Sociocentric SNA on fuzzy graph social network model

The paper presents a scalable and generalized approach to social network analysis using fuzzy graph theory. It proposes an intelligent sociocentric approach that calculates the degree of potential relationship of a social network of finite size by proposing a fuzzy graph social network model. It takes into account social entity functional and relational attributes simultaneously. It computes the degree of potential relationship of a social network in two steps. The first step computes the fuzzy pairwise relationship between all social nodes or entities by incorporating the proposed fuzzy node activeness index parameter with an online and offline communication relationship. The second step further uses all fuzzy pairwise relationships calculated in the first step to calculate the degree of potential relationship of a social network. It uses an astute function that utilizes weighted arithmetic and geometric means of the relationships between entities. It also uses two weights—betweenness and closeness centrality of an entity in the social network. The paper performs the experimental work on a small WhatsApp social network of undergraduate students in the university for 6 months. Hence, the paper proposes the degree of potential relationship in social networks, which may be used as a global parameter to compare different social networks by simultaneously incorporating social node's functional and relational attributes.


Introduction
Nowadays, Web 2.0 (Murugesan 2007) power aids like Blogs, Micro Blogs, Podcasts, Wikis, ePortfolios, Social Networks, and Social Bookmarking allow people to work together and express their ideas or experiences. People gain power and better chances in all fields by using these tools efficiently. Out of these tools, social networks are one of the most powerful, user-friendly, easy-to-use, most explored, all-purpose, and robust tools for all classes of people. A social network (Wasserman and Faust 1994;Hanneman et al. 2005;Musiał and Kazienko 2013) is a collection of social entities or nodes linked together with one or more kinds of relations. Social networks efficiently use structured, unstructured, multi-lingual, audio, video, and other types of data or information. Our previous work (Rani et al. 2018e) has explored the different dimensions of social networks with all possible issues, challenges, and their potential applications, viz. rumor identification (Srinivasan and Dhinesh Babu 2019), fake news detection (Sahoo and Gupta 2021), Twitter analysis (Noor et al. 2020), cognitive or mental behavior analysis (Bouarara 2021), recommendation system (Zhang et al. 2017), etc. Social network analysis (SNA) (Wasserman and Faust 1994;Hanneman et al. 2005) is ''a different aspect of research that mainly aims at relationships among the social nodes, on the structure and applications of these relationships, primarily in social and behavioral sciences.'' Broadly SNA is divided into two main categories-egocentric and sociocentric SNA. The egocentric analysis focuses on an individual node in the network and studies its effect, whereas sociocentric SNA focuses on a group of Communicated by Oscar Castillo.

Contribution
The significant contribution of this paper is that it proposes a novel intelligent fuzzy graph social network model (FGSN) by simultaneously incorporating the fuzzy concepts at the node and relational attributes. The main contributions of the paper are as follows: • It proposes one fuzzy parameter to a node and another to an edge depending on an online and offline communication. It captures the uncertainty at both node and connection levels. • It proposes a fuzzy pairwise relationship between all social nodes using the proposed fuzzy node activeness parameter with their online and offline communication relationship parameters. • It proposes the social network's sociocentric degree of potential relationship using all fuzzy pairwise relationships calculated in the above step with the proposed astute function, which utilizes weighted arithmetic and geometric means of relationships between entities. • It proposes the FGSN Model to capture uncertainty simultaneously at the node and edge attributes. • It experiments on the real dataset using betweenness and closeness centrality measures as node attributes. Closeness centrality provides better results, almost three times in average than the betweenness.
The paper organizes the rest of the paper as follows: Section 2 explores the correlated tasks done in this direction. Section 3 introduces the central concept of fuzzy graphs with other preliminary mathematical definitions explored in this work. Section 4 discusses the proposed work. Section 5 illustrates the experimental results with possible applications. Last section 6 ends the paper by concluding its future scope.

