)

(12)CE′(er)=CE(er)∑r=1mCE(er)

Combining the normalized degree centrality CD′(er) with the eigenvector centrality CE′(er), calculate the mixed center degree CF(er) of the node er(r=1,2,…,m):

(13)CF(er)=σCD′(er)+(1−σ)CE′(er)

According to [49], the value of σ is generally 0.5.

Suppose that m nodes in the network are divided into p subgroups, and there are s nodes in the subgroup Gk(1≤k≤p), the network weight of the node er(er∈ck,1≤r≤s) can be calculated as:

(14)wr1=CF(er)∑r=1sCF(er)

(2) Trust weights of each DM

Assigning weight to each DM is an important part of the decision-making process and plays a key role in obtaining the final solution. For online social networks, historical interaction information can provide a reliable source for accessing DMs.

Social influence is defined as the individual’s thoughts, feelings, attitudes or behaviors can affect others when interacting with other individuals or groups. DMs with higher social impact have the ability to influence the opinions of other members [47]. Using social network analysis techniques, the social impact of each DM in the group can be obtained. The more social relations a person has with others, the more influence the decision maker will be. In the social network analysis method, the centrality of the in-degree is used to quantify the social influence of DM in a network, and the social influence can reflect the importance of DM to a certain extent. Therefore, the social impact of er,r=1,2,…,m is defined by Eq. (15), and the trust weight is determined by Eq. (16).

(15)d′−(er)=1m−1∑h=1mTShr,∀r=1,2,…,m

(16)wr2=d′−(er)∑h=1md′−(er),∀r=1,2,…,m

Then the comprehensive weight of each DM can be calculated as w=θwr2+(1−θ)wr1.

The network weight between subgroups can be calculated from the reciprocal of the distance from each subgroup to the network center. The further the distance, the smaller the weight. The main steps for calculating weights are shown below [5]:

Step 1: Calculate the fusion centrality of the network.

(17)CF=1|M|∑r∈MCF(er)

where M={e1,e2,…,er,…,et} represents all nodes of the network.

Step 2: Calculate the fusion centrality of each subgroup.

(18)CFk=1|Sk|∑r∈SkCF(er)

where M={S1∪S2∪…∪Sk∪…∪Sp}, Sk={e1,e2,…,er,…,es}, |Sk| represents the number of all nodes of the subgroup ck, er is the nodes in the subgroup ck.

Step 3: Calculate the relative distance λk′ of CF(ck) and CF in community ck.

(19)λk′=1|CFk−CF|

Step 4: Standardize the weight λk′.

(20)λk′=λk′∑k=1pλk′

The trust relation between subgroups can be reconstructed by treating all nodes in a subgroup as a new node, and then calculating the trust weights between subgroups. Finally, the network weights and trust weights of the subgroups are integrated, and the comprehensive weights of the subgroups are obtained.

Next, the consensus measure will be computed based on Ht,t=1,2,…,K. Based on the work of Herrera-Viedma et al. [47], the main steps of the consensus measure are as follows:

Step 1: Calculate the similarity matrix. Let Hk=(hpij,k)n×n and Ht=(hpij,t)n×n correspond to the probability distribution of subgroup Gk and Gt based on the fuzzy preference relationship. Where hpij,k={hij,kl(pij,kl)|l=1,2,…,#hpij,k} or hpij,t={hij,tl(pij,tl)|l=1,2,…,#hpij,t} is a probability distribution based on hesitant fuzzy elements. For each pair of subgroups Gk and Gt(k<t,t∈{1,2,…,K}), the similarity matrix SMkt=(smijkt)n×n is calculated as follows:

(21)SMkt=(−⋯sm1nkt⋮⋱⋮smn1kt⋯−)

where smijkt is the similarity of the subgroups Gk and Gt with respect to the alternatives (xi,xj),i,j=1,2,…,n. According to definition 6, smijkt can be expressed as follows:

(22)smijkt=1−d(hpij,k,hpij,t)=∑l1=1#hpij,k∑l2=1#hpij,tpl1ij,k×pl2ij,t|hl1ij,k−hl2ij,t|

Step 2: Calculate the consensus matrix. By aggregating the similarity matrix of each pair of subgroups (Gk,Gt), a consensus matrix CM=(cmij)n×n can be obtained. Then this study normalizes the weights of the subgroup pairs (Gk,Gt), ωkt is defined as follows:

(23)ωkt=wGkwGt∑k=1K−1∑t=k+1KwGkwGt

cmij can be expressed as follows:

(24)cmij=∑k=1K−1∑t=k+1Kωktsmijkt

Step 3: Calculate the large-scale group consensus index (LGCI).

