# Distrust Behavior in Social Network Large-Scale Group Decision Making and Its Application in Water Pollution Management

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- The AHC method is utilized to decrease the complexity of LSGDM-SN with the FPR, in which both the trust relationships and distrust relationships among experts are incorporated into the clustering method. Then, the concept of preference similarity, trust similarity, and distrust similarity are proposed to compute the degree of overall similarity among experts. Meanwhile, the algorithm for the AHC of LSGDM-SN is presented.
- (2)
- Consensus feedback based on distrust behavior and social network analysis (SNA) is presented to encourage the subset to modify its FPR based on different distrust types. In the identification process, both the distrust score and the degree of difference are incorporated to measure the distrust degree of the subset. Based on the cases of distrust behaviors, two pieces of feedback advice are provided to the subset to adjust its FPR.
- (3)
- A score function of FPR is designed to choose the best alternative for water pollution management. By calculating the final score of the collective FPR, we rank all alternatives. Then, the one with the highest score is selected as the best solution for water pollution management. Finally, a framework for the proposed LSGDM-SN considering distrust behavior is depicted to visualize the decision process.

## 2. Preliminaries

#### 2.1. Fuzzy Preference Relation

_{1}, x

_{2}, …, x

_{n}} be the set of alternatives, where N = {1, 2, …, n}. E = {e

_{1}, e

_{2}, …, e

_{m}} denotes the set of experts in the LSGDM-SN problem, they utilize FPRs to express their evaluations toward alternatives.

**Definition**

**1**

**.**The FPR R of all alternatives X = {x

_{1}, x

_{2}, …, x

_{n}} is a fuzzy relation on the product set X × X with membership function μ

_{R}: X × X → [0, 1], μ

_{R}(x

_{i}, x

_{j}) = r

_{ij}, satisfying

_{ij})

_{n}

_{×n}is used to represent the FPR, where r

_{ij}is the preference degree of x

_{i}over x

_{j}. Particularly, 0 ≤ r

_{ij}< 0.5 signifies that the expert prefers x

_{j}to x

_{i}. If 0.5 < r

_{ij}< 1, it means x

_{i}is preferred to x

_{j}by experts. In particular, r

_{ij}= 0.5 means that the expert has no preference for x

_{i}and x

_{j}, and r

_{ij}= 1 indicates that x

_{i}is definitely preferred to x

_{j}.

**Definition**

**2**

**.**An FPR R = (r

_{ij})

_{n}

_{×n}has additive consistency if r

_{ij}+ r

_{jl}+ r

_{li}= 1.5, for ∀ i, l, j ∈ N.

**Definition**

**3**

**.**Let H = (h

_{ij})

_{n×n}be an FPR, it is an additive consistency FPR if it satisfies

_{ij}+ r

_{ji}= 1.

#### 2.2. SNA

- (1)
- Socio-matrix: The matrix G = (g
_{kq})_{m}_{×m}is utilized to present the relationships data among experts. If expert e_{k}has no relationship with e_{q}, the value of g_{kq}is equal to 0. Otherwise, the value of g_{kq}is 1 if there is an existing relationship between e_{k}and e_{q}. - (2)
- Graph: Social networks can also be represented by a graph, where the points are connected by straight lines. In the network diagram, e
_{k}→e_{q}indicates that there is a direct relationship from e_{k}to e_{q}. - (3)
- Algebraic: The algebraic can denote the social relationships among experts and the relationship combinations.

**Definition**

**4**

**.**Let t

_{kq}and d

_{kq}be the trust degree and distrust degree from e

_{k}to e

_{q}, respectively. The set of trust functions is denoted by tf

_{kq}= (t

_{kq}, d

_{kq}), where t

_{kq}, d

_{kq}∈ [0, 1].

**Example**

**1**.

_{kq})

_{m×m}between experts can be expressed by:

_{13}= (0.2, 0.6) signifies that the trust value from e

_{k}to e

_{q}is 0.2, while the distrust value from e

_{k}to e

_{q}is 0.6.