Related work
The research papers (Faust 2006;Lippold and Burns 2009;Perkins et al. 2009;Johnson et al. 2012;Rani et al. 2018bRani et al. , c, 2019c have compared various social networks from different angles. Faust (2006) has compared different social networks depending upon the number of nodes and edges in a network. This comparison is performed on the triad census, involving local properties, dyad distribution, and network density. Perkins et al. (2009) have compared social networks derived from ecological data. They have studied the disease transmission in rodents networks formed by rodent contact. They have explored two methodologies,-''Capture-Mark-Recapture'' and ''Radio tracking'' simultaneously on a diverse group of rodents. They have analyzed that the capture-mark-recapture performs better with a high rodent density. On the other hand, radio-tracking performs better with a low rodent density. In this, researchers have explored a particular kind of rodent social network. They have explored metrics-average contact rate, betweenness and closeness centrality measures, and connectedness score in the comparison task. This methodology cannot be generalized, so we need a generic method for the social network comparisons task. Lippold and Burns (2009) have explored ''a comparison between social networks of adults with intellectual disability and those with the physical disability.'' They have directly contrasted the relational attributes of people in two specific types of social networks formed due to their intellectual disabilities (IDs) and physical disabilities (PDs), respectively. They have explored the parameters, viz.-life experiences and practical and social support, based on statistical measurements. They have concluded that the PDs social networks have more nodes than IDs social networks. So, they have researched a particular case where functional, social support, and life experiences are incorporated. Johnson et al. (2012) have done ''the comparison of the email networks and the survey-based social networks in a bank.'' The authors have done a detailed bank study to match the formation of email networks with communication, advice-seeking, and friendship networks. The authors have used an egocentric analysis also. They have done an offline survey to map to their email networks.
We have also proposed four sociocentric direct methodologies for the comparison task of diverse social networks. In the first methodology, we have proposed a ''Quantitative solution using OWA operator'' (Rani et al. 2018b). We have proposed one generic quantitative approach for the comparison task incorporating six vital metrics at all levels of networks by using the Gephi tool. Here, we have also provided weightage to each metric according to their role in a social network and computed one global quantitative value for every social network for the comparison task. Moreover, we have used diverse social network data sets for the comparison task. This generic methodology is better than the previous ones because we have assigned weightage to all attributes based on their importance factor in the network. As social networks are dynamic, they have uncertainty factors. We have captured this uncertainty factor in our second proposed methodology (Rani et al. 2018c)-''qualitative solution using fuzzy membership function'' by using fuzzy sets (Zadeh 1965). This methodology is more intuitive and human-readable as here parameters are clearly outlined by fuzzy concepts and experimented on the diverse data sets. So, we have explored a fuzzy concept-based methodology for the comparison task of diverse social networks that works better than any traditional approach. In the past, all the above papers have not considered the user perception or view toward the social networks that we have tackled in our third proposed methodology. This user perception view is not incorporated in our earlier research work also. So, in our third methodology (Rani et al. 2019c) user perception, viz. user point of view toward particular social network has been focused. We have introduced a single global idealness variable computed using fuzzy k-means clustering for the comparison operation. The limitation of this is that it cannot tackle outliers.
In our recent work , we have explored fuzzy-k-medoids (Al-Zoubi et al. 2010;Sabzi et al. 2011) to tackle outliers. We have compared multiple social networks with static and dynamic parameters. We have explored static parameters, viz. safety, connectivity, instantaneity, ease of use, friendliness for new user, and benefits. These static parameters have used a constant score from one to ten. They have been given user opinion scores based on their experience on that parameter. We have explored the dynamic parameters-time spent and number of times opened, with a variable score. They have been rated according to the user experience on that social network. These have been normalized to map with static parameters. We have explored linear scales for these parameters, which determine nodes' inclination toward the respective social network and its influence on their growth. We have visualized our findings on a 3-dimensional conical model to show why one kind of social network is superior to other social networks. All the above papers have not considered the content or degree of relationship or communication that we consider in this paper. , 2014 have proposed an indirect methodology to compare social networks. They have computed m-size relationships by employing pairwise fuzzy relations between m nodes by applying the OWA operator, where m is a positive integer. These papers have boosted the analysis dimensions from two nodes to m nodes. They have taken the fuzzy adjacency matrix as the input and captured uncertainty only in the relationships of nodes or entities in the network. They have employed fuzzy membership mapping to express the degree of relationship within network nodes and OWA-based aggregation to calculate the relationships between m entities. Their approach is better than the traditional binary relationship because it conserves more details or statistics, although entity attributes have not been incorporated. However, they have not explained how and where they obtained these fuzzy pairwise relationship values. Neither, have they provided any quantification aspect for the relationships in social networks. This paper has taken one such parameter-the communication parameter upon we can directly quantify the relationship, as discussed in (Rani et al. 2019a). Communication structures prevailing among social nodes in the network are the chief parameters that make everyone's life easy and comfortable. The mode and frequency factors of communication can directly quantify the relationships among the social entities. The broad types of online and offline communication modes in the social network are the conditions for quantifying relationships between social entities. So, the communication structure always reflects the relationship between entities. Earlier works Brunelli et al. , 2014 have considered the symmetrical relationship between two entities in social networks. But in real life, it is not possible as no two relations are perfectly symmetric. So, it is always suitable and advisable to model the relationships between social entities as asymmetrical. The earlier work Brunelli et al. , 2014 has employed only one type of relationship. But in actual life, there always exist multiple simultaneous relationships between the entities in a network. This paper uses two asymmetrical relationships depending on the online and offline communication parameters. Besides this, the earlier work has presented a recursive approach, but discovering a recursive answer is not always practical in the presence of complexity, dynamism, and uncertainty dimensions. In earlier works Brunelli et al. , 2014, the OWA operator is explored to aggregate all network entities' relationships. The weights explored in their work are selected randomly and must have some physical significance. These weights should depend on both nodes and the connection or relational attributes of a network.