(1) Calculate the consensus degree of the pair of alternatives. The consensus degree cpij of the alternatives pair (xi,xj) can be obtained from the consensus matrix CM=(cmij)n×n. cpij can be expressed as follows:

(25)cpij=cmij,i,j=1,2,…,n,i≠j

(2) Calculate the consensus degree of the alternatives. The consensus degree cai, i=1,2,…,n of each alternative xi can be calculated as follows:

(26)cai=∑j=1,j≠incpijn−1

(3) Calculate the consensus degree of the preference relationship. This study refers to the consensus degree of the large group preference relationship as the consensus index LGCI, and the formula is as follows:

(27)LGCI=∑i=1ncain

The identification rules can obtain alternative comparison pairs, subgroups, and DMs continuously and accurately. Therefore, DMs can change their preferences in a precise way.

Identification Rule 1: Identify alternatives whose preferences need to be changed. The identified alternatives can be expressed as follows:

(28)ALT={xi|cai<LGCI¯,i=1,2,…,n}

Identification Rule 2: Identify the locations that need to be changed. For any xi∈ALT, the identified set of locations can be expressed as follows:

(29)Posi={(i,j)|xi∈ALT∧cpij<LGCI¯,i<j}

Identification Rule 3: Determine the ideal and non-ideal sets for each identified part. For (i,j)∈Posi, ideal set CLUij+ and non-ideal set CLUij− can be expressed as follows:

(30)CLUij+={Gk+(xi,xj)|k+=kargmax{∑t=1,t≠kKsmijtk}}

(31)CLUij−={G1,G2,…,Gk}∖CLUij+

Identification Rule 4: Identify the DMs who need to change their preferences. For the identified non-ideal set Gk−(xi,xj), this study can use a deeper consensus measure to determine the DMs with only the lowest level of consensus.

The DMs in the non-ideal set Gk−(xi,xj) can be expressed as {ek1,ek2,…,ekG(k−)}. G(k−)=#Gk−(xi,xj) is the number of DMs in Gk−(xi,xj). Note that the FPR of eky is Bky=(bij,ky)n×n. Similarly, the DMs in Gk+(xi,xj) can be represented as {et1,et2,…,etG(k+)}. G(k+)=#Gk+(xi,xj) is the number of DMs in Gk+(xi,xj). The preference of Gk+(xi,xj) is a PDHFE, which can be expressed by hpij,k+={hij,kl+(pij,k+l)|l=1,2,…,#hpij,k+}. The average preference for hpij,k+ can be calculated as follows:

(32)Expect(hpij,k+)=∑l=1#hpij,k+hij,kl+pij,k+l

Set thresholds β. The DMs who need to change their preferences (xi,xj) in Gk−(xi,xj) can calculate as follows:

(33)EXPSij−={eky|bij,ky−Expect(hpij,k+)|>β,ky∈{k1,k2,…,kG(k−)}}

For each eky∈EXPSij−, there are two directions rules as follows [10]:

Directions Rule 1: If bij,ky<Expect(hpij,k+), then eky needs to increase the preference value of (xi,xj).

Directions Rule 2: If bij,ky>Expect(hpij,k+), then eky needs to decrease the preference value of (xi,xj).

One way to achieve the direction rule is to provide a set of values for eky∈EXPSij−. Since each element in the initial eky belongs to S[0.1,0.9], it is better to set the range of values of all recommended values in this interval.

