The concept of bipolar-valued fuzzy relationships and their role in modeling bipolar-valued fuzzy social networks (BVFSN) are discussed. The goal of this study is to represent the bipolarity and fuzziness that are always present in the relationship between actors. Ranking the most central actors to achieve maximum spreading ability has been a difficult and critical topic thus far. Many centrality measures have been proposed to determine the significance of nodes in the central person detection process. In this article, we proposed two new centrality measures, BF-degree centrality and BF-closeness centrality, based on the natural characteristics of a bipolar-valued fuzzy social network. The proposed centrality measures highlight the fuzziness and bipolarity of the relationship. We consider the fuzzy stability index for direct connections here, as well as the strengths and weaknesses of the relationship. With the help of the effective and total effective strength of the path, we find the favorable path in this centrality measure. Moreover, to investigate the validity and reliability of these new centrality measures, we gathered Google Scholar data and built a G-S Research network. Experimental results show that the model and centrality measures can be used to objectively rank the most central node in the network.

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In the real-world scenario, one has to find a dominating person in a social network, conferences, meetings or any group discussion. The fuzzy graph (network) is one of the most powerful tools to find the strongest influential person in a network. This paper aims to develop a concept of fuzzy graphs (FGs) in the setup of bipolar-valued hesitant fuzzy sets (BVHFs). The concept of bipolar-valued hesitant fuzzy graph (BVHFG) is different from the concept of bipolar fuzzy graph (BFG). BVHFG is the generalization of hesitant fuzzy graph (HFG), which not only considers the satisfaction degree of units in a network but also considers the satisfaction degree to some implicit counter property of units with several bipolar fuzzy values. We first introduce the definition of BVHFG, represented by another class of imprecise membership grades that refers to BVHF membership grades. We shall subsequently see the scope of BVHF membership grades in BVHFG is greater than the scope of bipolar-valued membership grades in BFG. In addition, we also discuss the basic operations and functional properties of BVHFGs. Finally, we propose a numerical method to find the most dominating person using our proposed work. As the proposed method of ranking considers the degree of hesitation as well as bipolarity, this method has the edge over earlier work. To establish the importance of our method, we also find domination degrees for HFG and BVHFG using the same example and show that there is a significant change in the ranking of dominating persons.