The previous work has explored the fuzzy set theory to tackle the uncertainty only in the relationship between entities. Real social networks are full of dynamism, so there is also uncertainty in node or entity parameters. In (Yager and Rochelle 2008), Yager has presented the idea of ''Intelligent Social Network Modeling,'' where he has incorporated both entities attributes and relationship characteristics of social entities in the vector-valued form. He has provided a way to capture uncertainty simultaneously in both node and relationship attributes. He has explored a hybrid theory using the graph and fuzzy set theory. This unique combination may provide a good analysis of social networks. Today, we may use it to study, analyze, and visualize complexity in the relationship among social entities. Thus, we employ this ''Intelligent Social Network Modeling'' in this paper to make it an intelligent approach by using fuzzy graph theory.
We have predicted the Facebook group relationships using reactions to posts in our previous work (Rani et al. 2019b). It predicts a methodology for aggregating entities relationships with matching interests and views depending on how they react to shared posts on Facebook. The limitation of this methodology is that it is a particular solution. It is not providing a generalized solution for every social network. The ''Fuzzy set network model with OWA'' Brunelli et al. , 2014 relates to the real-world issues in social networks, but it has a lot of research gaps, as explained above. Hence, it may not be applied effectively to complex, dynamic, and uncertain real-life social networks systems. Thus, modeling social networks using fuzzy sets on the edges is an incomplete solution. It does not map the characteristics or attributes of entities in the real-world social network. The FGSN model is the one of solutions that can tackle uncertainty in both features-node and relationship that is missing in all the above papers in the literature. Hence, this paper proposes a FGSN model in place of the fuzzy adjacency matrix model to handle uncertainty in nodes attributes and their relationships attributes. The fuzzy adjacency matrix is no doubt suitable for small size social networks. But it is very complex for an extensive or dense network. It is not efficient and ideal in the case of scalable social networks. The fuzzy graph model of the social network is suitable for both small and large social networks. The other limitation of earlier work is that fuzzy weights are not attached to the nodes. This paper tackles this issue by providing fuzzy vertices in the graph by assigning one fuzzy parameter called a node activeness parameter to each of them. We select the betweenness and closeness centrality index as the weights to nodes. Both betweenness and closeness centrality indexes are helpful while calculating sociocentric relationships. But the experimental results demonstrate that the closeness centrality index is superior to the betweenness centrality index to obtain a better value for the sociocentric relationship degree.

Fuzzy graph and other preliminary definitions
In 1975, Rosenfeld (1975) introduced the concept of Fuzzy graph theory. He operated fuzzy relations on fuzzy sets (Zadeh 1965). In the fuzzy graph, we attach some uncertainty value to every node and edge that defines the participation value of every member (node and relationships). In literature, several researchers have discussed and explored the fuzzy graph in their research work (Koczy 1992;Samanta and Pal 2013;Samanta 2014;Malek et al. 2015;Das et al. 2021). Samanta and Pal (2013) have studied fuzzy graphs in ''Telecommunication System & Management.'' Samanta (2014) has also explored fuzzy graphs concerning the registration process of a new user to prevent fake users. Malek et al. (2015) have explored fuzzy graphs in social network analysis, especially in clustering and identifying the overlapping communities in a complex social network. Also, Koczy (1992) has explored the concept of fuzzy graphs for the evaluation and optimization of networks. Koam et al. (2020) have explored the fuzzy graphs application to identify crimes on-road and marine. Das et al. (2021) have explored fuzzy graphs in the identifications of COVID2019 affected central regions in India. Sinthamani (2021) has explored the application of fuzzy graphs to reduce traffic congestion. The formal definition of the fuzzy graph (FG) is given below in Definition 1.
