## Yixuan Yang , Sony Peng , Doo-Soon Park , Hye-Jung Lee and Phonexay Vilakone## |

V1 | V2 | V3 | V4 | V5 | V6v | V7 | V8 | V9 | |
---|---|---|---|---|---|---|---|---|---|

V1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |

V2 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |

V3 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |

V4 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |

V5 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 |

V6 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 |

V7 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 |

V8 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 |

V9 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 |

Table 2.

V1 | V2 | V3 | V4 | V5 | V6v | V7 | V8 | V9 | |
---|---|---|---|---|---|---|---|---|---|

V1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 |

V2 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 |

V3 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 |

V4 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 |

V5 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |

V6 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |

V7 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |

V8 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |

V9 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |

Based on the traditional formal concept lattice method [21], when Tables 1 and 2 are used as the input, after all rows and columns are intersected respectively, we can get two concept lattices, as shown in Fig. 3. Each node in the lattice is a concept node, with the blue text denoting the extent of the concept, and the red text denoting the concept's intent.

In our previous work [19], we proposed and proved a method to get the three-way concept lattice based on the above FCA method. The steps are to intersect the extent of the concepts in the two concept lattices in Fig.3 above in pairs, and combine the intent in pairs to obtain a new concept: OE concept (e.g., ({1, 2, 3, 4}, {1, 2, 3, 4}) and ({1, 2, 3, 4}, {5, 6, 7, 8, 9}) → ({1, 2, 3, 4}, ({1, 2, 3, 4}, {5, 6, 7, 8, 9}))), such that all OE concepts can form an OE concept lattice through a certain partial order structure, as shown in Fig. 4.

For each OE concept, we can think of a pair of node sets as a maximal balanced clique, if the first half of the pair is equal to the left subset of the second half (e.g., for the OE concept ({1, 2, 3, 4}, ({1, 2, 3, 4}, {5, 6, 7, 8, 9})), the first half (1, 2, 3, 4) and the left subsets of the second-half part ((1, 2, 3, 4), (5, 6, 7, 8, 9)) are equal), shows as Fig. 5. Therefore, the node set contained in this OE concept is a maximal balanced clique node set. In this context, there are all positive edges inside the two subsets and all negative edges between the two subsets.

Here, we will illustrate the problem statement and explain the proposed approach to detect absolute fair maximal balanced cliques in the signed attributed social network.

We have introduced some definitions in our previous work in [22], and this paper is the extension of the conference paper. Based on the content of [22], we have given a complete process for detecting maximal balanced cliques and absolute-fairness cliques in Sections 3 and 4. Meanwhile, we intend to apply this model to a real-life dataset and give an empirical study in Section 5.

To enhance comprehension, we present some definitions as follows.

At first, we have defined signed attributed social networks and absolute fair maximal cliques [10].

In comparison to a signed social network, a signed attributed social network includes attribute information of the nodes. It can be represented as G = (V, E, W, A), where V denotes the set of nodes, E denotes the edges present in the network, W indicates the weight set of edges, where “+” and “–” are used to denote a positive and negative weight, respectively, [TeX:] $$\mathrm{W}=\left\{\mathrm{W}^{+} \cup \mathrm{W}^{-}\right\},$$ and A denotes the set of attributes on the nodes.

Definition 2 (Criteria of fairness and maximal of absolute fair maximal clique). A clique in graph G = (V, E, A) is considered an absolute fair maximal clique if it meets the following criteria as specified:

– Fairness: It is fair, meaning the number of nodes associated with each attribute [TeX:] $$a i \in A$$ is equal.

– Maximal: It is maximal, meaning no clique [TeX:] $$\mathrm{C}^{\prime}$$ is a proper superset of C and also meets the fairness condition [22].

Fig. 6 shows a social network graph with nodes, edges, and the attributes of nodes (‘a’ and ‘b’ represent the attributes of nodes, such as node V1 belonging to Class ‘a’ and node V8 belonging to Class ‘b’). The clique in Fig. 6 is a fairness clique since the numbers of the attribute of nodes are the same (e.g., the number of attributes ‘a’ and ‘b’ are both equal to 4).

