Applications of Neutrosophic social network using max product networks
Abstract
Neutrosophic graphs deals with more complex, uncertain problems in real-life applications which provides more flexibility and compatibility than intuitionistic fuzzy graphs. The aim of this paper is to enrich the efficiency of the maximized network in accordance with time management and quality. Here we maximize three neutrosophic graphs into a fully connected Neutrosophic network using the Max product of graphs. Such a type of network is formed from individuals with unique aspects in every field of work among them. This study proposes the max product of three graphs and forming a single-valued neutrosophic graph to find the efficient time management in the flow of information on a single source time-dependent network of single-valued neutrosophic network. The proposed approach is illustrated with applications. Also, a spanning tree algorithm comparative study is done with Said Broumi et al. [15] and enhanced the result by minimum score function.
1Introduction
Graph theory, as a convenient mathematical tool, has a broad spectrum of uses in computer science, electrical engineering, system analysis, operations research, economics, networking, and transportation. The graph arises as a mathematical, graphical representation of the practical, real-life problems in those problems. A graph is a collection of sets (V, E), where V is a non-empty set of vertices connected by E, whose constituents are edges or links. Representing a problem as a graph provides a significant perspective and clarifies the situation. A network is typically a graph model with a set of nodes connected by edges or links. Networks give us a flexible framework for identifying and observing complex systems. The study of complex social networks is a crucial concept that comprises several disciplines. Complex systems network theory provides techniques for analyzing system of interaction structure, represented as networks. These complex networks are generally defined by simple graphs that consist of vertices representing the objects under exploration. For example, if a group of people, clustered entities in organizations, etc. are joined together by edges if they correspond, there is some relationship between them. Lotfi Zadeh introduced the fuzzy set theory in 1965 as a generalization of classical set theory that allows us to represent imprecise and vague phenomena. Fuzzy is an upper version of a crisp set, where every item or element has a varying membership grade [9]. It can illustrate that its elements have distinct membership grades between 1 and 0. Membership degrees are not like probability. Using the fuzzy relation, Kaufmann [1] presented the concept of a fuzzy graph. Rosenfeld introduced the concepts of fuzzy paths, fuzzy cycles, fuzzy bridges, fuzzy connectedness and fuzzy trees to a fuzzy graph and described some of its characteristics in [12]. Several mathematicians, like Rashmanlou and Pal [13], have done extensive research on fuzzy graphs and their applications in real-world problems. As a type 1 fuzzy set, Atanassov introduced a new type of fuzzy set, the intuitionistic fuzzy set [4]. Type 1 fuzzy sets have only a single membership grade; however, the intuitionistic fuzzy set always consider two independent membership grades: membership grade and non-membership grade for each element. Shannon and Atanassov [14] have described the concept of intuitionistic fuzzy set relationships and graphs for the first time. For further study on intuitionistic fuzzy graphs, interested readers may refer to [8]. Vague and Intuitionistic fuzzy graphs represent many real-world problems. Nevertheless, uncertainty due to conflicting information and vague information about real-world decisions Problem creation cannot be handled accurately by a fuzzy Graph or Intuitionistic fuzzy Graph. For this reason, Experts need other new concepts to deal with it scenario. Smarandachee explained Neutrosophic aggregation by extending the idea of fuzzy set [18]. It can handle any real-world problem with undefined, ambiguous, uncertain, and inconsistent data. Classical set, fuzzy set and Intuitionistic fuzzy sets are all expanded by Neutrosophic set. Every element in a Neutrosophic set has three membership grades: truth, indeterminate and false, These three membership grades are always distinct and fall within the range [0, 1].Smarandache’s proposed neutrosophic sets [6]. This is a complex mathematical method of dealing with imperfect, uncertain, and inconsistent data in real-world problems. They are a type of fuzzy set theory that includes intuitionistic fuzzy sets [8] as well as interval-valued intuitionistic fuzzy sets. To characterize Neutrosophic sets, the truth-membership values(T), indeterminacy-membership values(I), and falsity-membership values(F) are independent and lie within the real standard or nonstandard unit interval [0, 1]. Haibin Wang introduced the concept of single-valued neutrosophic sets (SVNS), a subclass of neutrosophic sets to make it easier to practice real-world applications. Single-valued Neutrosophic sets are generalizations of the intuitionistic fuzzy sets that are independent and the membership values ranging from [0,1]. Single-valued Neutrosophic sets are a subclass of neutrosophic sets, to make it easier to practice Neutrosophic sets in real-world applications. Related work in the extension of the single-valued Neutrosophic network is found in [2, 13, 14]. Yahya et al. [17] defined the max product of two intuitionistic fuzzy graphs. For the fundamental concepts of fuzzy related terminologies, interested readers may refer to [8]. We introduced the Max product of three single-valued Neutrosophic graphs and studied its characterization. Further extended our study on its applications, finding the effective minimal spanning tree. Also done a comparitive study with Broumi et al. [15] and enhanced the weight of minimal spanning tree using edge score function is defined by us. We follow the terminologies used in Broumi et al. [16], interested readers may refer it.