Definition 1 [Fuzzy graph (Rosenfeld 1975)] We denote the fuzzy graph ðG n Þ by 3-tuple V s ; r V ; l E h i : It consists of fixed finite set V s of n nodes, along with two membership mapping,r V : V s ! 0; 1 ½ and l E : represent the membership mapping values of the vertex v i and the edge v i ; v j À Á in G n , respectively. Where ð1 i; j n) and the simplest two-node fuzzy graph model with node n 1 and n 2 is depicted in Fig. 1.
We explore three other mathematical definitions in this paper from literature. They are following.
Definition 2 [Degree centrality (Jackson 2010)] Degree centrality is proportional to the number of direct connections of a social entity to other social entities in the social network G. Mathematically, we denote the degree centrality of an ith entity in social network G by DC i G ð Þ and is defined as follows: where D i (G) = number of direct connections of ith social entities in G. There are two types of degrees in the case of directed social network, viz.-IN degree and OUT degree. The IN degree is the number of incoming links to the social entities from other entities. Similarly, the OUT degree is the number of outgoing links from social entities to other social entities.
Definition 3 [Betweenness centrality (Freeman 1977)] Betweenness centrality b ð Þ is proportional to the shortest paths passing through that social entity. We denote the betweenness index of an ith entity in social network G by B i (G) (or B i ð Þ). We define it as the ratio of the geodesic shortest distance between kth and jth social entities, on which ith entity lies, to the number of the shortest distance between kth and jth entities. Mathematically, we define it as follows: where P k; j ð Þ is the total number of shortest distances from entity k to j entity, and P i k; j ð Þ is the geodesic shortest path among all k and j on which i entity lies. It defines the importance of a social entity in linking other entities in a social network. For normalization, it is divided by (n -1)(n -2) (for directed) or (n -1)(n -2)/2 (for an Definition 4 [Closeness centrality (Freeman 1978)] Closeness centrality (c) is inversely proportional to the sum of the shortest paths between the entity and all other entities in the social network. We denote the closeness centrality index of an ith entity in the social network G with N number of entities by C i (G) or C(i). We define it as the reciprocal of the global shortest distance of ith entity to all other jth entities. Mathematically, we define it as follows: where i 6 ¼ j, l(i,j) = global distance between i and j entities.
It is a measure that describes how close a social entity in social network G is to all other entities.

Proposed work
People are the social entities that communicate with their friends/relatives/known/unknowns by forming different social networks, viz. Facebook, E-mail, Google?, Hike, Viber, and WhatsApp. Broadly, these social entities in their social networks communicate either in an offline or online mode, or both methods or forms. The degree of the relationship depends on the amount, frequency, and way of communication. The nature and frequency or counts of communications between entities can directly quantify the relationships between social entities, as discussed in Rani et al. (2019a). We consider the frequency of offline and online data communicated between entities as characteristics of the connection between pairs of entities. Thus, the degree of relationship between the two entities is directly proportional to their degree of communication. More frequency of communication, whether online or offline, reflects more degree of relationship, whereas less frequency of communication reflects less degree of relationship. For the same purpose, the amount of frequency of communication is measured in terms of the number of online and offline communications, both with some weights aand 1 À a ð Þ, respectively, where a is the proposed online fuzzy parameter of a node, as we can represent it by fuzzy set. Hence, the function for calculating the degree of communication between an ordered pair of social entities s i and s j has in its numerators the following two variables, viz.
On ij and Of ij . We define On ij ; Of ij ; On iÀsum and Of iÀsum as follows: On ij = The number or count of online communications per day between s i ands j : On iÀsum = The number or count of total online communications per day between s i and all entities in a social network. Of ij = The number or count of offline communications per day between s i and s j Of iÀsum = The number or count of total offline communications per day between s i and all entities in a social network.