Then, to enhance comprehension, we have defined the maximal balanced clique (MBC) and absolute fairness maximal balanced clique (AFMBC) [22].

Definition 3 (Maximal balanced clique). A fully connected subgraph [TeX:] $$\mathrm{G}_1$$ of a signed network (which is described as G = (V, E, W)) is called a maximal clique. Moreover, if [TeX:] $$\mathrm{G}_1$$ is a maximal clique and for [TeX:] $$\forall(u, v) \in \mathrm {MBC},\mathrm{(u, v)} \in \mathrm{W}\left(\mathrm{W}=\left\{\mathrm{W}^{+} \cup \mathrm{W}^{-}\right\}\right),$$ is a MBC. To further refine the definition, an MBC can be divided into two sub-cliques [TeX:] $$\mathrm{C}_1 \text{ and } \mathrm{C}.$$ To ensure “balance”, a new requirement has been proposed such that for [TeX:] $$\forall \mathrm{u}, \mathrm{v} \in \mathrm{C}_1 $$ or [TeX:] $$\mathrm{u}, \mathrm{v} \in \mathrm{C}_2,(\mathrm{u}, \mathrm{v}) \in \mathrm{W}^{+} \text {, }$$ and when [TeX:] $$\forall \mathrm{u} \in \mathrm{C}_1, \mathrm{v} \in \mathrm{C}_2$$ or [TeX:] $$\mathrm{u} \in \mathrm{C}_2, \mathrm{v} \in \mathrm{C}_1$$ the u and v have negative edges, which are indicated as [TeX:] $$(\mathrm{u, v}) \in \mathrm{W}^{-}$$.

Definition 4 (Absolute fairness maximal balanced clique). An AFMBC can be defined as a pair of subcliques [TeX:] $$\mathrm{C}_1 \text{ and } \mathrm{C}_2$$ that satisfy both “fairness” and “balance,” so that the union of the two cliques, [TeX:] $$\mathrm{C} = \mathrm{C}_1 \cup \mathrm{C}_2,$$ is an AFMBC.

Definition 5 (Derived pseudo absolute-fairness maximal balanced clique). For the maximal balanced cliques in which the numbers of attributes are greater than the kinds of attributes and the numbers are unequal, we will derive the sub-absolute fairness maximal balanced clique (sub-AFMBC) as the outcomes.

Example: To illustrate, consider {(1,2,3,4), (5,6,7,8,9)} in Fig. 7 is an AFMBC. However, (5,6,7,8,9) is a clique, but it violates the fairness constraint such that we remove this clique and generate several new pseudo-AFMBC: (6, 7, 8, 9), (5,7,8,9), and (5,6,7,8). Finally, we obtain three AFMBCs that satisfy both fairness and balance: {(1,2,3,4), (6,7,8,9)}, {(1,2,3,4), (5,7,8,9)}, and {(1,2,3,4), (5,6,7,8)}. This process is demonstrated in Fig. 7.

Problem statement: Given a signed attributed social network G, by finding all AFMBCs in G, the cliques set Ω containing the AFMBCs is expressed as follows:

This section introduces the method for identifying AFMBC in signed attributed social networks.

In previous research [19], we demonstrated the equivalence between MBCs and the OE-concept in the OE-concept lattice using the three-way concept method. We used modified formal contexts, which include positive and negative formal contexts as subgraphs, to mine MBCs.

Our work has taken into account not only the “balanced” aspect but also “fairness.” Therefore, in addition to using the three-way concept method to obtain the MBCs, we have also analyzed the attributes of the nodes and ensured that the number of each attribute in each sub-clique of MBCs is equal.

The following are the technical steps of our method as specified:

Step 1: For a given G = (V, E, W, A), based on Definition 1, two adjacency matrixes of modified formal context F and [TeX:] $$\mathrm{F}^\mathrm{C}$$ are obtained.

Step 2: The MCDA method is applied to detect the MBCs by generating the corresponding OE-concept lattice.

Step 3: Travel all MBC and determine the fairness with each sub-clique of a pair, and filter the AFMBC that satisfies Definition 4, otherwise, derive the pseudo-AFMBC (based on Definition 5) from an MBC to make sure the number of attributes is equal.

