# Research Progress of Complex Network Modeling Methods Based on Uncertainty Theory

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## Abstract

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## 1. Introduction

- (1)
- The theoretical basis of complex network modeling research around set pair analysis and rough set and fuzzy set fusion includes: using a set pair connection degree to measure the similarity between complex network vertices and modeling complex networks based on set pair similarity; using rough sets with upper and lower approximate sets to construct a complex network of rough vertices (edges); using three-way decision methods (probabilistic rough sets) to model a complex network; and using the fuzzy set membership degree to describe the relationship between vertices and cliques or between cliques and then to model complex networks.
- (2)
- The modeling methods and applications of set pair analysis, rough set theory and fuzzy set theory in complex networks are summarized and analyzed, including: community discovery, link prediction, influence maximization and decision-making problems. A typical algorithm and an innovative algorithm are analyzed and compared.
- (3)
- The prospect of uncertainty theory in complex network modeling research and its possible extension in the field of uncertain hypergraphs is put forward.

## 2. SPA-Based Complex Network Modeling

#### 2.1. Theoretical Basis of Set Pair Modeling for Complex Networks

#### 2.1.1. The Basis of Set Pair Analysis

#### 2.1.2. Set Pair Similarity Measure between Complex Network Vertices

**Definition**

**1.**

#### 2.1.3. Complex Network Set Pair Relationship Matrix

**Definition**

**2.**

#### 2.1.4. Set Pair Relationship Community Description

#### 2.2. SPA-Based Complex Network Modeling and Application

#### 2.2.1. Community Discovery

- (1)
- Static community discovery

- (2)
- Dynamic community discovery

- (3)
- Overlapping community discovery

#### 2.2.2. Link Prediction

#### 2.2.3. Maximizing Impact

#### 2.2.4. Other Problems

## 3. Complex Network Modeling Based on RS

#### 3.1. Basics of RS Modeling Complex Network

#### 3.1.1. Basic Theory of RS

#### 3.1.2. RS Complex Network

_{V}: In the complex network, for vertex set U

_{V}, there are X

_{V}and R

_{V}, where X

_{V}is a subset of vertex set U

_{V}, and R

_{V}is an equivalence relation of U

_{V}. When X

_{V}is the rough set of R

_{V}, the vertex set of the complex network is said to have rough characteristics, and the complex network is called the rough vertex complex network RCN

_{V};

_{E}: In the complex network, for edge set U

_{E}, there are X

_{E}and R

_{E}, where X

_{E}is a subset of the edge set U

_{E}, and R

_{E}is an equivalence relationship of the U

_{E}. When X

_{E}is the rough set of R

_{E}, it is said that the edge set of the complex network has rough characteristics, and the complex network is the rough edge complex network RCN

_{E};

_{V}and rough edge complex network RCN

_{E}are collectively referred to as rough complex networks.

#### 3.1.3. Rough Complex Network Accuracy Metrics

_{V}(or RCN

_{E}) to the measure of the upper approximate complex network ¯) be the precision of the rough vertex (or edge) complex network, denoted as:

_{V}and the rough edge complex network RCN

_{E}precision, namely:

#### 3.1.4. Coarse Clustering Coefficients for Rough Complex Network

_{i}, $\frac{{k}_{i}\left({k}_{i}-1\right)}{2}$ is the maximum number of edges that may exist between the ${k}_{i}$ vertices, and ${M}_{i}$ is the actual number of edges between the ${k}_{i}$ vertices.

#### 3.2. Rough Set Modeling Method for Complex Networks

#### 3.2.1. Rough Decision-making Model

- (1)
- Rough path and defensive decision making

- (2)
- Rough and complex network decision-making method

#### 3.2.2. Community Discovery

- (1)
- Non-overlapping community discovery

- (2)
- Overlapping community discovery

#### 3.2.3. Other Problems

## 4. Modeling Method Based on Fuzzy Set Theory

#### 4.1. Fuzzy Set Theory Modeling Basis

#### 4.1.1. Fuzzy Membership Function

- (1)
- The membership function must have an upper bound of 1 and a lower bound of 0. That is, the value range of the membership function is [0,1];
- (2)
- For each sample, its membership must be unique. That is, for a fuzzy set, an element can only correspond to one degree of membership.