2Preliminaries
2.1Single-valued Neutrosophic Graph (SVNG) [16]
Let
2.2Strong SVNG [16]
The SVNG G’ is called strong SVNG if
2.3Complete SVNG [16]
The SVNG G’ is called complete SVNG if
2.4Regular SVNG [16]
The SVNG G’ is called regular SVNG if
∑s′≠r′T μ′ ((s′, r′))= constant;
∑s′≠r′I μ′ ((s′, r′))= constant;
∑s′≠r′F μ′ ((s′, r′))= constant;
2.5Degree of a vertex of SVNG [7]
Let be the SVNG. The degree of a vertex v ∈ V′ is denoted by dG′ (v) = ((dT)G′
(v) , (dI)G′(v),(dF)G′ (v)). Here ,
2.6Max product of two Intuitionistic fuzzy graph [17]
Let
The Max product of two Intuitionistic fuzzy graph
where
3Max product of three single valued Neutrosophic graphs(MPTSVNG)
3.1Definition
Let
The Max product of three neutrosophic graph
where
Fig. 1
Graph
Fig. 2
Graph
3.2Definition
The Max Product of Three single-valued Neutrosophic Graph is defined as
(i)
(ii)
(iii)
3.3Example
Let
Fig. 3
Graph
Fig. 4
Max product of three single-valued Neutrosophic network
3.4Edge score function of a SVNG
Indeterminancy value(I) does not depend on both true(T) and falsity(F) value because I is not a complement of T and F and the values of T, I, F are independent of each other.
We defined a Edge Score Function (ESF) of Single-valued Neutrosophic Graph is
3.5Theorem
The MPTSVNGs
proof: Let
Case (i): For every vertex
Case (ii):
[by 2.1 (i)]
=
[by defn 3.2 (ii)]
Case (iii):
[by 2.1 (i)]
=
[by 2.1 (ii)]
=
=
[by 2.1 (iii)]
Case (iv):
=
[by 2.1 (i)]
=
=
[by 2.1 (ii)]
=
[by 2.1 (iii)]
3.6Definition
Let
3.7Definition
Let
In the above example, Degree of each vertex in the max product network is
Hence
Total degree of each vertex in the max product network is
Similarly,
The advantage of using the Max product of three single-valued Neutrosophic graphs in a business network relates to the ease of information sharing and speed with which an understanding of data can be portrayed effectively by individuals with unique aspects in every field of work among them. however, their limitations occur in the operations of two networks in the max product of graphs comparatively to three max product of single-valued Neutrosophic graphs. When two networks are maximized, the aspects of sharing information to fewer particular members in the network. whereas in a Max product of three single-valued Neutrosophic networks, the information sharing is maximized and not limited to a few aspects. It is essential to know how to improve the number of edges in social networks to increase the flow of information.
