#### Abstract

Network connectivity is an important factor in data transmission, information sharing, and network defense, and it is a crucial indicator which determines the performance of a network. When a network node or channel contains uncertainties in both positive and negative aspects, the entire network can be modeled with bipolar fuzzy graphs. This paper analyzes the influence of each vertex on the connectivity of the entire network through the definition of the connected state on the bipolar fuzzy graph. The solid results are given, and the new concepts are applied in campus network connectivity analysis. This approach helps to analyze the hidden dangers of the bipolar network, find the weaknesses in the connection, and prevent network attacks in advance.

#### 1. Introduction

In information science, the connectivity of the network and the connected state of each vertex determine the efficiency of the entire network and the corresponding algorithm. For example, in a protein molecular network, the connected state of the apex determines the molecular activity, which affects the working mechanism of the biological system. In federated learning, the status of each device is not like silos which works all the time. In a certain period of time, devices tend to be in a disconnected state, which leads to the need to select a collection of devices that are in working state during each iteration of federated learning. For various reasons of actual needs, network connectivity has become a hot topic in the field of information technology (see Cheung and Bell [1], Yemini et al. [2], Burla et al. [3], Denison et al. [4], Hummer et al. [5], Ma et al. [6], Gong et al. [7], Gao et al. [8, 9], and Kumar et al. [10]).

Due to the complexity of the network itself, in many application scenarios involving the uncertainty of vertices and edges, it is necessary to borrow tools and methods of fuzzy mathematics. Since the network itself uses vertices to represent sites and edges represent the connecting channels between sites, this network is a graph structure. When it is necessary to describe the uncertain features in the network, the entire model becomes a fuzzy graph model; that is, the membership function of vertices and edges is used to describe this uncertainty in detail. On the other hand, this uncertainty in fuzzy mathematics can often be divided into positive and negative properties. It is well known that the positive membership function cannot reflect the negative uncertainty of objectives. For example, the membership function stands for “like” and means “dislike” while cannot infer . A reasonable approach is to use a negative membership function to describe “dislike.”

In modeling, the positive uncertainty corresponds to the positive membership function value, and the negative uncertainty corresponds to the negative membership function value. In this setting, the network model becomes a bipolar fuzzy graph, and different types of uncertainties can use different membership functions and ultimately determine various frameworks of the bipolar fuzzy graph. Common ones include bipolar intuitionistic fuzzy graphs, bipolar Pythagorean fuzzy graphs, and so on (see Akram et al. [11], Atef et al. [12], Mahapatra et al. [13], Talebi et al. [14], Gong and Hua [15], Binu et al. [16], Amanathulla et al. [17], Kalathian et al. [18], Akram and Sattar [19], and Mahapatra et al. [20]).

Very recently, several works contribute to bipolar fuzzy sets and bipolar fuzzy graphs from the perspective of theory and application. Mahmood and Ur Rehman [21] introduced a new notation of bipolar complex fuzzy set which is the fusion of bipolar fuzzy set and complex fuzzy set. Jana [22] studied the multiple attribute group decision-making with bipolar fuzzy numbers in terms of multiple attribute border approximation area comparison approach. Poulik et al. [23] determined the upper and lower boundaries of Randic index of bipolar fuzzy graphs. Ali et al. [24] proposed the theory of an innovative hybrid model called the fuzzy bipolar soft expert sets and applied it to COVID-19 applications. Mehmood et al. [25] raised a new trick to solve LR-type fully bipolar fuzzy linear programming problems with equality restrictions. Ozcelik and Nalkiran [26] introduced an evaluation based on the distance from average solution equipped with trapezoidal bipolar fuzzy sets. Cornejo et al. [27] obtained features of the solvability of bipolar max-product fuzzy equations with the standard negation. More related books on bipolar fuzzy graph can be referred to Mathew et al. [28], Pal et al. [29], Akram et al. [30], Chakraverty et al. [31], Chakraverty [32], and Volosencu and Aceves-Fernandez [33].

Binu et al. [34] introduced the connectivity status of vertices in fuzzy graph and classified all the vertices by means of its connectivity status. It inspires us to extend these concepts to more fuzzy settings, and in this work, we focus on bipolar fuzzy setting. The contributions of this work are three-fold: first, we introduce the bipolar connectivity status of each vertex and entire network; then, some theoretical results are inferred in light of fuzzy theory and graph theory; and finally, the given tricks are used in campus network connectivity analysis which aims to prevent network attacks in advance.

The organization of the rest sections are as follows: we present the basic existing concepts on bipolar fuzzy graph in Section 2; new definitions, theoretical results, and proofs are determined in Section 3; a numerical experiment is manifested in Section 4 which simulates university campus network; and finally, the conclusion and remarks are given in the last section.

#### 2. Definitions in Bipolar Fuzzy Graph Setting

Let be a universal set and be a bipolar fuzzy set in with two membership functions and . If is a bipolar fuzzy set in where , for any , and for any , then is a bipolar fuzzy graph (BFG) (the corresponding originally graph is called a crisp graph of fuzzy graph). In what follows, we always use and instead of minimum and maximum operations, respectively.