Fig. 8 illustrates the framework of our methodology, which includes all the above steps.

The method is explained in pseudocode and presented as follows.

For implementing the proposed algorithm, we used a real-life signed attributed social network: namely, a research team at a manufacturing company [23] (https://toreopsahl.com/datasets/#FreemansEIES). We used Python 3.8 and Intel Core i5 processors on MacOS to execute the experiment.

Intra-organizational networks contain two company members’ social networks. In our work, we have utilized one of the networks of 77 researchers. The network concerned constitutes 77 researchers and their opinions as to one another's knowledge and abilities. The scores range from 0–6 and are used to represent weight levels in the network. A score of 0 indicates no knowledge of the person, while scores of 1 and 2 indicate strong disagreement and disagreement, respectively. Then, scores of 3 and 4 indicate neutrality, whereas scores of 5 and 6 indicate agreement and strong agreement, respectively. To preprocess the data, scores of 1 and 2 are treated as extremely negative ratings and labeled as -1, whereas scores of 5 and 6 are treated as extremely positive ratings and labeled as 1. Scores that fall between these extremes are labeled as 0.

The information on researchers include location, tenure, and organizational level. In our work, we have chosen the tenure (short-tenured, less than 61 months; long-tenured, greater than or equal to 61 months) and the organization level (management: global department manager, local department manager, and project leader; general: researcher) as the attributes to analyze this network.

The concept of “group polarization” was initially introduced by Stoner in 1961 in the field of sociology. Later on, Sunstein [24] defined “group polarization” as a phenomenon where group members tend to hold certain opinions or attitudes at the beginning, and after deliberation, they continue to move in the direction of their initial biases, eventually leading to the formation of an extreme point of view.

In fact, company managers prefer employees to reduce group polarization as much as possible in the work process. Therefore, managers may need to make necessary interventions on social networks. However, if the intervention method is only a single-shielded communication due to the internal structure of the social network, it is not conducive to crossing the conceptual gap. Therefore, maintaining an openness to multiple concepts and allowing isolated “small worlds” in social networks to perceive each other are a meaningful way to finally de-polarize them [25]. This empirical study seeks to find as many small groups as possible with polarized tendencies (i.e., MBCs). The goal is mainly based on the professionalism ratings of 77 employees of the research company and aims to ensure that positions or work experience in the cliques have “fairness” (i.e., absolute-fairness balanced cliques) and to enumerate all possible cooperative groups, establishing opportunities for cooperation and communication between isolated “small worlds,” so that polarization is reduced within the company's employees.

Fig. 9 illustrates the signed attributed social network of this dataset. Each node represents one employee, and the solid lines (gray) and dashed lines (blue) represent the positive and negative relationship between employees, respectively.

Taking this dataset as input, after calculating with our method, we get two MBCs: ({28, 63, 73, 74, 76}, {15, 49}), ({10, 55, 68}, {1, 2}). This means that these two MBCs are possible to have group polarization. To address this issue, we have considered the “fairness” of the cliques by considering working hours (tenure) and work positions (organization level) as attributes, respectively. Then, we propose some grouping suggestions for cooperative groups to reduce the problem of group polarization by letting employees on the list work cooperatively with a possibility of occurrence.

These MBCs with two attribute labels are shown in Figs. 10 and 11. These figures represent two subgraphs of the network graph shown in Fig. 9. Similarly, each node represents one employee, while solid lines (gray) and dashed lines (blue) represent the positive and negative relationship between employees, respectively.

Table 3 shows the number of cliques in different situations. It includes the number of cliques detected by sub-networks that contain only positive edges, networks that contain only negative edges, and unsigned networks (which do not distinguish between positive and negative, with both weights -1 and 1 being regarded as 1). At the same time, we also derive one of the goals of our paper—MBCs in signed social networks.

Table 3.

Sub-graph with positive edges | Sub-graph with negative edges | Sub-graph with unsigned edges | Signed graph | |
---|---|---|---|---|

Number of cliques | 205 | 139 | 85 | 2 |

According to Table 3, it is obvious that there are many cliques in this social network, whether they are positive, negative, or regardless of trust. However, only two maximal cliques are balanced (shown in the last column in Table 3), and they also account for two of the groups where group polarization is possible. The goal of the algorithm in this paper is to find these specific cliques and analyze them. Table 4 provides a detailed analysis of these two maximal balanced cliques and derives the sub-cliques from them.