Let
The number of vertices and edges present in NSVNG1 * NSVNG2 is
We proposed the Max product of three graphs, which is a more effective than two. Let
i) (a, b, c) , (a′, b, c) are adjacent in
Therefore, the number of edges present in this case is
ii) (a, b, c) , (a, b′, c) are adjacent in
Therefore, the number of edges present in this case is
iii) (a, b, c) , (a, b, c′) are adjacent in
Therefore, the number of edges present in this case is
The number of vertices and edges present in
4Applications
Social networks are platforms where users share their experiences and interact with each other. Unlike traditional web pages, users are not only passive consumers but also content producers and spreaders. Thus, it is necessary to benefit from the interactions between users in order to spread the information in the shortest and most effective way in social media networks. This research explores the idea of prioritization of social network connections by representing a social media network as single valued Neutrosophic network.
Nowadays, the use of social networks are progressing very fast. Social networks can be used for many purposes. Many types of social networks are available. These social networks are prepared to grow their business rapidly, and hence the providers of social networks try to increase their networks. Over the past few years, online social networking has exploded in popularity as a means for people to share information and build connections with others. For communication, marketing, and spreading of news, etc., it becomes a vital instrument. In the social network market, there is a substantial competitive situation, so all social network organizations are trying to enhance their networks in the maximization. So maximization of networks directly depends on how many users and edges or relationship are there between users. In social networks, it is essential to know how to improve the number of edges. A user of a social network wants to connect to another user by nature. Therefore, it is necessary to connect to the right persons of other network to increase the flow of information. However, the given data in social networks are not precise all the times. Therefore, Neutrosophic network systems capture these uncertainties with a degree of memberships.
4.1Example
Social networks are crucial to fostering company culture, collaboration and information flow. Let us consider a three group of networks of an organization namely
Let us assume that the network
Table 1
Responsibilities of nodes
| Nodes | Respective position of nodes |
| a1 (0.2, 0.3, 0.1) | General Manager |
| a2 (0.1, 0.2, 0.4) | Executive Director |
| b1 (0.2, 0.3, 0.1) | Digital Marketer |
| b2 (0.2, 0.3, 0.1) | Chief Operating Officer |
| b3 (0.2, 0.3, 0.1) | Information Officer |
| c1 (0.2, 0.3, 0.1) | Proactive Individual |
| c2 (0.2, 0.3, 0.1) | Chief Financial Officer |
The roles of each node is different from one another, when these nodes are maximized into a max product, the above network
For example if a1(0.2,0.3,0.1) is the general manager of the network
Therefore, making a collaboration with every node of the network maximizes the output to improve on the organization’s development. These small network of organizations
The maximized single-valued Neutrosophic network
The flow of information from one node to another node in the maximized network takes place in a effective time management. The truth-membership degree, the indeterminacy-membership degree and the falsity-membership degree of each link are given by an effective time management of the node in collaboration.
From the above maximized single-valued Neutrosophic network model we find the minimal spanning tree to make the network more flexible with the minimum possible weights of the edges with score function are found and thus the minimal cardinality of edges increase in the profits of financial marketing of the organizations.
4.2Minimal spanning tree algorithm
In this section, we provide an another application of Max product using minimal spanning tree algorithm for single valued neutrosophic undirected graph (network) by edge cardinality score function (ECSF).
Input: Adjacency matrix for max product weighted network (MPWN)
Output: Minimal Spanning tree network of MPWN
Step 1: Frame the adjacency matrix for the given MPWN using edge cardinalitty of score function
Step 2: Since the MPWN is an undirected graph, the adjacency matrix is a symmetric matrix. Hence consider the upper triangular entries of the adjacency matrix. In this entries, identify the smallest positive non zero ECSF value. Name it as x1 and mark the corresponding edge name it as e1 in the MPWN. Next in all unmarked values, identify the smallest positive non zero ECSF value. Name it as x2 and mark the corresponding edge name it as e2 in the MPWN.
Step 3: Proceed the iteration until all elements are either marked as zero or all the non zero elementes are marked. If any edge ei of MPWN produces a cycle with all previously marked edges then mark the corresponding ECSF value of ei is 0. In this case exclude the edge ei in the network.
Step 4: At last, we received (n - 1) edge minimal spanning tree network of MPWN from the marked edges of MPWN
Let
Fig. 5
Single valued Neutrosophic network with minimum score function.
(1)
Fig. 6
Identification of minimal spanning tree.
