Abstract
Over the past 20 years, the emergence of social media and its developments have rapidly changed communication and information technology. Social media plays a necessary part in accessing information and communication. In spite of its many advantages, users have also been facing many threats on social media platforms. This research aims to analyze and resolve these threats by using the new concepts of complex cubic T spherical fuzzy sets (CCuTSFS) that have a broad structure including degrees of membership, neutral-membership, and non-membership. It does a better job of modeling uncertainty than any other preexisting structure. Furthermore, we defined the concepts of the Cartesian product (CP) between CCuTSFSs, complex cubic T spherical fuzzy relation (CCuTSFR) and the types of CCuTSFRs with appropriate examples. This study looked at the relationship between different types of security and threats on social media for the first time in fuzzy set theory. The proposed methods demonstrate how to control the effects of threats by using valid security methods. Finally, the benefits of the presented strategies are explained in the comparative study.
1. Introduction
Uncertainty is an essential element of life, resulting from a range of factors ranging from a lack of judgment to a lack of knowledge. The nature of modeling and computing approaches is frequently precise, unavoidable, and crisp. In crisp set theory, a statement has only two options: true or false. Thus, decisions based on crisp knowledge are clear and unambiguous. Information is frequently complicated, uncertain, and confusing because people cannot deal with uncertainty. Mathematics is largely concerned with the precise and proper representation of data. After the introduction of crisp knowledge, some life-changing novelty in mathematics was made by the introduction of fuzzy sets and fuzzy logic.
In 1965, Zadeh [1] established the theory of fuzzy sets (FS) and fuzzy logic, which is one of the most prosperous theories for representing data uncertainty. Every element in the set is assigned a level of membership by an FS. The membership level is a function that takes values from the [0, 1] unit interval. Klir and Folger [2] introduced relations between classical sets. The classical set theory only deals with yes and no situations, hence a relation of classical sets enumerates the existence and nonexistence of a relationship. Mendel et al. [3] proposed the fuzzy relation (FR) for FSs. Unlike classical relations, FRs are not limited to yes-or-no problems. Based on their membership level, they can indicate the level, intensity, and degree of good connection between any two FSs. The higher the degree of membership, the better the relationship, whereas the lower the degree of membership, the worse the relationship. FR is a more comprehensive concept than a classical relationship because it deals with issues in both situations. Zadeh [4] introduced the idea of an interval-valued fuzzy set (IVFS) in 1975. The IVFSs are a more broader version of the FS. These sets describe the membership degree in the form of the subset from the unit interval. Because it can be difficult for an expert to accurately explain the level of certainty using a real number. As a result, an interval that represents the amount of confidence is a solid option. Bustince and Burillo [5] invented the interval-valued fuzzy relation (IVFRs), and they generalized the classical relation and FRs. Deschrijver and Kerre [6] derived the connection between different extensions of fuzzy set theory. Bustince et al. [7] employed matrices to create IVFSs and used them to detect edges. De Baets and Kerre [8] defined the application of FRs.
Atanassov [9] proposed the intuitionistic fuzzy set (IFS), which consists of membership and nonmembership degrees and satisfies the condition that the sum of the two degrees is less than or equal to 1. It is the generalization form of FS. Burillo and Bustince [10] invented the intuitionistic fuzzy relation (IFR), which studies the relationship between two IFSs. It is a broader form of the FR. Further, Atanassov [11] introduced a novelty concept called the interval-valued intuitionistic fuzzy set (IVIFSs) by showing the degree of membership and nonmembership of an IFS in the form of intervals. De et al. [12] used the IFSs in medical diagnosis. Yager [13] created the notion of the Pythagorean fuzzy set (PyFS) by changing the constraint of IFSs, which increases the space of membership and nonmembership by imposing some new restriction, i.e., the sum of the squares of membership and nonmembership must be interval, according to the innovative requirements. Zhou et al. [14] discussed and implemented a new PyFS divergence measure in medical diagnosis. Yager [15] recognized the constraints of assigning degrees to objects in PyFSs and proposed the concept of q-rung orthopair fuzzy set (qROFSs). By removing the limits imposed by the earlier set theories, these sets allow professionals and researchers to freely award membership and nonmembership degrees. The total of the nth power of membership and nonmembership degrees in qROFSs must be in the unit interval, with being a natural number.