_{k}trusts e

_{q}, and e

_{q}trusts e

_{y}, then e

_{k}may trust e

_{y}to some extent. That is, the expert e

_{q}acts as an intermediary to achieve trust propagation from e

_{k}to e

_{y}. In a social network, trust can be propagated by one or more intermediaries. However, if there is more than one middleman to propagate information, the information may be distorted. Therefore, in order to ensure the accuracy of information propagation, this paper assumes that information can only be transmitted by one intermediary. Taking Figure 1 as an example, there is no direct trust and distrust from e

_{k}to e

_{y}, but trust and distrust can be propagated through e

_{k}→e

_{q}→e

_{y}.

_{k}→e

_{q}→e

_{y}is the information propagation path from e

_{k}to e

_{y}, and the indirect trust function value from e

_{k}to e

_{y}can be obtained using the uninorm trust propagation operator

_{kq}, d

_{kq}) is the trust function from e

_{k}to e

_{q}, and (t

_{qy}, d

_{qy}) is the trust function from e

_{q}to e

_{y}.

_{k}to e

_{y}, as shown in Figure 2. There are two trust function propagation paths from e

_{k}to e

_{y}: e

_{k}→e

_{q}

_{1}→e

_{y}and e

_{k}→e

_{q}

_{2}→e

_{y}. We propose the trust aggregation operator in the process of trust propagation according to the weighted average operator.

**Definition**

**5.**

_{k}to e

_{y}in path γ, the indirect trust function value from e

_{k}to e

_{y}is defined as

**Definition**

**6.**

_{kq}= (t

_{kq}, d

_{kq}) (k, q ∊ E) be the trust function from e

_{k}to e

_{q}, and tso

_{q}and dso

_{q}are the trust score and distrust score from e

_{k}to e

_{q}, respectively. The values of tso

_{q}and dso

_{q}are calculated by

_{q}and dso

_{q}are within the interval [0, 1].

## 3. Large-Scale Expert Clustering Based on AHC Approach

**Definition**

**7.**

_{k}and e

_{q}, respectively. Let PS

_{kq}= (ps

_{kq})

_{n×n}be the preference similarity matrix among experts e

_{k}and e

_{q}. Based on the Manhattan distance, the value of ps

_{kq}is calculated by

_{kq}is within the interval [0, 1], and ps

_{kq}= ps

_{qk}.

**Definition**

**8.**

_{kq})

_{m×m}be the complete matrix of the trust function, where tf

_{kq}= (t

_{kq}, d

_{kq}) is the trust function value given by e

_{k}to e

_{q}. The trust similarity ts

_{kq}and distrust similarity ds

_{kq}between experts e

_{k}and e

_{q}are defined as follows

_{k}is the trust score of expert e

_{k}, and dso

_{k}denotes the distrust score of expert e

_{k}.

**Definition**

**9.**

_{kq}, ts

_{kq}, and ds

_{kq}be the preference similarity, trust similarity, and distrust similarity between experts e

_{k}and e

_{q}, respectively. The overall similarity os

_{kq}between experts e

_{k}and e

_{q}is defined as follows

_{kq}, the higher the similarity between experts e

_{k}and e

_{q}.

^{p}(p = 1,…, υ), and experts in subset su

^{p}are denoted as ${e}_{k}^{p}$ (k = 1,…,|su

^{p}|).