We propose the fuzzy membership degree of relationship between a pair of social entities s i and s j denoted by R ij or degree s i ; s j À Á ; including above all four variables, as follows: if some communication between s i and s j 0; if no communication between s i and s j Here, a i is an online mode index (OLI) of entity s i per day called the activeness degree of ith entity, where a i 2 0; 1 ½ : It defines the fraction of time a social entity s i is in the online mode per day. It measures the degree of activeness of an entity in a network. It depends on the total time or fraction of hours spent by an entity in the social networks per day in online mode ðT H on Þ; whether communicating or not. We propose a i as follows: if s i in the online mode for the whole day T H on 24 ; if s i in the online mode to some extent 0; if s i in the offline mode for the whole day A social entity is in any of two possible states at a particular time-online and offline. So 1 À a i ð Þis an offline mode index (OFI) of an entity s i . This section proposes a novel Intelligent FGSN that simultaneously incorporates the fuzzy concepts at the node and relational attributes. We use ideas from the intelligent social network model of Yager andRochelle (2008), Yager (2009) in addition to fuzzy graphs in social networks. Our proposed intelligent model is more capable and robust for handling uncertainty, as the social entity, i.e., node attributes, is also added to the parameters of the relationship. This fuzzy graph model approach is scalable, as we can employ it for both small and large social networks. In this intelligent model, we assign three weights to the entities. The first is the fuzzy weight or parameter as discussed and defined in Eq. (5). We denote it by a; indicating the entity's online activeness degree or index. Here, a represents the personal attribute of the node that defines the fuzzy presence of a node in the network at any instant. We use it to propose the fuzzy relationship values between each pair of entities connected directly to the network by their communication parameter. The second one is called the betweenness degree or index of a node. We denote it by b that calculates the whole group's potential relationship, i.e., sociocentric relationship. The last one is the closeness degree or index of a node which is denoted by c that also calculates the sociocentric relationship.
Formally, we propose a novel Intelligent FGSN model with six-tuples, S c ; E; a S c ; b S c ; c S c ; l E : It consists of a finite set of social entities S c forming the social network, a subset of the domain S ¼ s 1 ; s 2 ; s 3; . . .; s n È É with two membership functions sets (a S c and l E ) and two weight-sets ðb S c andc S c Þ: The membership function set, a S c is a set containing the activeness fuzzy parameter values of each entity in a social network. Its values make the presence of an entity fuzzy in a social network. We calculate its values by using Eq. (5). The weight-sets b S c andc S c ; contain all values of each entity's betweenness and closeness index forming the social network. We use the Gephi tool to calculate their values. Mathematically, we describe them as follows: The edge set E is the set of an ordered pair of entities, having communication between them and expressed as E S c Â S c : It depicts the relationship between an ordered pair of entities with membership degree set l E ; between 0 and 1. It is a fuzzy relation defined on set S c depending on online and offline communication frequency. Formally, we define it as follows: We compute it using proposed Eqs. (4) and (5). It makes an edge fuzzy in the network.
We divide the proposed work mainly into two steps. The first step calculates all fuzzy pairwise relations between all ordered pairs using the proposed Eqs. (4) and (5). The second step evaluates the sociocentric relationship among the group. The sociocentric relationship is the total potential relationship between all entities in the social network. So, we define a sociocentric relationship as the relationship degree between all entities participating in social networks of size m, where m [ 2 is any natural number. Formally, we denote the degree of relationship between m entities {1, 2, 3, …, m} by membership function To calculate this, we have taken as input the relationship degree between each pair of entities calculated in the first step using proposed Eqs. (4) and (5).