Let be a bipolar fuzzy graph. is a path with length *k* from to in which is a sequence of different vertices, where for . Then,and

Set

Then, the strength of connectedness between arbitrary in bipolar fuzzy graph is denoted by

If and for any in , then the bipolar fuzzy graph is connected. If and for any , then the bipolar fuzzy graph is a complete bipolar fuzzy graph.

All edges in bipolar fuzzy graph can be divided into three classes in light of connectedness. An edge is strong if and . If and , then edge is called -strong. If and , then the pair is called -strong.

The connectivity index of the bipolar fuzzy graph is defined by Poulik and Ghorai [35], which is formulated bywhere and are positive connectivity index and negative connectivity index of bipolar fuzzy graph .

#### 3. Connectivity Status in a Bipolar Fuzzy Graph

The difference between the general frame fuzzy graph and the bipolar fuzzy graph mainly depends on the membership function that describes the fuzzy characteristics of the negative pole contained in the bipolar fuzzy graph. The membership function determines the fuzzy structure characteristics of the negative pole and thus has a decisive influence on the connectivity of the vertices of the entire fuzzy graph. In this section, we introduce the connectivity parameters for bipolar fuzzy graphs.

*Definition 1. *Let be a bipolar fuzzy graph and be the ordered vertex set of . The connectivity status of a vertex is defined bywhere and are positive connectivity status and negative connectivity status of vertex , respectively. The connectivity status of bipolar fuzzy graph is denoted bywhere and are positive connectivity status and negative connectivity status of bipolar fuzzy graph , respectively.

For convenience, we write , , and as , , and for short. The minimum positive connectivity status, maximum positive connectivity status, minimum negative connectivity status, and maximum negative connectivity status of bipolar fuzzy graph are defined by , , and , respectively. A bipolar fuzzy graph is positive connectivity status regular if and positive connectivity status regular if . Two vertices are said to be of positive equistatus (respectively, negative equistatus) in bipolar fuzzy graph if they have the same positive connectivity status (respectively, negative connectivity status) in . Two vertices are said to be of equistatus if they are both positive equistatus and negative equistatus.

*Example 1. *As depicted in Figure 1, , , for , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , and .

Furthermore, we infer , , , and . and are positive equistatus vertices; and are positive equistatus vertices; and are negative equistatus vertices; and are negative equistatus vertices; however, no two vertices are equistatus.

Our first result reveals the relationship between connectivity index and connectivity status of the bipolar fuzzy graph.

Theorem 1. *Let be a bipolar fuzzy graph and be the ordered vertex set of . Suppose that and for any . Then,*

Proof of Theorem 1. We only prove the negative part, i.e.,

Under the hypothesis of Theorem 1, we infer

For a complete bipolar fuzzy graph, we have the following statements.

Theorem 2. *Let be a bipolar fuzzy graph and . Assume that are four vertices (allow some of them to be the same vertex) satisfying , , , and . Then, the following statements hold:*(1)* and *(2)*, , , and *(3)*If , then and are equistatus*

Proof of Theorem 2.(1)We only verify the negative part. Since is a complete bipolar fuzzy graph, we have where , and hence(2)Clearly, , , , and . Thus, the results follow from the definition of , , , and .(3)It needs to confirm that and with the same positive and negative membership function are both positive equistatus and negative equistatus. We only check the correctness of negative equistatus below.

First, we rank all the vertices according to their negative membership function values: . Since , we suppose and are adjacent vertices in the queue. Without loss of generality, assume . If , then in terms of , we are done by

If , then and

In all, the Proof of Theorem 2 is finished.

Before introducing the next theorem, we need to extend the concept of saturated fuzzy cycle raised by Mathew et al. [36] to bipolar setting.

*Definition 2. *Let be a connected bipolar fuzzy graph. is -saturated (respectively, -saturated) if each vertex is incident at least one -strong edge (-strong edge). is called saturated if it is both -saturated and -saturated. That is to say, at least one -strong edge and one -strong edge are incident on every vertex in the bipolar fuzzy graph.

*Example 2. *As depicted in Figure 2, , , , , , , , , , , , , , and.

Clearly, and are -strong edges; and are -strong edges; and is not strong edge. Hence, is saturated.

Thereom 3. *Let be a saturated fuzzy cycle with n vertices. For , set , , , and as the positive membership value of -strong edge, positive membership value of -strong edge, negative membership value of -strong edge, and negative membership value of -strong edge incident to , respectively. Let be the edge of all -strong edges in C. Set , , , and (allow some pair of vertices are the same). Then,*(1)

*,*(2)

*If -strong edges of*(3)

*C*have equal positive and negative membership values, then all vertices in*C*are of equistatus*, , , and*

Proof of Theorem 3. Since every vertex in *C* has degree exactly 2 in C and for a fixed vertex in C, the connectivity of pairs of vertices (contain ) is determined by -strong edges except one pair with exactly -strong edge. Hence, we can obviously check the statement (1) in view of graph theory.