Algorithm 1. The detailed AHC method. |

Input$:\text{}\mathrm{the}\text{}\mathrm{FPRs}\text{}\mathrm{of}\text{}m\text{}\mathrm{experts},\text{}\mathrm{complete}\text{}\mathrm{matrix}\text{}\mathrm{of}\text{}\mathrm{trust}\text{}\mathrm{function}\text{}\mathit{TF}={\left({\mathit{tf}}_{\mathit{kq}}\right)}_{m\times m},\text{}\mathrm{the}\text{}\mathrm{parameter}\text{}\mathit{\lambda},\text{}\mathrm{the}\text{}\mathrm{number}\text{}\mathrm{of}\text{}\mathrm{subsets}\mathit{\upsilon}$.Output: subsets su^{1}, …, su^{p}, …, su^{υ}.Step 1: Regard each expert e_{k} (k =1, …, m) as one initial subset.Step 2: Calculating the values of ps_{kq}, ts_{kq}, ds_{kq} and os_{kq} for each initial subset based on Equations (8)–(11).Step 3: Selecting the maximum os_{kq}, then classify initial subsets e_{q} and e_{k} into one new subset.Step 4: Deleting subsets e_{q} and e_{k} in Step 3, the overall similarity is recalculated based on Definition 9.Step 5: Steps 3 and 4 are repeated constantly. If all subsets are merged into one subset, the process is ended.Step 6: Setting the number of subsets$\upsilon $, output the clustering result su^{1}, …, su^{p}, …, su^{υ}. |

**Remark**

**1.**

## 4. Consensus of LSGDM-SN Based on Distrust Behavior and SNA

#### 4.1. Consensus Measure

**Definition**

**10.**

^{p}, ${\omega}^{p,1}$, ${\omega}^{p,2}$,…, ${\omega}^{p,\left|{\mathit{su}}^{p}\right|}$ be the set of weights of experts in subset su

^{p}. The FPR ${R}^{p}{=(r}_{\mathit{ij}}^{p}{)}_{n\times n}$ of subset su

^{p}is computed by

^{p}| is the number of experts in subset su

^{p}. The weight of expert ω

^{p}

^{,k}can be obtained by ω

^{p}

^{,k}= 1/|su

^{p}|.

^{p}represents the weight of subset su

^{p}and it is obtained by φ

^{p}= 1/υ. Clearly, the value of φ

^{p}is within [0, 1] and ${{\displaystyle \sum}}_{p=1}^{\upsilon}{\phi}^{p}$= 1.

**Definition**

**11.**

^{p}and collective CLC can be determined by

#### 4.2. Consensus Feedback Based on Distrust Behavior and SNA

**Definition**

**12.**

^{p,1}, …, dso

^{p,k},…, ${\mathit{dso}}^{{p,|\mathit{su}}^{p}|}$ be the distrust scores of experts in subset su

^{p}, and ${R}^{p}{=(r}_{\mathit{ij}}^{p}{)}_{n\times n}$ be the FPR of subset su

^{p}, $\overline{R\text{}}{=\left({\overline{r}}_{\mathit{ij}}\right)}_{n\times n}$ be the collective FPR, the distrust degree dd

^{p}of subset su

^{p}toward moderator is defined by

^{p}with the minimum consensus level is found, the moderator works to provide modification advice for the subset. Given the threshold of distrust degree β, the corresponding modification strategy for the subset su

^{p}with minimum consensus level is summarized in the following two cases.

**Case**

**1.**

^{b}≤ β, the subset trusts in the moderator to some extent. That is, the subset is willing to adopt the suggestions from the moderator. Therefore, the FPR of the subset is changed as follows.

**Case**

**2.**

^{b}> β, the subset absolutely distrusts the suggestion of the moderator in the feedback process. In this case, we assume that the subset can be influenced by the subset with the maximum consensus level. Hence, the FPR of the subset is modified by

^{p}

^{,(}

^{τ}

^{)}} means the subset su

^{p}with maximum consensus level in iteration τ.

**Remark**

**2.**

#### 4.3. Selection Process

_{i}is

#### 4.4. The Proposed Framework of LSGDM-SN

## 5. Case Study

_{1}, x

_{2}, x

_{3}, and x

_{4}. Subsequently, 20 experts with social relationships from different sectors of society are invited to participate in the LSGDM-SN [2,4,5,6]. The detailed evaluation of 20 experts on the four alternatives is shown in Table 2. Moreover, the social assessments between the 20 experts are depicted in Table 3. Additionally, some related parameters are predefined as follows.