Let k be the total number of pairs of entities in social networks, excluding the self-communicating edges and the edges that are not communicating. We find the maximum possible value of k by the binomial coefficient ( m C 2 ). We propose that the calculation of the sociocentric relationship is best when we employ the arithmetic mean (AM) and geometric mean (GM) with some weights factor. There are some limitations when we consider the AM and GM independently, as discussed in detail with an example in the paper (Rani et al. 2018d). Thus, we propose to use both AM and GM in the group analysis. Using a node's betweenness or closeness as an essential weight factor is always better, so we assign weights according to their betweenness or closeness centrality measure index. The weight depends on the importance of all the entities participating in a network. The importance here is the simple function of each entity's betweenness or closeness index in a network. Further, the weights here use both AM and GM of betweenness or closeness centrality of all entities. Hence, we propose an astute function, which uses a hybrid approach with the benefits of both the AM method and the GM method with weights function using weights b and c assigned to each entity in a network. b is the betweenness centrality index of each entity in a social network. c is the closeness centrality index of each entity with other entities in a social network. The astute function directly finds the degree of relationships among m entities of FSN, depending on their online and offline communication frequency. The simple process using both AM and GM without using b and c is defined as follows: Here,q is the or-ness coefficient (0 q 1). q ¼ 0:5 if no specific purpose is specified, which means an equal weight of AM and GM. The value of q varies from application to application. As a special or particular case, if q ¼ 1, the proposed function becomes similar to the AM function. Similarly, if q ¼ 0, the proposed function becomes identical to the GM function. Hence, any intermediate value will blend both the AM and GM functions, making this function more suitable for any social network. To make it a more intelligent approach, we propose to use the weighted AM and GM rather than applying simple AM and GM with some q parameters. Formally, we propose to compute the degree of sociocentric relationship among m social entities l S c À Á by the astute function using both AM and GM with weights borc as parameter d as follows: where d can be b or c The above-defined sociocentric relationship among m social entities l S c d ð Þ as defined in Eq. (7) is astute because it captures uncertainty in both entity and edge attributes of entities in the social network. It calculates the degree of potential relationships between m numbers of entities by employing the entity and the connection characteristics of the entities. Figure 2 shows an intelligent FGSN with four entities s 1 , s 2 , s 3 , and s 4 with fuzzy weights a i ; b i ; c i and R ij : Here, a i ; b i ; and c i represent the activeness, betweenness, closeness parameters, respectively, of the s i entity in the network, and R ij represents the degree of relationship between s i and s j entities in the social network conditions 1 i 4 and 1 j 4: Here, we need to note that if two entities do not have an edge or relationship, we take its degree of relationship as zero.
Hence, this section has proposed the sociocentric SNA on FGSN model that incorporates the entity's attributes and relationship attributes between entities, making it more robust and intelligent to use. The algorithm for the proposed methodology is as follows:

Experimental work
Nowadays, students are the most influential class who wisely use the maximum benefits of technology and social networks. Clafferty (2011) has aimed to facilitate a social networking environment for students. They have revealed that the social networking environment is an effective and valuable strategy for students. Students explore and use these social network tools very interactively and intelligently. We need to analyze their social network to study the most influencing or potential group in class depending on their communications. A student usually communicates Sociocentric SNA on fuzzy graph social network model 13209 with their friends/classmates using different social networks like WhatsApp, Facebook, Google ? , Viber, Hike, email, etc. WhatsApp messenger social network is the most influential and popular among them, as explored in papers (Yeboah and Ewur 2014;Rani et al. 2018aRani et al. , 2019a. Recently students are the main class, mainly exploring and studying WhatsApp for their academic, personal and professional purpose (Clafferty 2011;Yeboah and Ewur 2014;Rani et al. 2018aRani et al. , 2019c. The paper (Rani et al. 2019a(Rani et al. , 2019c has also concluded, ''WhatsApp is the most preferred social media platform among students. It has also concluded that communication is the topmost parameter on which we can do relationship analysis of the social network.'' We can also see the popularity of What-sApp from the website 1 statistics which provides an idea of the most popular global mobile messenger apps as of October 2021, based on the number of monthly active users. As of that month, 1.8 billion users access WhatsApp messenger monthly. Also, the growth rate achieved by WhatsApp after its launching years is exponential that too in significantly less time. Its growth rate is much faster than Facebook and Twitter, as clearly presented in Fig. 3. The main reason is that it provides us lot of features. It is the most accessible, straightforward, user-friendly, and fastest method to transmit SMS, images, video, audio or video calls, etc. Rani et al. (2018d) have also explored that youths are more user-friendly and comfortable with its usage. It is also more desirable for the students as it is easier and fastest to use. It is also more effective, reliable, robust to use, and money efficient. The students use it to inform, share, and discuss personal information exchanges and educational learning purposes. We have concluded that ''The Idealness of WhatsApp is much superior to that of Facebook'' from papers (Rani et al. 2019c. It is concluded from the papers (Rani et al. 2019a(Rani et al. , 2019c that students choose ''Facebook in terms of time spent, the number of times opened and friendly to new users.'' The reason is that Facebook's newsfeed is psychologically more addictive than What-sApp chats. However, when it comes to parameters, viz. people connectivity, ease of usage, educational purposes, transfer of files, and safety, WhatsApp is superior to Facebook with some significant margin. So, we have experimented with the proposed methodology on the small size ''Student WhatsApp fuzzy graph social network'' (SWFSN) for 1 week, 1-month, and 6-month data using Python (Python 3.9.7 on macOS) and SNA Gephi SNA tool. In SWFSN, entities or nodes are the students, and the edges between any two nodes are the relationships between the students. The communication parameter between nodes defines the relationships between the students. The relationship between students on WhatsApp is directly calculated based on the communication characteristics. For experimental results, we take the small size SWFSN with five students, student1 s 1 ð Þ, student2 s 2 ð Þ, student3 s 3 ð Þ, student4 ðs 4 Þ, and student5 ðs 5 Þ. We collect data from students by providing them data collection survey form. But, we have shown 1 week of data for the understanding purpose. Their online and offline communication relationship values are collected through Google survey form and stored in an excel file. Depending on the data, the directed SWFSN for 7 days, i.e., 1 week, is created, visualized, and analyzed using the Gephi tool. The directed SWFSN visualization for only 7 days with their b and c values is shown in Figs. 4,5,6,7,8,9 and 10, respectively. We use the following color coding scheme in all these figures.