#### 4.1.2. Fuzzy Clustering Algorithm

- (1)
- Initialization: take the fuzzy weighting index m = 2, the number of clusters C (2 ≤ C ≤ n), where n is the number of data sample points, the iteration stop threshold is $\epsilon $, the initial cluster center value is P
^{(0)}, and the number of iterations l = 0; - (2)
- Calculate the partition matrix U composed of the values of membership degrees U
^{(l)}:For any i, k, ${d}_{ik}$ represents the distance between the sample point ${x}_{k}$ and the i-th class. If ${d}_{ik}{}^{\left(l\right)}>0$, then the membership degree ${\mu}_{ik}$ of the sample point ${x}_{k}$ and the i-th class is:

- (3)
- Update the cluster center value:

- (4)
- If $||{P}^{\left(l+1\right)}-{P}^{\left(l\right)}||<\epsilon $, the algorithm stops; otherwise, go to step (2).

#### 4.1.3. Modularity

#### 4.2. Fuzzy Set Theory Modeling Method

#### 4.2.1. Community Discovery

- (1)
- Topological structure

- (2)
- Fuzzy clustering

#### 4.2.2. Other Problems

## 5. Challenges and Prospects

- (1)
- In traditional graph theory, an edge can only connect two vertices, and it is impossible to model and analyze higher-dimensional situations. A hypergraph is a graph model whose edges can connect multiple vertices. Now, some scholars have constructed an uncertain hypergraph model and applied it to the research of practical problems. However, research in this area still has a number of areas for development;
- (2)
- A three-way decision is an effective rough decision-making model, and the current research on this is far from sufficient. The question of how to reduce time complexity and space complexity as much as possible while ensuring high precision will be a long-term research hotspot;
- (3)
- Some scholars have combined rough set theory and fuzzy set theory to propose a series of models and algorithms. However, the cross-integration of other uncertainty theories and their modeling in complex networks remains to be studied.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**The changing trend of literature related to set pair analysis, rough set and fuzzy set theory.

**Figure 4.**Schematic diagram of non-overlapping community discovery and overlapping community discovery.

Paper Title | Years | Major Contributions |
---|---|---|

Set Pair Social Network Analysis Model and Its Application [17] | 2011 | For the first time, set pair analysis was used in a social network, and a set pair social network analysis model and its related theorems are proposed. |

Set Pair Community Mining and Situation Analysis Based on Web Social Network [19] | 2011 | The set pair connection degree was used in analyzing the sameness, difference and opposites of neighbor nodes, and a set pair community mining algorithm for context analysis is proposed. |

The α Relational Communities of Set Pair Social Network and Its Dynamic Mining Algorithms [15] | 2013 | In the set pair social network, the concept of α-relation community with a given threshold was proposed, and the static and dynamic α-relation community mining algorithms based on set pair analysis theory are given, respectively. |

Research on Community Detection Algorithm Based on the Measure of Set Pair Similarity [20] | 2016 | A similarity measurement method between vertices based on the weighted aggregation coefficient connection degree and a new similarity-based hierarchical clustering algorithm VSFCM were proposed, which are used in the algorithm research of community discovery. |

Research on Community Discovery Based on k-shell [21] | 2018 | Aiming at the problems existing in the VSFCM algorithm, combining the k-shell decomposition method with set pair analysis, the algorithm KPCM and the algorithm KPCMV were proposed and applied to community discovery. |

Measuring Similarity Between Vertices and Its Application in Social Network [22] | 2017 | A new metric, the weighted aggregation coefficient connection degree, was proposed for a traditional social network and applied to social network link prediction and community discovery. The symbolic network is characterized as a system of similarities, differences and antithesis, and a new measure of similarity between vertices was proposed, which is applied to link prediction and dynamic community discovery in a symbolic network. Using the connection degree to describe the similarities, differences and antithesis between vertices of the topic attention network, a new measure of similarity between vertices was proposed and applied to the community discovery and influence maximization research of a topic-attention network. |