^{1}= {e

_{7}, e

_{10}, e

_{13}}, su

^{2}= {e

_{2}, e

_{3}, e

_{4}, e

_{9}, e

_{11}, e

_{17}, e

_{19}}, su

^{3}= {e

_{6}, e

_{15}}, su

^{4}= {e

_{1}, e

_{5}, e

_{8}}, su

^{5}= {e

_{14}, e

_{18}}, su

^{6}= {e

_{16}, e

_{20}}, su

^{7}= {e

_{12}}. This is depicted in Figure 4.

^{1,(0)}= 0.94, CLS

^{2,(0)}= 0.95, CLS

^{3,(0)}= 0.92, CLS

^{4,(0)}= 0.94, CLS

^{5,(0)}= 0.91, CLS

^{6,(0)}= 0.92, and CLS

^{7,(0)}= 0.88. Through Equation (15), we have CLC

^{(0)}= 0.92. Because CLC

^{(0)}< $\overline{\mathit{CLC}}$, we continue the feedback process.

^{7,(0)}= min{CLS

^{p}

^{,(τ)}|p = 1,2,…,7} = 0.88, the consensus based on distrust behavior and SNA is conducted on subset su

^{7,(0)}. By Equation (16), we have dd

^{7,(0)}= 0.33. Since dd

^{7,(0)}> β = 0.3, the subset su

^{7,(0)}absolutely distrusts the suggestion of the moderator. Then, the FPR of subset su

^{7,(0)}is modified according to Equation (18). The new FPR R

^{7,(0)}can be found in Table 4. After four iterations, the consensus feedback is ended. The detailed consensus based on distrust behavior and SNA is expressed in Table 4.

_{1}) = 0.50, fs(x

_{2}) = 0.51, fs(x

_{3}) = 0.46, and fs(x

_{4}) = 0.52. Therefore, the ranking of the four alternatives is x

_{4}> ≻x

_{2}> ≻x

_{1}> ≻x

_{3}. Consequently, the best solution is x

_{4}.

## 6. Comparative Analysis and Discussion

#### 6.1. The Impact of Distrust Behavior on Alternative Ranking

^{b}is only based on Equation (17). Then, by Equation (19), the final score of each alternative can be derived. The detailed results without the identification and management of distrust behavior are given in Table 5.

#### 6.2. The Impact of Weight Determination on Decision Result

#### 6.3. Sensitivity Analysis of Consensus Threshold

#### 6.4. The Comparison with Other Related Studies

## 7. Conclusions

- (1)
- A novel AHC method considering preference similarity and social similarity is proposed to decrease the complexity of LSGDM-SN with FPRs. Several definitions, including preference similarity, trust similarity, and distrust similarity, are proposed to compute the degree of overall similarity among experts. Subsequently, the AHC algorithm dealing with the LSGDM-SN problem is designed.
- (2)
- The consensus feedback for detecting and managing distrust behavior is presented, which encourages the subset to modify its FPR based on different distrust types. To identify the distrust behavior, both the distrust score and the degree of difference are incorporated to measure the distrust degree of the subset. Based on the value of distrust degree, two pieces of modification advice are provided to subset to modify it FPR.
- (3)
- A score function is defined to derive the alternatives ranking in water pollution management. After computing the values of scores for all alternatives, we rank them. The optimal alternative is obtained based on the maximum value score. Finally, the LSGDM-SN framework considering distrust behavior is described to visualize the decision process.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Socio-Matrix | Graph | Algebraic |
---|---|---|

G = $\left(\begin{array}{ccccc}0& 0& 1& 0& 0\\ 1& 0& 0& 0& 0\\ 0& 1& 0& 1& 0\\ 0& 0& 0& 0& 1\\ 1& 0& 0& 1& 0\end{array}\right)$ | e_{1}Be_{3}e _{2}Be_{1}e _{3}Be_{2}, e_{3}Be_{4}e _{1}Be_{5}e _{5}Be_{4}, e_{5}Be_{1} |