Cuong and Kreinovich [16] established the concept of picture fuzzy set (PFS) by the inclusion of a neutral degree in an IFS. The membership, neutral, and nonmembership degrees take on values from the unit interval, and the sum of all these degrees is accommodated within the unit interval [0,1]. Mahmood et al. [17] flourish the concept of spherical fuzzy set (SFS) by changing the constraint such that the sum of squares of membership, neutral, and nonmembership degrees consists of the unit interval. They also introduced the notion of T SFS (TSFS). The sum of membership, neutral, and nonmembership degrees must be contained in the unit interval in TSFSs when raised to the power , where is a natural number. Guleria and Bajaj [18] activated the Eigen SFSs in decision making problems. Ullah et al. [19] defined the similarity measures for TSFS and applied them in pattern recognition. Ullah et al. [20] conceive the correlation coefficients for TSFSs and used them in clustering and multiattribute decision making. Van Dinh et al. [21] presented a picture of fuzzy database applications and theories. Dutta [22] used the PFSs in the field of medical diagnosis.
After FS, a new theory of complex fuzzy sets (CFSs) was introduced by Ramot et al. [23], which is a broad form of FS. The CFS represents the membership degree in the form of a complex number. Instead of a real number in the unit interval, the degree of membership in a CFS is a complex number in a unit circle in the complex plane. The CFSs make it easier to represent multidimensional issues, particularly those that are time dependent. In addition, they defined the complex fuzzy relation (CFR). Lie et al. [24] investigated the application of CFSs. Greenfield et al. [25] introduced the concept of an interval-valued complex fuzzy set by changing the degree of membership of a CFS from a single number to an interval (IVCFS). Nasir et al. [26] recently defined interval-valued complex fuzzy relations as a tool for analyzing relationships between two or more IVCFSs. Alkouri and Salleh [27] proposed the notion of a complex intuitionistic fuzzy set (CIFS). In the complex plane, CIFS limit the sum of the degree of membership and nonmembership to the unit disc. Jan et al. [28] provide the CIFR which was applied to cybersecurities and cybercrimes in oil and gas industries. The concept of an interval-valued complex intuitionistic fuzzy set (IVCIFS) was established by Garg and Rani [29]. Using this relationship between two or more IVCFS, Nasir et al. [30] introduced the idea of interval-valued complex intuitionistic fuzzy relation (IVCIFR). Dick et al. [31] suggested the complex Pythagorean fuzzy set (CPyFS) by modifying the codomain of CIFS. Garg et al. [32] further expanded on CPyFS to create a complex q-rung orthopair fuzzy set (CqROFSs). The complex-valued membership and nonmembership degrees are assigned to the elements of a set in CqROFSs, when increased to the power , the sum of modulus of membership and nonmembership degrees must be within the unit interval, where is a natural number.