For statement (2), if the membership value of arbitrary -strong edge is equal, then the statement (1) implies that and are constants for each , where .

Furthermore, the statement (3) is also valid by means of statement (1).

Imitating the vertex classification based on the bipolar connectivity index in Poulik and Ghorai [35], we define the vertex type based on the connectivity status in the following.

*Definition 3. *Let be a bipolar fuzzy graph and .(1)If and , then is called a bipolar connectivity status-reducing vertex(2)If and , then is called a bipolar connectivity status-enhancing vertex(3)If and , then is called a bipolar connectivity status-neutral vertexLet . is a bipolar connectivity status-reducing vertex set if all its vertices are bipolar connectivity status-reducing vertex; is a bipolar connectivity status enhancing vertex set if all its vertices are bipolar connectivity status-enhancing vertex; and is a bipolar connectivity status-neutral vertex set if all its vertices are bipolar connectivity status-neutral vertex.

*Example 3. *As depicted in Figure 3, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , and .

Furthermore, we get , , , , , , , , , , , , , , , , .

, , , , , , , , , , , , , , , , and .

, , , , , , , , , , , , , , , , and .

, , , , , , , , , , , , , , , , and .

, , , , , , , , , , , , , , , , and .

Note that and ; and ; and ; hence , , and are bipolar connectivity status-reducing vertices. By and , we infer that is a bipolar connectivity status-enhancing vertex. However, does not belong to any type of vertex.

#### 4. Campus Network Bandwidth Allocation Application

The information management office of each university is responsible for the distribution and scheduling of its network resources, including the purchase of bandwidth, which is allocated to various colleges and common institutions such as libraries, gymnasium, school hospital, and cafeterias. According to the geographic location of each institution in the school, the network topology is shown in Figure 4, and the institution corresponding to each vertex is shown in Table 1.

It can be seen from the topological representation graph that the entire university is divided into two areas: the east area and the west area, centered on the library and the administrative building. The west area is dominated by the schools of humanities and arts, and the east area is dominated by the departments of science and engineering. There are student dormitories and canteens on both sides. It should be noted that this is a simplified topology graph. In the actual campus structure, there are also parks, teaching buildings, lakes, and so on in the circles enclosed by the east and west district schools, which we does not show these facilities all in Figure 3.

Assume that and for all vertices in this example. Considering the network cable layout of the channel between the vertices, router settings, uplink capacity and downlink capacity, link delay time, arrangement time, and other positive and negative elements, the membership function values of each edge are shown in Figure 4. The connectivities of all pairs of vertices are listed in Table 2.

In light of Table 2, the connectivity status of vertices is determined in Table 3.

For the bandwidth allocation problem, the average minimum connectivity in the entire network is required so that the connectivity of each vertex is not lower than this minimum threshold. Assuming that the vertex where the college is located also represents the router set by the department, the edge weight represents the corresponding route and the threshold is set to . That is to say, it requires and . Note that the vertices do not satisfy , and the vertices do not satisfying . Therefore, we must focus on these three vertices to enhance their connected state value to achieve the minimum connection constraint. The connected state analysis of the vertices can be used to obtain the correct direction for designing a balanced network.

#### 5. Conclusion

In computer networks, there are many negative uncertain factors, such as the delay of the network apex, the capacity limitation of the uplink and the downlink, the status of the queuing sequence, and so on. Since bipolar fuzzy graphs have negative membership functions that specifically describe these negative uncertain factors, bipolar fuzzy graphs are more flexible in specific network applications than general fuzzy graphs.

In this paper, we introduce the connectivity status in the bipolar fuzzy graph which reflects the positive and negative connectedness in a connected network for each vertex, as well as the whole network. Its main function is to help us analyze the connectivity strength of each vertex in the bipolar network and find the weakest link in the entire network through the comparison between the vertices to identify security risks, strengthen network disadvantages in time, and prevent possible attacks against network weaknesses.

The following contents can be a topic for continued research:(1)The connectivity status should be further considered in special bipolar graph settings, for instance, bipolar fuzzy incidence graph, bipolar intuitionistic fuzzy graph, and bipolar Pythagoras fuzzy graph(2)The bipolar connectivity status in specific graph structures should be discussed, such as bipolar planar graph and bipolar cube graph

#### Data Availability

All experiment data are included in this article.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest in publishing this paper.

#### Acknowledgments

This work was supported by Guangdong Basic and Applied Basic Research Foundation of China (No. 2021A1515110834), Natural Science Foundation of China (No. 51574232), Guangdong Natural Science Foundation of China (No. 2020A1515010784), and Guangdong University of Science and Technology University Major Scientific Research Achievement Cultivation Program Project (No. GKY-2020CQPY-2).