Table 4 denotes the outputs of the empirical study. All the absolute-fairness maximal cliques are enumerated in this table. Company managers can group cooperative groups according to this strategy, with the result of eliminating the problem of group polarization between these two MBCs. At the same time, it also fully considers the fairness aspect of employees' different job positions and work experiences, which in turn is more conducive to teamwork

In this paper, we have presented a method of enumerating AFMBCs, applied it to signed attributed social networks, and conducted an empirical study on real-life datasets. Firstly, we have applied the threeway concept method and list to obtain all MBCs. Then, we have filtered and derived sub-cliques conforming to fairness to guarantee the “fairness” of the detected MBC. Through our empirical study, we have demonstrated the legitimacy and meaning of our method.

In the future, we intend to emphasize more efficient methods to adapt to real-life scenarios, especially in terms of applying them to dynamic social networks and other issues.

She is working at Dongfang Jingyuan Microelectronics Technology in China. The interests of her research are social network analysis, community detection, and formal concept analysis. She earned her B.Sc. degree from the Department of Software Engineering at Taiyuan University of Technology, China in 2017. She completed her M.Sc. degree from the Department of Software Engineering at Shaanxi Normal Uni-versity, China in 2020. She completed her Ph.D. in Software Convergence at Soon-chunhyang University in Asan, South Korea.

In 2018, she obtained her undergraduate degree in ITE from the Royal University of Phnom Penh, in Cambodia. Currently, she is a candidate pursuing a combined master and Ph.D. program in the Department of Software Convergence at Soonchunhyang University in South Korea. She is focusing on data mining, recommendation system, and mobile development in AI.

In 1988, he completed his Ph.D. in Computer Science at Korea University. At present, he holds the position of Chair Professor in the Department of Computer Software Engineering at Soonchunhyang University, South Korea, and he is also the Dean of the Graduate School at Soonchunhyang University. He has held several other positions, including Director of BK21 FOUR Well-Life Big Data at Soonchunhyang University from 2020 to 2022, Director of Wellness Service Coaching Center at Soonchunhyang University from 2014 to 2021, President of KIPS (Korea Information Processing Society) from 2015 to 2015, and Director of the Central Library at Soonchunhyang University from 2014 to 2015. He was also the Editor in Chief of Journal of Infor-mation Processing Systems at KIPS from 2009 to 2012, and Dean of the Engineering College at Soonchunhyang University from 2002 to 2003. He has served as an orga-nizing committee member of various international conferences such as CSA 2022, BIC 2022, MUE 2022, WORLDIT 2022, BIC 2021, MUE 2021, WORLDIT 2021, and CSA 2020. His research interests are focused on data mining, big data processing, and parallel processing. He is affiliated with several professional organizations, including IEEE, ACM, KIPS, KMS, and KIISE.

In 1997 and 2000, she obtained her B.S. and M.S. degrees, respectively, from the Department of Computer Science Engineering at Soonchunhyang University. She received her Ph.D. degree in the same department at Soonchunhyang University, Asan, South Korea, in 2006. Currently, she serves as an assistant professor at the Institute for Artificial Intelligence and Software at Soonchunhyang University, Asan, South Korea. Her research areas include data mining, parallel processing, big data processing, and computer education.

He completed his Ph.D. in Computer Sciences and Engineering at Soonchunhyang University in 2010. He obtained his Master's degree in Computer Application (Software System) from Guru Gobind Sigh Indraprastha University in India in 2010, and his Bachelor’s degree in Mathematics and Computer Sciences from the National University of Laos in 2003. Since 2021, he has served as the Head of the Robotic Engineering Division at the Department of Computer Engineering and Information Technology at the National University of Laos. Additionally, he has been the coordinator of the K-Lab Vientiane Project, funded by NIPA from the Government of the Republic of Korea, since 2021. His research interests focus on data mining, parallel processing, automation, robotics, deep learning, and artificial intelligence.

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