Nasir et al. [33] recommended the concept of a complex picture fuzzy set (CPFS). Ali et al. [34] defined the concepts of a complex spherical fuzzy set (CSFS) and, in addition, proposed the concepts of a complex T spherical fuzzy set (CTSFS) as a generalization of the CSFS. Jun et al. [35] developed some logic operations of the cubic set and introduced the idea of the cubic fuzzy set (CuFS) by combining both interval-valued fuzzy numbers and fuzzy numbers. Cubic fuzzy set is the improved form of both FS and IVFS because they cover the same set of information. Kaur and Garg [36] defined the generalization of cubic fuzzy with t-norm operators. Kim et al. [37] defined the new innovative results of cubic fuzzy relations (CuFR). Garg and Kaur [38] concocted the concept of a cubic intuitionistic fuzzy set (CuIFS), which is expressed by two sections, i.e., an interval-valued intuitionistic fuzzy set and another by an intuitionistic fuzzy set. The CuIFS includes more information than the general IVIFS and IFS because it is composed of both these sets. Talukdar and Dutta [39] developed the cubic Pythagorean fuzzy set (CuPyFS) with the application of multicriteria decision making. They are the generalization form of CuIFS. They improve the limitation. Zhang et al. [40] introduced the concept of cubic q-rung orthopair fuzzy set (CuqROFS). They are the generalization form of CuPyFS. The constraint of the CuqROFS increased the space, so the sum of the power will be less than or equal to 1. Gumaei and Hussain [41] proposed a new operator of cubic picture fuzzy set (CuPFS) with application. The cubic picture fuzzy set also includes the neutral degree. They are the three levels of degree, i.e., cubic membership, cubic neutral, and cubic nonmembership. Devaraj and Aldring [42] developed the cubic spherical fuzzy set (CuSFSs). The cubic spherical fuzzy set solves more complexities. Chinnadurai et al. [43] introduced the notion of a complex cubic fuzzy set (CCuFS). The CCuFS covered both sets of information, i.e., the complex fuzzy set and the cubic set. Zhou et al. [44] used the CCuFS in multiattribute decision making. Chinnadurai et al. [45] proposed the idea of the complex cubic intuitionistic fuzzy set (CCuIFS), which is a combined form of the cubic membership degree and cubic nonmembership degree. They are covered by both internal and external results. They are investigating the relationship between studying the two or more Cartesian products (CP) of CuFSs. He [46] gave the idea of social media security. Social media security is the more effective way they work to combat the social media threat.
In this article, we introduced the novel concepts of the complex cubic T spherical fuzzy set (CCuTSFS) and CCuTSFR. In addition, the CP between the two or more CCuTSFSs is considered. In addition, different types of CCuTSFRs are defined, such as reflexive, irreflexive, symmetric, antisymmetric, transitive, equivalence classes, and many more. Every definition with appropriate examples and results of CCuTSFRs has also been defined. The CCuTSFR is an improved form of the CFR, CCuFR, CIFR, CCuIFR, CPyFR, CCuPyFR, CqROFR, CCuqROFR, CPFR, CCuPFR, CSFR, CCuSFR, CTSFR, and CuTSFR. The CCuTSFR is more efficient of the T spherical fuzzy relation (TSFR) and complex T spherical fuzzy relation (CTSFR) because this structure covers both the results of the TSFR and IVTSFR. The CCuTSFR is discussing all of the three stages; membership, neutrals, and nonmembership with complex numbers. The CCuTSFR is the multidimensional structure that is described in both amplitude and phase terms. The phase term is used to define the time frame or periodicity. The CCuTSFR increases the space with the power of if and , then the CCuTSFR are converting to the CCuSFR and CCuPFS, respectively. In this manuscript, we examine the relationship between social media security against particular threats. The CCuTSFR, including all of the stages if the neutral will be equal to zero, then the CCuTSFR are changed to the CCuIFR, and if the nonmembership will also be equal to zero then the CCuTSFR are converting to the CCuFR, and the space As a result, the CCuTSFRs are the better framework to use because it can encompass all aspects of the situation, including positive, no, and bad effects with time. The concepts of CCuTSFR are used to determine the best security methods. The novel idea of CCuTSFRs covers the present and the future. Further, this structure can be extended to other fuzzy set theory model that can be applied to many other fields such as economics, statistics, sports, computer science, medical, information technology, etc.
The arrangements of this paper are defined as follows.
In Section 2, a few predefined ideas of fuzzy algebra are described. In Section 3, the new concepts of CCuTSFS, CCuTSFR, and their types are defined. In Section 4, an application of the CCuTSFRs to study the relationship between social media security and threats is proposed. In Section 5, the proposed methods with the predefined structures are compared. Section 6 concludes the research.
2. Preliminaries
In this section, we defined some basic definitions of CuFS, CuFR, CCuFS, CIFS, CIVIFS, CPFS, and CqROFS.