Study on the Measure Methods of Similarity between Vertices in Network [16] | ||

Research on Network Community Discovery Methods Based on Topic Concern [23] | 2017 | A topic attention model was constructed, and the similarity between vertices of the model is characterized using the set pair connection degree. With a focus on the characteristics of the topic-attention network, a new measure of similarity between vertices, TANCD, was proposed, and a topic community discovery algorithm based on TANCD was proposed. Combining deep-learning technology with natural language processing, a topic community discovery algorithm based on representation learning was proposed. |

A Study on the Influence Propagation Model in Topic Attention Network [24] | 2017 | |

Study on the Measure Methods of Similarity between Vertices in Network [16] | 2017 | |

Representation Learning of the Topic-Attention Network [25] | 2019 | |

Research and Application of Three-way Clustering Based on Set Pair Information Granule [26] | 2021 | This paper constructs a three-way clustering algorithm SPKM based on set pair information granules and applies the set pair three-way clustering algorithm to community mining in a complex network. |

Paper Title | Years | Major Contributions |
---|---|---|

Research on Complex Network Attack Modeling and Security Assessment Method [30] | 2013 | A rough path generation algorithm is proposed. On this basis, the ant colony algorithm is used to further mine k key vulnerable paths to the attack target. |

Rough Decision Analysis Model Based on New Feature Dominance Relationship [31] | 2015 | A decision analysis model based on an extended rough set is constructed, the rough approximation relationship of decision classes is obtained under the new feature of relationship, and the classification decision rules are given. |

Decision Methods and Applications of Rough Complex Network Based on Network-Based [34] | 2016 | Combining rough set theory with complex networks for the first time, the concept of the rough complex network is proposed, and the concepts of positive field, negative field and boundary field of the rough complex network are given. A scale-free benefit risk assessment model is constructed, combined with game theory, to conduct a game analysis of the third-party payment rough network operation risk. A decision method of rough and complex networks is given by defining the network basis of rough and complex networks, and it is used to solve the operation risk decision analysis of third-party payment rough and complex networks. |

Research on Risk Analysis and Decision Models of the Third-party Payment Rough Network [35] | 2016 | |

A Knowledge Discovery Model for Third-party Payment Network based on Rough Set Theory [33] | 2017 | |

Concept Design and Construction Algorithm of Rough Complex Network [32] | 2017 | |

Benefit Risk Evaluation of Third-party Payment Network Based on Rough Set [36] | 2018 | |

Research on the Statistical Characteristics and Definition of the Complex Network with Uncertainty [29] | 2018 |

Paper Title | Years | Mean Work |
---|---|---|

Three-way Decision-based Overlapping Community Detection [37] | 2017 | Using the idea of a three-way decision, a non-overlapping community and overlapping community discovery algorithm based on a three-way decision is proposed [37]. A three-way division [38] is performed for the overlapping communities that appear during the granulation process to obtain non-overlapping communities; aiming at the problem of community merging in the process of hierarchical clustering, a community discovery method based on variable granularity hierarchical clustering is proposed. |

Three-way Decision Based on Non-overlapping Community Division [38] | 2017 | |

Research on Non-overlapping Community Division Based on Three-way Decision Theory [39] | 2018 | |

VGHC: A Variable Granularity Hierarchical Clustering for Community Detection [40] | 2020 | |

Application of Rough Set and Ant Colony Algorithm on Community Discovery [41] | 2012 | A model of network community structure discovery is constructed based on a rough set, and information centrality is used to measure the relationship between vertices. |

Research of Community Mining in Social Network Based on Granular Computing [42] | 2015 | Based on granular computing of a rough set model, a community mining algorithm based on granular computing is constructed. |

Research of Community Mining in Social Network Based on Granular Computing [43] | 2020 | An overlapping community discovery algorithm is proposed based on a rough set and density peaks [43]; an overlapping community discovery algorithm is proposed based on a rough set and distance dynamic models [44]. |

Overlapping Community Detection Method Based on Rough Set and Distance Dynamic Model [44] | 2020 | |

Research on Overlapping Community Detection Algorithm Based on Rough Set [45] | 2021 | |

A Rough Connectedness Algorithm for Mining Communities in Complex Network [46] | 2016 | A new algorithm based on a rough set is proposed to detect disjointed, overlapping and hierarchically nested communities in a network by constructing the granularity of neighborhood vertices and representing them as a rough set [46]; an overlapping community detection algorithm is proposed based on link granularity information and a rough set [47]. |