Let Rndij,k+=Round(Expect(hpij,k+)), where Round is a method to find the nearest discrete membership in the range of S[0.1,0.9]. Then the recommended set eky can be expressed as follows:

RSeky(xi,xj)={bij,ky′|bij,ky′≠bij,ky∧bij,ky′∈(S[0.1,0.9]∧[min{bij,ky,Rndij,k+},max{bij,ky,Rndij,k+}])} By the opposite nature of each other, the preference eky and (xi,xj) can be automatically adjusted and updated.

After completing the above adjustment process, the final step is to modify the original social network.

Step 1: Let (i,j) be the position that needs to be modified, E′={e′1,e′2,…,e′v} is a set of DMs whose opinions need to be modified.

Step 2: Calculate the TSrh value according to Eq. (5), and use it as an index to evaluate the trust degree between members. Find the most trusted member E′={e′1,e′2,…,e′v} in the ideal set CLUij+, denoted as F′={f′1,f′2,…,f′v}.

Step 3: Modify the original social network. According to E′={e′1,e′2,…,e′v} and F′={f′1,f′2,…,f′v}, connect ei′ and fi′, where i=1,2,…,v.

A network corresponding to Fig. 3 is constructed in the complex network analysis tool Pajek, as shown in Fig. 4, where nodes v1−v20 correspond to nodes 1–20 in Fig. 3, respectively.

Using the Lauvain package included in the Pajek software, automatic classification is performed to form the classification results as shown in Fig. 5. The subgroup is represented by the set {G1,G2,…,GKr}, which is divided into three subgroups. The subgroup consisting of yellow nodes is G1, the subgroup consisting of green nodes is G2, and the subgroup consisting of red nodes is G3, i.e., G1={v1,v3,v5,v6,v9,v10,v11,v18}, G2={v2,v12,v14,v15,v16,v19}, G1={v4,v7,v8,v13,v17,v20}.

For the subgroup G1, the network is shown in Fig. 6.

According to the original data required by the software and the original data required for the centrality of the feature vector, the network weight of each node in the subgroup G1 can be obtained according to the Eq. (14), as shown in Table 1.

According to the network weights and trust weights, the weight of each DM is aggregated. And this study takes θ=0.2, then, the comprehensive weights of the nodes in the subgroup G1 are shown in Table 1.

According to the calculation results by Pajek software and Section 3.2, the network weight of each subgroup is {0.05,0.04,0.19}.

Considering each subgroup as a node, Fig. 3 can be abstracted into the structure shown in Fig. 7, where the direction of the edge represents the relationship between nodes, and the thickness of the edge represents the connection strength.

The trust weights between subgroups are {0.40,0.26,0.34}.

Similarly, by aggregating the network weights and trust weights of each subgroup, the comprehensive weight of each subgroup can be obtained {0.33,0.22,0.45}.

According to their preference for alternatives provided by the 20 DMs through FPR, the preference relationship of each subgroup also forms a probability distribution based on fuzzy preference relations [51]. According to Section 3.4, the corresponding Gk,k=1,2,3 has three PDFPRs, which can be expressed as Hk=(hpij,k)n×n,k=1,2,3.

For the subgroup G1, the PDFPR can be expressed as follows:

hp12,1={0.1(0.14),0.3(0.10),0.4(0.25),0.6(0.25),0.7(0.13),0.9(0.13)}

hp13,1={0.4(0.38),0.6(0.49),0.9(0.13)},

hp14,1={0.2(0.13),0.4(0.10),0.6(0.29),0.7(0.22),0.8(0.26)},

hp23,1={0.1(0.10),0.3(0.28),0.6(0.27),0.7(0.13),0.8(0.10),0.9(0,12)}

hp24,1={0.4(0.29),0.5(0.10),0.7(0.26),0.8(0.23),0.9(0.12)},

hp34,1={0.3(0.14),0.4(0.23),0.6(0.13),0.7(0.15),0.9(0.35)}.