Definition 1 (see [35]). For a non-empty set , a cubic fuzzy set (CuFS) on U is defined aswhere represents the IVFS defined on and represents the FS. Moreover, are called the lower and upper degrees of membership respectively, and is called membership degree. Such that , and also
Therefore, a CuFS can be written as
Definition 2. (see [37]). Let us take two CuFSs in a nonempty set Then, their CP is defined aswhereThe subset of the CP of two CuFS is called a cubic fuzzy relation (CuFR).
Example 1. Let two CuFSs and on is defined asThen, their CP isFrom the subset of is called relation
Definition 3. (see [43]). For a nonempty set U, a complex cubic fuzzy set (CCuFS) on U is defined aswhere represent the lower and upper degrees of membership, respectively. Thus, such that , and Moreover, are called the amplitude term of membership degree, and satisfy the inequalities , and , are called the phase term of degree of membership, and satisfy the inequality.
Definition 4. (see [27]). For a nonempty set U, a complex intuitionistic fuzzy set (CIFS) on U is defined aswhere is a complex-valued mapping i.e., , which represents the membership and nonmembership degree of the CIFS, respectively. Where C is the set of complex numbers. Thus, and Moreover, are called amplitude term of the membership and nonmembership degree, respectively, and are called phase term of the membership and nonmembership degree, respectively. On condition that and
Definition 5. (see [29]). For a nonempty set U, a complex interval-valued intuitionistic fuzzy set (CIVIFS) on U is defined aswhere , and are called the lower and upper membership degrees, while are called the lower and upper nonmembership degrees. and given that,Equivalently, CIVIFS can be written as
Definition 6. (see [32]). For a nonempty set U, a complex q-rung orthopair fuzzy set (CqROFS) on U is defined aswhere is a complex-valued mapping i.e., , which represents the membership and nonmembership degree of the qROFS, respectively. Where C is the set of complex numbers. Thus, and Moreover, are called amplitude terms of the membership and nonmembership degree, respectively, and are called phase terms of the membership and nonmembership degree, respectively. On condition that, 0 and n
Definition 7. (see [33]). For a nonempty set U, a complex picture fuzzy set (CPFS) on U is defined aswhere is a complex-valued mapping i.e., , which represents the membership, neutral, and nonmembership degree of the CPFS, respectively. Thus, , , and Moreover, are called amplitude term of the membership, neutral, and nonmembership degree, respectively, and are called phase term of the membership, neutral, and nonmembership degree, respectively. On condition that 0 and
3. Complex Cubic T Spherical Fuzzy Relation
This section introduces the novel concepts of cubic T spherical fuzzy set (CuTSFS), complex cubic T spherical fuzzy set (CCuTSFS), the CP of two CCuTSFS, complex cubic T spherical fuzzy relation (CCuTSFR), and its types.
Definition 8. For a nonempty set , a cubic T spherical fuzzy set (CuTSFS) on is defined aswhere represent the interval-valued TSFS and represent the TSFS of . Where are called the membership, neutral, and nonmembership degrees, respectively. Such that , , and Moreover, on condition that , n
Definition 9. For a nonempty set a complex cubic T-spherical fuzzy set on is defined aswhere are called the lower and upper membership, neutral, and nonmembership degrees, respectively, while are called membership, neutral, and nonmembership degrees defined as , . is the set of complex numbers and the complex numbers can be written as , and . Therefore, CCuTSFS can be expressed asMoreover, [0,1] are the elements of membership degree, [0,1] are the elements of neutral degree, [0,1] are the elements of nonmembership degree, such that and n .
Definition 10. Let us take two CCuTSFSs in a nonempty set Then, their CP is defined aswhere
Example 2. Let two CCuTSFSs and for on is defined asThen, their CP is
Definition 11. A complex cubic T spherical fuzzy relation (CCuTSFR) denoted by is a subset of the CP of two complex cubic T spherical fuzzy sets (CCuTSFSs).