Experts | FPR | Experts | FPR |
---|---|---|---|

e_{1} | $\left(\begin{array}{cccc}0.5& 0.4& 0.5& 0.9\\ 0.6& 0.5& 0.4& 0.2\\ 0.5& 0.6& 0.5& 0.2\\ 0.1& 0.8& 0.8& 0.5\end{array}\right)$ | e_{2} | $\left(\begin{array}{cccc}0.5& 0.7& 0.1& 0.1\\ 0.3& 0.5& 0.4& 0.2\\ 0.9& 0.6& 0.5& 0.7\\ 0.9& 0.8& 0.3& 0.5\end{array}\right)$ |

e_{3} | $\left(\begin{array}{cccc}0.5& 0.7& 0.8& 0.9\\ 0.3& 0.5& 0.8& 1.0\\ 0.2& 0.2& 0.5& 0.2\\ 0.1& 0& 0.8& 0.5\end{array}\right)$ | e_{4} | $\left(\begin{array}{cccc}0.5& 0& 0.1& 0.6\\ 1.0& 0.5& 0.2& 0.2\\ 0.9& 0.8& 0.5& 0.2\\ 0.4& 0.8& 0.8& 0.5\end{array}\right)$ |

e_{5} | $\left(\begin{array}{cccc}0.5& 0.1& 0.9& 0.3\\ 0.9& 0.5& 0.5& 0.6\\ 0.1& 0.5& 0.5& 0.1\\ 0.7& 0.4& 0.9& 0.5\end{array}\right)$ | e_{6} | $\left(\begin{array}{cccc}0.5& 0.3& 0.5& 0.1\\ 0.7& 0.5& 0.3& 0.5\\ 0.5& 0.7& 0.5& 0.3\\ 0.9& 0.5& 0.7& 0.5\end{array}\right)$ |

e_{7} | $\left(\begin{array}{cccc}0.5& 0.9& 0.7& 1.0\\ 0.1& 0.5& 0.4& 0.1\\ 0.3& 0.6& 0.5& 0.3\\ 0& 0.9& 0.7& 0.5\end{array}\right)$ | e_{8} | $\left(\begin{array}{cccc}0.5& 0.7& 0.4& 0\\ 0.3& 0.5& 0.7& 0.5\\ 0.6& 0.3& 0.5& 0.9\\ 1.0& 0.5& 0.1& 0.5\end{array}\right)$ |

e_{9} | $\left(\begin{array}{cccc}0.5& 0.9& 0.8& 0.2\\ 0.1& 0.5& 0.7& 0.7\\ 0.2& 0.3& 0.5& 0.8\\ 0.8& 0.3& 0.2& 0.5\end{array}\right)$ | e_{10} | $\left(\begin{array}{cccc}0.5& 0.7& 0.9& 0.5\\ 0.3& 0.5& 0.9& 0.6\\ 0.1& 0.1& 0.5& 0.5\\ 0.5& 0.4& 0.5& 0.5\end{array}\right)$ |

e_{11} | $\left(\begin{array}{cccc}0.5& 0.5& 1.0& 0.2\\ 0.5& 0.5& 0.2& 0.9\\ 0& 0.8& 0.5& 0.7\\ 0.8& 0.1& 0.3& 0.5\end{array}\right)$ | e_{12} | $\left(\begin{array}{cccc}0.5& 0.6& 0.5& 0.6\\ 0.4& 0.5& 0.5& 0.8\\ 0.5& 0.5& 0.5& 0.5\\ 0.4& 0.2& 0.5& 0.5\end{array}\right)$ |

e_{13} | $\left(\begin{array}{cccc}0.5& 0.2& 1.0& 0\\ 0.8& 0.5& 0.2& 0.7\\ 0& 0.8& 0.5& 0.4\\ 1.0& 0.3& 0.6& 0.5\end{array}\right)$ | e_{14} | $\left(\begin{array}{cccc}0.5& 0.2& 0.3& 0.9\\ 0.8& 0.5& 0.4& 0.4\\ 0.7& 0.6& 0.5& 0.3\\ 0.1& 0.6& 0.7& 0.5\end{array}\right)$ |