An Overlapping Community Detection Algorithm Based on Rough Clustering of Links [47] | 2020 | |

Rough Net Approach for Community Detection Analysis in Complex Network [48] | 2020 | A new rough network model is constructed, and new quality measures are proposed for exploratory analysis of a community structure in a single network and multiple networks. |

Paper Title | Years | Mean Work |
---|---|---|

Fuzzy Overlapping Communities in Network [49] | 2011 | The concept of a fuzzy overlapping partition is proposed. |

Fuzzy Clustering in a Complex Network Based on Content Relevance and Link Structures [50] | 2016 | Considering the membership degree of the cluster to which the vertex belongs, a clustering algorithm (FCAN) based on fuzzy set theory is proposed. |

Overlapping Community Detection with Node Structure and Attribute [51] | 2016 | Reference [51] proposed a complex network fuzzy overlapping community structure detection method based on vertex topology. However, the average execution time of this algorithm is slightly longer, and it is more suitable for sparse networks. Therefore, the literature [52] proposed a fuzzy overlapping community structure detection model based on two-stage clustering to solve the above problems. |

Automatic Detection and Simulation of Complex Network Fuzzy Overlapping Community Structure [52] | 2020 | |

Community Discovery of Complex Network Based on Fuzzy Density Peak Clustering Algorithm [53] | 2018 | In [53], a community discovery algorithm based on the clustering of fuzzy density peaks (CDFDPC) was proposed. The algorithm uses the F_DPC algorithm to determine the core community, and then the fuzzy clustering idea is used to determine the membership degree of each point to complete the distribution of the remaining vertices. Reference [54] proposed a density peak clustering algorithm based on generalized nearest neighbor similarity and designed a multi-step assignment strategy. This strategy effectively avoids the associated errors in the data point allocation process of the traditional DPC algorithm and increases the robustness of the algorithm. |

Research on Density Peaks Clustering and Application in Community Detection [54] | 2022 | |

Overlapping Community Division Based on Rough Fuzzy Clustering Algorithm [55] | 2020 | Based on the rough fuzzy clustering algorithm, combined with the concept of signal transmission, an overlapping community structure mining algorithm, RFCMST, based on rough fuzzy clustering and signal transmission is proposed. |

Fuzzy Overlapping Community Partitioning Algorithm Based on Vertex Vector Representation [56] | 2021 | A fuzzy community partition algorithm based on vertex vector representation is proposed. |

Research on Design of Fuzzy Clustering Algorithm Based on Q Function Optimization for Weighted Directed Complex Network [57] | 2016 | On the basis of the traditional algorithm, a new Q function suitable for fuzzy partitioning of a weighted directed complex network is constructed, a fuzzy clustering algorithm for a complex network is designed, and the algorithm is improved for the unstable results of the FCM clustering algorithm. |

Fuzzy Analysis and Information Mining on Overlapping Communities in Directed Network Based on Matrix Decomposition [58] | 2019 | In order to develop a fuzzy community analysis method for a directed network, a new fuzzy metric that can describe the association of directed point groups is introduced, and a new modular index suitable for fuzzy structures of a directed network is constructed. |

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## Share and Cite

**MDPI and ACS Style**

Wang, J.; Wang, J.; Guo, J.; Wang, L.; Zhang, C.; Liu, B.
Research Progress of Complex Network Modeling Methods Based on Uncertainty Theory. *Mathematics* **2023**, *11*, 1212.
https://doi.org/10.3390/math11051212

**AMA Style**

Wang J, Wang J, Guo J, Wang L, Zhang C, Liu B.
Research Progress of Complex Network Modeling Methods Based on Uncertainty Theory. *Mathematics*. 2023; 11(5):1212.
https://doi.org/10.3390/math11051212

**Chicago/Turabian Style**

Wang, Jing, Jing Wang, Jingfeng Guo, Liya Wang, Chunying Zhang, and Bin Liu.
2023. "Research Progress of Complex Network Modeling Methods Based on Uncertainty Theory" *Mathematics* 11, no. 5: 1212.
https://doi.org/10.3390/math11051212