Example 3. From equation (23), the subset of is
Definition 12. Let be a CCuTSFR on a CCuTSFS (i)A CCuTSFR is said to be reflexive, if(ii)A CCuTSFR is said to be irreflexive, if Implies(iii)A CCuTSFR is said to be symmetric, if(iv)A CCuTSFR is said to be antisymmetric, if(v)A CCuTSFR is said to be transitive, if(vi)A CCuTSFR is said to be equivalence, if it is(1)Reflexive(2)Symmetric(3)Transitive(vii)A CCuTSFR is said to be preorder, if it is(1)Reflexive(2)Transitive(viii)A CCuTSFR is said to be partial order, if it is(1)Reflexive(2)Antisymmetric(3)Transitive(ix)A CCuTSFR is said to be CCuTS strict order, if it is(1)Irreflexive(2)Transitive(x)Let and be two CCuTSFRs. Then, is composite relation if(xi)A CCuTSFR is said to be CCuTS linear order, if it is(1)Reflexive(2)Antisymmetric(3)Transitive(4)Complete
Example 4. Let a CCuTSFS Ă on is defined asThen, its self-CP is(1)The reflexive relation is(2)The symmetric relation is(3)The transitive relation is(4)The equivalence relation is(5)The partial order relation is(6)The relation is a CCuTS composite fuzzy relation between CCuTSFR and .
Definition 13. The equivalence class of modulo is written as
Example 5. The CCuTSF equivalence relation is givenNow, their equivalence classes are
Theorem 1. A CCuTSFR is a CCuTS symmetric fuzzy relation on a CCuTSFS if and only if = .
Proof:. Suppose that is a CCuTS symmetric fuzzy relation on a CCuTSFS , thenButHence,
Conversely, suppose that thenSo, is a CCuTS symmetric fuzzy relation on
Theorem 2. A CCuTSFR is a CCuTS transitive fuzzy relation on a CCuTSFS if and only if
Proof:. Suppose that is a CCuTS transitive fuzzy relation on a CCuTSFS , assume thatthere exists an element Then, by the transitivity of Hence,
Conversely, let us assume that then the composition of CCuTSFR implies that,Hence, is CCuTS transitive FR.
Theorem 3. If is a CCuTS equivalence FR on a CCuTSFS then
Proof:. As is a CCuTS equivalence FR on a CCuTSFS The CCuTS symmetric FR implies thatThe CCuTS transitive FR implies thatAlso, the CCuTS composite FR implies thatTherefore, (1).
Conversely, suppose thatThen there exists an element insince is a CCuTS equivalence FR. Hence, the CCuTS transitive FR implies thatTherefore, (2).
Equations (23) and (56) prove that
4. Application
In this section, the application of the proposed concepts is discussed. The CCuTSFS, CCuTSFR, and their types are used in this study of social media applications. We observe cybersecurity and cyber threats on social media.
4.1. Security of Social Media Platforms
Social media security is a critical aspect of modern business or personal success. It encompasses a wide range of technologies, strategies, and procedures. Social media has been constructed by the relations between users and bodies on the basis of modern Internet technology and platform carriers. Social media security has enabled organizations to safeguard the information that is shared over the network. All industries, organizations, and businesses require some kind of social media security clarification to save them from increasing cyber threats in today’s wild world. In the following sections, we discussed some threats that are faced by social media and the social media security methods. The flow diagram for the application activity is shown in Figure 1.
The above flowchart can be explained as(i)Collect the securities and threats methods.(ii)Give the values of every set of securities and threats.(iii)Find the CP between two CCuTSFS.(iv)Read the information.