e_{15} | $\left(\begin{array}{cccc}0.5& 0.6& 0.8& 0.2\\ 0.4& 0.5& 0.7& 0.3\\ 0.2& 0.3& 0.5& 0.9\\ 0.8& 0.7& 0.1& 0.5\end{array}\right)$ | e_{16} | $\left(\begin{array}{cccc}0.5& 0.4& 0.4& 0.8\\ 0.6& 0.5& 0.7& 0.7\\ 0.6& 0.3& 0.5& 0.5\\ 0.2& 0.3& 0.5& 0.5\end{array}\right)$ |

e_{17} | $\left(\begin{array}{cccc}0.5& 0.8& 0.4& 0.8\\ 0.2& 0.5& 0.7& 0.1\\ 0.6& 0.3& 0.5& 0.1\\ 0.2& 0.9& 0.9& 0.5\end{array}\right)$ | e_{18} | $\left(\begin{array}{cccc}0.5& 0.6& 0.7& 0.1\\ 0.4& 0.5& 0.5& 0.8\\ 0.3& 0.5& 0.5& 0.1\\ 0.9& 0.2& 0.9& 0.5\end{array}\right)$ |

e_{19} | $\left(\begin{array}{cccc}0.5& 0.2& 0.3& 0.2\\ 0.8& 0.5& 0.7& 0.9\\ 0.7& 0.3& 0.5& 0.3\\ 0.8& 0.1& 0.7& 0.5\end{array}\right)$ | e_{20} | $\left(\begin{array}{cccc}0.5& 0.4& 0.8& 0.2\\ 0.6& 0.5& 0.9& 0.2\\ 0.2& 0.1& 0.5& 0.2\\ 0.8& 0.8& 0.8& 0.5\end{array}\right)$ |

e_{1} | e_{2} | e_{3} | … | e_{18} | e_{19} | e_{20} | |
---|---|---|---|---|---|---|---|

e_{1} | (1, 0) | (0.9, 0.3) | (0.8, 0.7) | (0.6, 1.0) | (-, -) | (-, -) | |

e_{2} | (-, -) | (1, 0) | (0.8, 0.3) | (-, -) | (0.7, 0.9) | (-, -) | |

e_{3} | (0.3, 0.3) | (0.1, 0.6) | (1, 0) | (0.8, 0.4) | (-, -) | (-, -) | |

… | |||||||

e_{18} | (-, -) | (-, -) | (-, -) | (1, 0) | (0.1, 0.1) | (0.8, 0.5) | |

e_{19} | (-, -) | (-, -) | (0.7, 1.0) | (0.4, 0.9) | (1, 0) | (-, -) | |

e_{20} | (-, -) | (0.7, 0.2) | (0.1, 0.7) | (0.3, 0.4) | (-, -) | (1, 0) |

τ | CLC ^{(τ)} | su^{p,(τ)} | su^{p,(τ)} < β | FPR |
---|---|---|---|---|

0 | 0.92 | su^{7,(0)} | No | $\left(\begin{array}{cccc}0.50& 0.58& 0.48& 0.52\\ 0.42& 0.50& 0.50& 0.67\\ 0.52& 0.50& 0.50& 0.46\\ 0.48& 0.33& 0.54& 0.50\end{array}\right)$ |

1 | 0.93 | su^{5,(0)} | Yes | $\left(\begin{array}{cccc}0.50& 0.42& 0.53& 0.50\\ 0.58& 0.50& 0.51& 0.57\\ 0.47& 0.49& 0.50& 0.29\\ 0.50& 0.43& 0.71& 0.50\end{array}\right)$ |

2 | 0.94 | su^{6,(0)} | Yes | $\left(\begin{array}{cccc}0.50& 0.44& 0.56& 0.52\\ 0.56& 0.50& 0.68& 0.50\\ 0.44& 0.32& 0.50& 0.38\\ 0.48& 0.50& 0.62& 0.50\end{array}\right)$ |

3 | 0.94 | su^{3,(0)} | No | $\left(\begin{array}{cccc}0.50& 0.50& 0.57& 0.50\\ 0.50& 0.50& 0.53& 0.50\\ 0.43& 0.47& 0.50& 0.49\\ 0.50& 0.50& 0.51& 0.50\end{array}\right)$ |