For the subgroup G2, the PDFPR can be expressed as follows:

hp12,2={0.3(0.29),0.4(0.15),0.6(0.21),0.7(0.15),0.9(0.20)},

hp13,2={0.2(0.21),0.4(0.30),0.6(0.18),0.7(0.31)},

hp14,2={0.2(0.15),0.3(0.21),0.4(0.18),0.5(0.15),0.8(0.31)},

hp23,2={0.1(0.30),0.3(0.11),0.4(0.21),0.6(0.18),0.8(0.20)},

hp24,2={0.2(0.30),0.3(0.21),0.6(0.29),0.7(0.20)},

hp34,2={0.1(0.20),0.3(0.11),0.4(0.51),0.6(0.18)}.

For the subgroup G3, the PDFPR can be expressed as follows:

hp12,3={0.1(0.17),0.2(0.17),0.4(0.38),0.6(0.27)},

hp13,3={0.2(0.08),0.4(0.35),0.6(0.17),0.7(0.22),0.8(0.17)},

hp14,3={0.1(0.35),0.3(0.08),0.4(0.17),0.6(0.39)},

hp23,3={0.3(0.19),0.4(0.30),0.5(0.16),0.6(0.17),0.8(0.17)},

hp24,3={0.3(0.08),0.4(0.35),0.6(0.17),0.8(0.39)},

hp34,3={0.4(0.08),0.6(0.17),0.7(0.57),0.8(0.17)}.

According to Section 3.4, the consensus degree of the three subgroups can be calculated. First, the similarity matrixes between subgroups are calculated, and the results are as follows:

SM12=(−0.73950.79900.73840.7395−0.69920.72040.79900.6992−0.69350.73840.72040.6935−),

SM13=(−0.74580.81150.70320.7458−0.75940.79110.81150.7594−0.78980.70320.79110.7898−),

SM23=(−0.72740.78740.72240.7274−0.74120.73850.78740.7412−0.68660.72240.73850.6866−).

According to Eq. (13), the normalized weights between subgroups are {0.22,0.47,0.31}.

Second, the consensus of the group is calculated, and the consensus matrix is as follows:

CMround1=(−0.73870.80130.71690.7387−0.74040.75920.80130.7404−0.73660.71690.75920.7366−)

The consensus degree between alternative pairs is cpij=cmij,i,j=1,2,3,4.

Third, the consensus degree at the level of the alternative is ca1=7523, ca2=7461, ca3=7594, ca4=7376.

Finally, the consensus degree of the group is LGCI=0.7488.

Set LGCI¯=0.8. Because LGCI<LGCI¯, the feedback adjustment mechanism can be activated. According to Rule 1 and Rule 2 in Section 3.3.1, the preference relationship on (x2,x3) needs to be modified. According to Rule 3, ideal subgroups and non-ideal subgroups can be obtained:

CLU23+={G3}, CLU23−={G1,G2}.

Let the parameter β=0.3 and calculate Expect(hp23,3)=0.495, then this study can identify the DMs that need to readjust the preference value in subgroup G1 and subgroup G2.

DMs in subgroup G1:e5,e9,

DMs in subgroup G2:e15,e16.

Find the most trusted person for DMs e5,e9,e15,e16 in subgroup G3. The most trusted persons for e5,e9,e15,e16 are e8,e17,e8,e8, respectively. The new preference value of the DM is changed to the preference value of the most trusted person [52].

First round of adjustment:

According to Section 3.2.1, this study can get a new social network based on the Louvain method and a new subgroup classified as shown in Figs. 8 and 9.

As can be seen from the Fig. 9, three new subgroups can be obtained in this adjustment: G1={v1,v3,v5,v8,v18}, G2={v2,v6,v12,v14,v15,v16,v19}, G3={v4,v7,v9,v10,v11,v13,v17,v20}. Then the trust weight and the network weight are integrated to determine the relative weight between the subgroups as shown in Table 2. And the weight of each subgroup is {0.21,0.25,0.54}.

For the subgroup G1, the PDFPR can be expressed as follows:

hp12,1={0.1(0.16),0.4(0.42),0.6(0.25),0.9(0.17)},

hp13,1={0.4(0.43),0.6(0.16),0.7(0.24),0.9(0.17)},