Find the next smallest non-zero score function value 0.5833 which is highlighted in the following matrix and mark the corresponding edge e4 (a2b2c2- a2b1c2) in the MPSVWN. Find the next non-zero score value 0.5833(is equal to the previous value) and is highlighted in the following matrix and mark the corresponding edge e5 (a2b3c1 - a2b2c1) in the MPSVWN. Find the next non-zero score value 0.5833 (is equal to the previous value) and is highlighted in the following matrix and mark the corresponding edge e6 (a2b2c2 - a2b3c2) in the MPSVWN. Find the next smallest non-zero score function value 0.6 which is highlighted in the following matrix and mark the corresponding edge e7 (a1b1c2 -a2b1c2) in the MPSVWN. Find the next smallest non-zero score function value 0.6166 which is highlighted in the following matrix and mark the corresponding edge e8 (a1b2c1 - a2b2c1) in the MPSVWN. Find the next smallest non-zero score function value 0.6166 (is equal to the previous value) and is highlighted in the following matrix and mark the corresponding edge e9 (a1b2c2 - a2b2c2) in the MPSVWN.Find the next smallest non zero score function value 0.6166 and its score value is reduced to 0, since the corresponding edge (a2b2c1- a2b2c2) created a cycle with all marked edges in the MPSVWN.
Fig. 7
Minimal spanning tree
Find the next smallest non-zero score function value 0.65 which is highlighted in the following matrix and its score value is reduced to 0 since the corresponding edge (a1b1c1- a1b2c1) created a cycle with all marked edges in the MPSVWN. Find the non-zero score function value 0.65(is equal to the previous value) and is highlighted in the following matrix and its score value is reduced to 0 since the corresponding edge (a1b1c1- a1b2c2) created a cycle with all marked edges in the MPSVWN. Find the next smallest non-zero score function value 0.65 which is highlighted in the following matrix and its score value is reduced to 0 since the corresponding edge (a1b2c1- a1b2c2) created a cycle with all marked edges in the MPSVWN. Find the next smallest non-zero score function value 0.65 which is highlighted in the following matrix and its score value is reduced to 0 since the corresponding edge (a1b2c1- a1b3c1) created a cycle with all marked edges in the MPSVWN. Find the smallest non-zero score value 0.65 which is highlighted in the following matrix and mark the corresponding edge e10 (a1b2c2 –a1b3c2) in the MPSVWN. Find the next non-zero score value 0.65 which is highlighted in the following matrix and mark the corresponding edge e11 (a2b3c1 - a1b3c1) in the MPSVWN. As we received the minimal spanning tree with 11 edges, rest of the score values will not be used further. The selection of minimal spanning tree from
Table 2
Adjacency matrix of Fig. 5
| a1b1c1 | a1b1c2 | a2b1c1 | a2b1c2 | a1b2c1 | a1b2c2 | a2b2c1 | a2b2c2 | a2b3c1 | a1b3c2 | a2b3c2 | a1b3c1 | |
| a1b1c1 | — | 0.6666 | 0.5666 | 0 | 0.65 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| a1b1c2 | 0.6666 | — | 0 | 0.6666 | 0 | 0.65 | 0 | 0 | 0 | 0 | 0 | 0 |
| a2b1c1 | 0.5666 | 0 | — | 0.5666 | 0 | 0 | 0.5166 | 0 | 0 | 0 | 0 | 0 |
| a2b1c2 | 0 | 0.6666 | 0.5666 | — | 0 | 0 | 0 | 0.5833 | 0 | 0 | 0 | 0 |