4 | 0.95 |

Collective FPR | fs(x_{i}) | Ranking of Alternatives |
---|---|---|

$\left(\begin{array}{cccc}0.50& 0.48& 0.55& 0.49\\ 0.52& 0.50& 0.54& 0.53\\ 0.45& 0.46& 0.50& 0.40\\ 0.51& 0.47& 0.60& 0.50\end{array}\right)$ | fs(x_{1}) = 0.505fs(x _{2}) = 0.523fs(x _{3}) = 0.453fs(x _{4}) = 0.520 | x_{2} > ≻x_{4} > ≻x_{1} > ≻x_{3} |

Consensus Iteration | fs(x_{i}) | Ranking of Alternatives |
---|---|---|

5 | fs(x_{1}) = 0.511fs(x _{2}) = 0.519fs(x _{3}) = 0.451fs(x _{4}) = 0.518 | x_{2} > ≻x_{4} > ≻x_{1} > ≻x_{3} |

**Table 7.**The consensus iteration and ranking of alternatives with different $\overline{\mathit{CLC}}$.

θ | τ | fs(x_{1}) | fs(x_{2}) | fs(x_{3}) | fs(x_{4}) | Ranking of Alternatives |
---|---|---|---|---|---|---|

0.92 | 0 | 0.509 | 0.527 | 0.453 | 0.512 | x_{2} > ≻x_{4} > ≻x_{1} > ≻x_{3} |

0.93 | 1 | 0.506 | 0.526 | 0.448 | 0.520 | x_{2} > ≻x_{4} > ≻x_{1} > ≻x_{3} |

0.94 | 3 | 0.509 | 0.519 | 0.455 | 0.517 | x_{2} > ≻x_{4} > ≻x_{1} > ≻x_{3} |

0.95 | 4 | 0.504 | 0.518 | 0.457 | 0.521 | x_{4} > ≻x_{2} >≻x_{1} > ≻x_{3} |

0.96 | 7 | 0.509 | 0.518 | 0.453 | 0.520 | x_{4} > ≻x_{2} > ≻x_{1} > ≻x_{3} |

0.97 | 10 | 0.508 | 0.520 | 0.454 | 0.518 | x_{2} > ≻x_{4} > ≻x_{1} > ≻x_{3} |

0.98 | 14 | 0.508 | 0.519 | 0.453 | 0.520 | x_{4} > ≻x_{2} > ≻x_{1} > ≻x_{3} |

0.99 | 22 | 0.507 | 0.520 | 0.453 | 0.519 | x_{2} > ≻x_{4} > ≻x_{1} > ≻x_{3} |

1 | 344 | 0.508 | 0.519 | 0.453 | 0.520 | x_{4} > x_{2} > x_{1} > x_{3} |

Method | Clustering Method | Social Relationship | FPR | Distrust Behavior | Water Pollution Problem |
---|---|---|---|---|---|

Lu et al. [40] | K-means clustering | √ | × | × | × |

Liu et al. [15] | DM clustering | × | × | × | × |

Wang et al. [23] | Louvain algorithm | √ | × | × | × |

Meng et al. [4] | Trust-based density peaks clustering | √ | × | × | × |

Our method | AHC method | √ | √ | √ | √ |

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## Share and Cite

**MDPI and ACS Style**

Lu, Y.; Liu, G.; Xu, Y.
Distrust Behavior in Social Network Large-Scale Group Decision Making and Its Application in Water Pollution Management. *Water* **2023**, *15*, 1638.
https://doi.org/10.3390/w15091638

**AMA Style**

Lu Y, Liu G, Xu Y.
Distrust Behavior in Social Network Large-Scale Group Decision Making and Its Application in Water Pollution Management. *Water*. 2023; 15(9):1638.
https://doi.org/10.3390/w15091638

**Chicago/Turabian Style**

Lu, Yanling, Gaofeng Liu, and Yejun Xu.
2023. "Distrust Behavior in Social Network Large-Scale Group Decision Making and Its Application in Water Pollution Management" *Water* 15, no. 9: 1638.
https://doi.org/10.3390/w15091638