4.1.1. Threats
The different types of social media platforms attract a variety of threats towards them, which tend to steal users’ identities or attack their privacy. Some threats are discussed that are prominent over social media. The objective of an attacker influences how they tackle a social media threat. Furthermore, each threat has been allocated to the membership, neutral, and nonmembership degrees. The membership degree is assigned to the professional according to their performance. The membership degree of threats shows the weakness, neutral degree demonstrates the neutral effect of threats and nonmembership degree indicates the strength of the threats. The different types of social media threats are shown in Table 1.(1)Identity Theft (IT). Personal information becomes registered on one or more social media sites. This data become important as hackers and identity thieves use this information to reorganize passwords, claim for loans, or other various goals.(2)Phishing Attacks (PA). Phishing happens when someone conveys a fraudulent message sketch to trick a human victim into providing sensitive personal information like passwords or credit card numbers.(3)Social Engineering (SE). Employees may be contacted by attackers in an attempt to persuade them to give personal details, prove credentials, or send money to the attacker. In a more sophisticated approach, an attacker can mimic a high-ranking executive to persuade the intended victim to send money to the attacker’s accounts.(4)Malware Attacks (MA). They are becoming increasingly popular on social media platforms these days. When criminals create malicious software and install it on someone else’s device without their consent, in order to steal personal details or cause device destruction for financial gain.(5)Image Retrieval and Analysis (IRA). The attackers here use several face and image identification technologies to learn more about the target and its correlated description. It not only affects the target but also his or her friends and family. The aim of this attack is to accumulate photographs and videos from the target.(6)Celebrity Name Misuse (CNM). This is one of the most widely used social media platforms for threats nowadays. In several cases, hackers have been known to create a new account in the name of a celebrity. Fake accounts like this can be used to propagate false information.
Therefore, the CCuTSFS summarizing the threat is given as follows:
4.1.2. Security
The social media techniques, procedures, and applications against threats are discussed. The social media security platform was built to protect the channels that are most important to your organization. Employees who openly share too much personal and corporate information on social media are the source of the bulk of social media threats. Because these accounts are personal, corporations cannot prevent users from using social media. Consumers, on the other hand, maybe educated on the best ways to protect their data and credentials. Avoiding social media risks requires education. The values of membership, neutrality, and nonmembership are assigned to each security mechanism. The strength is determined by the security’s membership degree. Thus, a higher amplitude term and a higher phase term value of its membership degree indicate better security; no effect of amplitude term with a phase term value of its neutral degree indicates no effect of security; and lower amplitude term and phase term values of its nonmembership degree indicate lower security. Some of the major social media security methods are explained with their fuzzy grades. The different types of social media security are shown in Table 2.(1)Unique Password (UP). Make sure that each of your social media accounts have its own password. Make account passwords long and strong. Do not utilize the information shared on social media accounts to create a password. A strong password is one that is easy to remember but difficult for others to guess. Employees should not share passwords with another people. It is essential to not use the same passwords for Twitter as for Facebook, Instagram, and other social media tools.(2)Enable Multifactor Authentication (MFA). This should be standard security procedure for everybody who uses the Internet today. Anyone attempting to log into an account is required to provide a code transmitted to an external device.(3)Update Security Settings (USS). Update the security framework on all digital and social channels consistently. There are many fantastic step-by-step privacy instructions available online to assist with setting up secure. They are often renovated, so revise them from time to time to make sure that nothing has changed.(4)Use the Block Button (UBB). The links sent by spammers should be ignored. Always report the account as spam for the sake of others who are less knowledgeable. The social networking service will monitor it and, if enough people take the same actions, remove the account. Use ad blockers on corporate devices. The most immature method is to block that person on social media.(5)Declining Friend Request (FR). Friend’s invitations from known people should be accepted. The request should be declined even if the users have several mutual friends. By looking for verified accounts, it is critical to identify the genuine social media profiles of celebrities, public figures, and businesses.
Therefore, the CCuTSFS summarizing the security is given as follows:
4.1.3. Calculations
We use the following mathematics to examine the effectiveness and ineffectiveness of every social media security measure and threat. We have the following two CCuTSFSs and , corresponding to the set of security and threats, respectively.
For each social media threat and security, we assign degrees of membership, neutral, non-membership and interval-valued degrees of membership, neutral, and nonmembership. These functions in the set and represent both the present and future influences of each threat and security. The CP between the CCuTSFS and is used to define the effectiveness of each security method against a particular threat. Table 3, shows the CP between the CCuTSFS and .
