# Complex Neutrosophic Hypergraphs: New Social Network Models
## 1 Introduction  
Fuzzy sets (FSs) were originally defined by [PERSON] [1] as a novel approach to represent uncertainty arising in various fields that was questioned by many researchers at that time. A FS is characterized by a truth membership function \(\mu\) which ranges over \([0,1]\). To generalize the notion of FSs, intuitionistic fuzzy sets (IFSs) were proposed by [PERSON] [2] because it is not always true that the falsity degree of an element in a FS is \(1-\mu(x)\) as there may be some hesitation part. Therefore, the truth (t) and falsity (f) membership functions are used independently to characterize an IFS such that the sum of truth and falsity degrees should not be greater than one. Fuzzy sets give the degree of membership of an element in a given set (the non-membership of degree equals one minus the degree of membership), while IFSs give both a degree of membership and a degree of non-membership, which are more-or-less independent from each other. [PERSON] et al. [3] introduced different types of centroid transformations of IF values. Furthermore, [PERSON] et al. [4] defined various new operations for generalized IF soft sets. As an extension of IFSs, [PERSON] [5] introduced the concept of neutrosophy to study the nature, origin, and neutralities, and the neutrosophic set (NS). A NS is characterized by truth (t), indeterminacy (i), and falsity (f) membership functions. A NS is used as a powerful mathematical tool to deal the inconsistent data that exists in our daily life. For the practical use of NSs in science and engineering, [PERSON] [5] and [PERSON] et al. [6] introduced single-valued neutrosophic sets (SVNSs). A SVNS propose an additional choice to handle indeterminate information. [PERSON] [7] proposed a decision-making method by using the weighted correlation coefficient or the weighted cosine similarity measure of SVNSs to rank the alternatives and proposed an illustrative example to demonstrate the application ofthe proposed decision-making method. The same author defined SVN minimum spanning tree and its clustering method [8]. [PERSON] [9] also proposed a multicriteria decision-making method using aggregation operators for simplified NSs. The existing models such as FSs, IFSs, SVNSs cannot handle imprecise, inconsistent, and incomplete information of periodic nature. These theories are applicable to different areas of science, but there is one major deficiency in these sets, i.e., a lack of capability to model two-dimensional phenomena. To overcome this difficulty, the concept of complex fuzzy sets (CFSs) was introduced by [PERSON] et al. [10]. A CFS is characterized by a membership function \(\mu(x)\) whose range is not limited to [0, 1] but extends to the unit circle in the complex plane. Hence, \(\mu(x)\) is a complex-valued function that assigns a grade of membership of the form \(v(x)e^{\mu(x)}\), \(t=\sqrt{-1}\) to any element \(x\) in the universe of discourse. Thus, the membership function \(\mu(x)\) of CFS consists of two terms, i.e., amplitude term \(v(x)\) which lies in the unit interval [0, 1] and phase term (periodic term) \(\alpha(x)\) which lies in the interval \([0,2\pi]\). This phase term distinguishes a CFS model from all other models available in the literature. Opposing to a fuzzy characteristic function, the range of CFS's membership degrees is not restricted to \([0,1]\), but extends to the complex plane with unit circle. [PERSON] et al. [11] discussed the union, intersection, and compliment of CFSs with the help of illustrative examples. A systematic review of CFSs was proposed by [PERSON] and [PERSON] [12]. To generalize the concept of CFSs, complex intuitionistic fuzzy sets (CIFSs) were introduced by [PERSON] and [PERSON] [13] by adding non-membership degree \(v(x)=s(x)e^{\mu(x)}\) to the CFSs subjected to the constraint \(r+s\leq 1\). The CIFSs are used to handle the information of uncertainty and periodicity simultaneously.