• We show the concept of the degree centrality of each student with the different color-coding schemes by using three colors-Blue, green, and red. We use that the blue color indicates a higher degree of centrality than the green one, which is higher than the red one. So, the red color depicts the lowest degree of centrality, whereas the blue represents the most elevated. • We encode the width of the link or edge, i.e., relationship degree between students, with the different colors-blue, green, and orange. Here, the blue color represents more weights on edges or more degree of the relationship than the green, which is more than orange. • We use the concept that the width of the link or degree of relationship is directly proportional to the frequency of communication. The higher the width of the connection between two entities, the more communication relations between them, and the lower the width of the link between two entities, the lesser is the frequency of communication. • We use a darker color to depict more degree of communication, whereas lighter color represents a minor degree of communication.
We provide 1st day online and offline communication frequencies in Tables 1 and 2, showing the number of online and offline communication frequencies between all Fig. 3 WhatsApp Faster growth rate pairs of students, respectively. We compute the online fuzzy index (OLI) a, students' activeness in its online mode, using Eq. (5) as social network size is taken as five so i = {1, 2, 3, 4, 5}. Here (1 -a i ) is called the ith student's offline index (OFI). It signifies the non-activeness of a student in the group. We mention the collected values of online hours spent by all students in Table 3. We calculate the values of the OLI (a) and OFI 1 -a and provide them in Table 3. We calculate the pairwise fuzzy relationship between five students using Eqs. (4) and (5) for 1st day and provide them in Table 4. It uses both online and offline communication values mentioned in Tables 1 and 2, along with activeness and non-activeness index values simultaneously mentioned in Table 3.
We use the Gephi tool to calculate the betweenness (b i Þ and closeness c i ð Þ index value of each ith student in SWSN for 1st day and mention their values for all five students in Table 5. We normalize these values to lie in the interval       Fig. 4. Further, we mention their b and c values in the text on the respective student node. The first-day communication shows that s 4 has maximum closeness and betweenness index than others, and we encode their more degree by blue color. We encode the maximum frequency of communication between s 1 and s 2 which has the potential color blue with a maximum edge width. The s 1 and s 2 student nodes have the same degree, and that is why they are encoded by green color in Fig. 4. We encode green as the second potential color after blue. s 2 has more closeness index value than s 1 : The students s 3 and s 5 have a minimum degree with a minimum number of direct communication links. They have minimum betweenness and closeness index values encoded with red color. The red color represents its least potentiality in all three colors. We use a similar encoding with three colors: blue, green, and orange for the edges or degree of communication between edges. Here, blue or its version-grey color edges show more degree of communication than the green one, which is more than orange color edges. More width of the same color on edges shows more degree of communications than less width of the same color. We employ the same color coding for nodes and edges for all other 6 days as we have done for the first day. We have provided data for the first day only. But their visualization directly reflects the communication characteristics in figures from Figs. 5,6,7,8,9 and 10. We employ the formula proposed in the last section in Eq. (7) to compute the degree of group relationship for the first 7 days. The degree of group relationship value calculated with c is 0.206436278, more significant than the value (0.130021957) using b for the 1st day. We calculate the degree of sociocentric relationship values for the SWFSN of students for 1 week, i.e., 7 days, using b and c, and mention in Table 6. The average sociocentric relationship value for a week using the betweenness index and closeness index is 0.077983966 and 0.227699880, respectively. Therefore, the value in an average case using the closeness centrality index is around three times more than the betweenness centrality index.