The CP describes the relationship between each set of elements, i.e., the condition and impact of a security on a threat. The levels of a membership indicate how effective a social media security system is in detecting a specific threat over a period of time. The level of abstinence reflects whether a security system has no effect or has a neutral effect in the face of a threat over a time period. The degree of nonmembership levels denotes a security inefficiency in the face of a certain threat with a time period. For example, the ordered pair describes the unique password against image retrieval and analysis. The UP protects the IRA because of this reason that unauthorized person does not access to the user’s device. Furthermore, it explains the present and future effects and impact of the ordered pair. The UP overcomes the threat of the IRA in the present because security effectiveness is better than the degree of ineffectiveness. A given ordered pair predicts the future security in the form of an interval.
5. Comparative Analysis
In this part, we compare the presented structure of CCuTSFR with some other preexisting structures such as FR, CuFR, CCuFR, IFR, CuIFR, CCuIFR, PyFR, CuPyFR, CCuPyFR, qROFR, CuqROFR, CCuqROFR, PFR, CuPFR, CCuPFR, SFR, CuSFR, CCuSFR, TSFR, and CuTSFR.
5.1. Comparison of FR, CuFR, and CCuFR with CCuTSFRs
The structure of FR and CuFR discuss only the membership degree with only one dimension. They are not capable of solving the multidimensional problem. The CuFR expressed the membership degree in both the present and future, but the FR described the membership degree only present aspect. These structures are unable to model periodicity. The structures of CCuFR discuss only the membership degree, which only shows the effectiveness of security measures on the threats. Consequently, they cannot offer a complete solution to the problem. The CCuTSFR examines all three levels, i.e., membership, neutral, and nonmembership with a complex number. They are capable to solve the periodicity. We consider the following two CCuFSs and , which represent the threat and security sets, respectively.
The CP of
CCuFR shows only the effectiveness of security against a particular threat. The membership grade effect of the first element on the second element in an ordered pair. As a result, these structures have limitations and consequently provide restricted information. Meanwhile, the CCuTSFR gives the complete information.
5.2. Comparison of IFR, CuIFR, and CCuIFR with CCuTSFR
The structures of IFR, CuIFR, and CCuIFR discuss the membership degree and nonmembership degree. The membership degree shows the effectiveness of security measures on the threats, and nonmembership shows the ineffectiveness of security measures on the particular threat. Therefore, they cannot show the neutral values of security measures against the threat. The IFR and CuIFR can solve only one-dimensional problems. The IFR and CuIFR are unable to multivariable difficulties. But CCuIFR can solve the periodicity. The CCuTSFR is superior to this structure because they are also discussing the neutral effect of the first element to the other. So, the structure of CCuTSFR discusses all stages, i.e., membership, neutral, and nonmembership with both aspects of the present and future. We consider the two CCuIFSs, , and , which describe the threat and security sets, respectively.
The CP of
It is clear from the CCuIFR, that it describes the effectiveness and ineffectiveness of the security measures against a certain threat. They are not discussing neutral value.
5.3. Comparison of PyFR, CuPyFR, and CCuPyFR with CCuTSFR
These structures discuss membership and non-membership. These structures are used to show the effectiveness and ineffectiveness of the security measures against a certain threat. These are the generalization forms of CuIFR. As the framework of IFS has its limitations in assigning the membership and nonmembership because their sum exceeds from [0,1] in many cases. For example, the ordered pair (0.8, 0.5) cannot be considered as an intuitionistic fuzzy number (IFN) because their sum exceeds 1, but sum This shows that the range of the Pythagorean fuzzy set (PyFS) is greater than that of IFS. The PyFR and CuPyFR are only one-dimensional. The CCuPyFR is discussing both amplitude and time frame. The CuPyFR and CCuPyFR have certain limitations as compared to the CCuTSFR because these structures do not describe the neutral effect.