| a1b2c1 | 0.65 | 0 | 0 | 0 | — | 0.65 | 0.6166 | 0 | 0 | 0 | 0 | 0.65 |
| a1b2c2 | 0 | 0.65 | 0 | 0 | 0.65 | — | 0 | 0.6166 | 0 | 0.65 | 0 | 0 |
| a2b2c1 | 0 | 0 | 0.5166 | 0 | 0.6166 | 0 | — | 0.6166 | 0.5833 | 0 | 0 | 0 |
| a2b2c2 | 0 | 0 | 0 | 0.5833 | 0 | 0.6166 | 0.6166 | — | 0 | 0 | 0.5833 | 0 |
| a2b3c1 | 0 | 0 | 0 | 0 | 0 | 0 | 0.5833 | 0 | — | 0 | 0.65 | 0.65 |
| a1b3c2 | 0 | 0 | 0 | 0 | 0 | 0.65 | 0 | 0 | 0 | — | 0.65 | 0.7166 |
| a2b3c2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.5833 | 0.65 | 0.65 | — | 0 |
| a1b3c1 | 0 | 0 | 0 | 0 | 0.65 | 0 | 0 | 0 | 0.65 | 0.7166 | 0 | — |
Table 3
Adjacency matrix of Fig. 6
| a1b1c1 | a1b1c2 | a2b1c1 | a2b1c2 | a1b2c1 | a1b2c2 | a2b2c1 | a2b2c2 | a2b3c1 | a1b3c2 | a2b3c2 | a1b3c1 | |
| a1b1c1 | — | 0.6666 | 0.5666 | 0 | ——0.65 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| a1b1c2 | 0.6666 | — | 0 | 0.6 | 0 | ——0.65 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| a2b1c1 | 0.5666 | 0 | — | 0.5666 | 0 | 0 | 0.5166 | 0 | 0 | 0 | 0 | 0 |
| a2b1c2 | 0 | 0.6 | 0.5666 | — | 0 | 0 | 0 | 0.5833 | 0 | 0 | 0 | 0 |
| a1b2c1 | 0.65 | 0 | 0 | 0 | — | ——0.65 0 | 0.6166 | 0 | 0 | 0 | 0 | ——0.65 0 |
| a1b2c2 | 0 | 0.65 | 0 | 0 | 0.65 | — | 0 | 0.6166 | 0 | 0.65 | 0 | 0 |
| a2b2c1 | 0 | 0 | 0.5166 | 0 | 0.6166 | 0 | — | ——\!\!\!\!—\!\!\!\!—\!\!\!\!—0.6166 | 0.5833 | 0 | 0 | 0 |
| a2b2c2 | 0 | 0 | 0 | 0.5833 | 0 | 0.6166 | 0.6166 | — | 0 | 0 | 0.5833 | 0 |
| a2b3c1 | 0 | 0 | 0 | 0 | 0 | 0 | 0.5833 | 0 | — | 0 | ——0.65 0 | 0.65 |
| a1b3c2 | 0 | 0 | 0 | 0 | 0 | 0.65 | 0 | 0 | 0 | — | 0.65 | 0.7166 |
| a2b3c2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.5833 | 0.65 | 0.65 | — | 0 |
| a1b3c1 | 0 | 0 | 0 | 0 | 0.65 | 0 | 0 | 0 | 0.65 | 0.7166 | 0 | — |
4.3Comparative study
In this section, we do the comparative study with Broumi et al[15] presented algorithm to find the minimal spanning tree of the following single-valued neutrosophic undirected graph which is shown in Fig. 8.
Fig. 8
Broumi et al. Neutrosophic graph.
We defined a edge score function using our proposed algorithm, we get the same minimal spanning tree with minimum weight,
0.55+0.3333+0.5166+0.6666+0.3833 = 1.7666 as shown in Fig. 10. But using the algorithm proposed by Broumi et al[15] with their score function minimum weight of the minimal spanning tree is 0.5+0.2+0.433+0.6+0.267=2 shown in Fig. 9.
Fig. 9
Broumi et al. minimal spanning tree.
Fig. 10
Broumi et al. minimum spanning tree by edge score function.
Comparatively, we get the improved minimum weight of the minimal spanning tree using the score function defined in this work.
5Conclusion
Single valued Neutrosophic Network gives more enhanced structure than Neutrosophic Networks which helps to deal with more ambiguous conditions. The authors studied the max product of a single-valued neutrosophic graph structure and discussed the real-world application of the maximized network with a minimum spanning tree algorithm which is generated to achieve the minimum efficient time to complete the tasks in a social network. In the future, the study will be extended to other operations of three single-valued neutrosophic Graphs.
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