Further, we calculate the degree of sociocentric relationship values for the SWFSN of students for 1 month and 6 months, respectively, using b and c and represent in Figs. 11 and 12 with blue and brown color, respectively. We get the same conclusion from all experiments, whether 1 week, 1 month, or 6 months. Using the closeness centrality index, we get a better relationship between five students than the betweenness index value. It is almost three times better than the betweenness centrality index, as    seen from Figs. 11 and 12. Hence, the closeness centrality index is a better parameter than the betweenness index while quantifying the group's relationship. This fact is better justified by the actual theoretical fact that if each student is directly connected, then the loss of information while communication between students is minimal and will be secure and reliable in terms of confidentiality. But if other students connect students, there is always a loss of information at each link during the communication. In the latter case, there is a possibility that the meaning of information may get changed or reversed. Therefore, closeness index is an appropriate approach for calculating the sociocentric relationship. This methodology is scalable and applicable for the extensive or dense social networks, for better and more accurate results. This work is advantageous for assigning projects for fruitful outcomes to potential fixed-size groups of students depending on their degree of sociocentric relationship in that group. It can be used to find the most influential subgroup in a social network. We calculate the degree of sociocentric relationship values for the most influential subgroup SWFSN of students over 6 months using c and provide in Table 7. It is observed that the most influential subgroup of size two that includes students s 1 and s 2 ; (1, 2), because this pair has the maximum value (0.84) of the degree of relationship value out of all possible teams or  pairs. If a project is assigned to two students, this is the best group. It is also observed that the most influential subgroup of size three that includes students s 1 ; s 2 and s 4 ; (1, 2, 4), because this subgroup has the maximum value (0.54) of the degree of relationship value out of all possible subgroups of three sizes. For assigning a project to a group of three students, this is the optimal group. Similarly, it is demonstrated from Table 7 that the most influential subgroup of size four includes students s 1 ; s 2 ; s 4 ; and s 5 ; (1, 2, 4, 5) because this subgroup has the maximum value (0.49) of the degree of relationship value out of all possible subgroups of size four. If a project is assigned to four students, this is the potential group for that assignment. Similarly, this proposed work is advantageous for the indirect comparison of social networks, making potential groups or committees in an academic/non-academic organization depending on their communication interactions. We may use this to recommend a possible group to form any trust or society. It may be a suitable methodology for community detection in social networks. It may also apply to the medical field in selecting or recommending an excellent team of doctors with good agreement with their positive communication. Even we can deploy our model to find the best potential family or subgroup in a family with a reasonable consensus. In future work, we may use this model with the type of communication-positive, negative, and neutral as proposed in the paper (Rani et al. 2019a).
Fuzzy computing and quantum computing are today's disruptive technologies. Fuzzy computing provides intelligent and cost-effective solutions for real-life applications, where it tackles uncertainty and complexity, as we have discussed in the previous sections of this paper. In contrast, quantum computing is a future cutting-edge technology that will provide high-speed automatic solutions to highly complex and big data analytics problems, as discussed in the article (Wang et al. 2022). Hence, their hybrid version quantum fuzzy computing (Schmitt et al. 2009;Reiser et al. 2016; Ishikawa and Kikuchi 2021; Nȃdȃban 2021) will provide fast, automatic, and accurate solutions for highly complex and big data problems by simultaneously capturing uncertainty at both the functional and structural levels. Hence, we may use this model on quantum computers for fast processing by tackling uncertainty in complex and vast social networks parameters.

Conclusion
We have successfully proposed the suitable fuzzy graphs social network model that calculates the degree of potential relationship in complex and uncertain social networks. The experiment results conclude that in sociocentric analysis, the closeness centrality gives a better value, almost three times in average case than the betweenness. We get the same conclusion from all experiments, whether 1 week, 1 month, or 6 month. We get a better relationship value than the betweenness index value using the closeness centrality index. It is almost three times better than the betweenness centrality index. So, we plan to focus only on the closeness extended centrality index by incorporating the weights of the communication relationship between the entities in future work. Moreover, the paper has considered two types of broad communication modes-online and offline. But for better, accurate, and stable results, it will be considering all the possible types of communications-online and offline with text, picture, audio, and video communication structure in the subsequent work. Besides this, we will extend the study for dense social networks. Also, we will try to add the trust factor as another potential parameter along with the communication. Further, we may use this methodology to generate one global parameter to compare social networks by simultaneously incorporating social entities' functional and relational attributes.