5.4. Comparison qOFR, CuqOFR, and CCuqOFR with CCuTSFR
The qOFR, CuqROFR, and CCuqROFR describe the only success and failure effects. They do not describe the values of success and failure. CuPyFS improved the limitations that occurred in CuIFS, but there are still some duplets that cannot be categorized as IFN or PyFN. For example, the duplet is neither an IFN nor a PyFN because sum and sum If for = 3, then their duplet sum exists to the range of [0,1]. These structures are the generalization of CuPyFR and CCuPyFR, but these structures have certain limitations as compared to CCuTSFR.
5.5. Comparison of PFR, CuPFR, CCuPFR with CCuTSFR
These structures discuss all values such as membership, neutral, and nonmembership. These structures show the effectiveness of security measures against the threat; the neutral structure shows the ordered pair that has no effect; and nonmembership shows the ineffectiveness of the security against threat. The PFR and CuPFR are not showing the complex value with time frame i.e., PFR and CuPFR are only single dimensions. The CCuPFR describes the time frame or periodicity. The structure CuPFR and CCuPFR are using . So, there are some limitations to selecting the degree of membership, neutral, and degree of nonmembership because their sum exists to the unit interval of [0, 1]. The CCuTSFR has improved the limitations to choosing the membership. neutral, and nonmembership degrees are using .
5.6. Comparison of SFR, CuSFR, and CCuSFR with CCuTSFR
The structure of SFR, CuSFR, and CCuSFR improve the limitations of CuPFR using 2. A CuSFR based on membership, neutral, and nonmembership degrees, respectively, with the constraint that the sum of the square must exceed 1. The CuSFR increased the range of a CuPFR. These structures, SFR and CuSFR are discussing only one-dimensionally. The CCuSFR is the generalization form of a PFR, CuPFR, and CCuPFR. But the CCuSFR faces some limitations in choosing the degrees of membership, neutral, and nonmembership as compared to the CCuTSFR.
5.7. Comparison of TSFR and CuTSFR with CCuTSFR
The structure TSFR and CuTSFR improve the limitations of CuPFR and CuSFR using = n. The CuTSFR increased the range of the CuSFR and CuPFR. The CCuTSFR is the generalization form of the CuSFR and CCuSFR. Some ordered pairs exist that their sum of the squares exceeds the unit interval then occurs the CuTSFR for using the parameter These structures TSFR and CuTSFR are discussing only one-dimensional. They have not defined the duration of time. The CCuTSFR has defined both amplitude and time frame. The CCuTSFR is the better form of all the preexisting frameworks. The summary of all fuzzy preexisting structures is defined in Table 4.
6. Conclusion
In this paper, we explored the novel concepts of the complex cubic T spherical fuzzy set (CCuTSFS), complex cubic T spherical fuzzy relation (CCuTSFR), and CP of two CCuTSFSs. Moreover, numerous types of CCuTSFR are also discussed, including CCuTS-reflexive-FR, CCuTS-irreflexive-FR, CCuTS-symmetric-FR, CCuTS-transitive FR, CCuTS-equivalence classes-FR, and many more. The CCuTSFS is the generalization form of the TSFR and IVTSFR. The novel concept of the CCuTSFRs is the more generalized form of all the predefined structures. Because this structure covers all levels i.e., membership, neutral, and nonmembership with complex number, this structure describes all levels of both present and future aspects. They are better at dealing with fuzziness. The goal of these new frameworks and novel modeling procedures is to solve social media security problems. The proposed study is used to analyze the relationship between various types of security methods and threats. They define the effectiveness, neutral, and ineffectiveness degrees of both the present and future. The advantage of these structures is that they are used to define all the three stages membership, neutral, and nonmembership with both the amplitude and phase term. The presented framework is compared to the other previous methods. As a result, the CCuTSFR is more advanced than all the other preexisting structures. In the future, the CCuTSFR will be used for better outcomes. This idea can be used for several interesting and diverse applications.
Data Availability
No data set was generated or analyzed during the current study.
Conflicts of Interest
The authors declare that there are no conflicts of interest with this study.
Acknowledgments
The authors are grateful to the Deanship of Scientific Research, King Saud University for funding through Vice Deanship of Scientific Research Chairs.