Distances between Bipolar Fuzzy Graphs

Wei Gao

School of Information Science and Technology, Yunnan Normal University, Kunming, China.

DOI: 10.4236/oalib.1113152                

Abstract

Bipolar fuzzy graphs use positive and negative membership functions to characterize the uncertainty of structured fuzzy data. This article extends the concept of network distance to bipolar fuzzy graphs, provides corresponding definitions, and obtains several remarks.

Keywords

Fuzzy Graph, Bipolar Fuzzy Graph, Distance

1. Introduction

By integrating graph theory with fuzzy systems, fuzzy network enables irregular data or structures to collaboratively fuse computer networks with uncertain information (Gao et al. [1] and Sun et al. [2]). With the great focus on uncertain information and increasing decentralization requirements, fuzzy network gains more and more concern in both mathematics and computer science (Binu et al. [3], [4] and [5]). Due to the widespread presence of irregular information with uncertainty in nature and human society, fuzzy graphs and corresponding fuzzy networks have yielded rich results in various settings from both theoretical and practical perspectives (Binu et al. [6] and Zhou et al. [7]).

Unfortunately, the membership function (MF) of vertices or edges in fuzzy graphs does not satisfy the complementary law, which is the fundamental distinguishing between probability theory and fuzzy theory. Recall that the probability function satisfies the complementary law. For example, for any event $E$ , the probability of event $E$ occurring is $P\left(E\right)=0.2$ . Then, we immediately get $P\left(\overline{E}\right)=0.8$ which is standard for the probability of the opposite event happening. However, it becomes completely different when it comes to fuzzy theory. Let $A$ = {apple, banana, orange}, and MF ${\mu }_1$ represents Bill’s preference for various fruits. Suppose $A$ = {(apple, 0.9), (banana, 0.6), (orange, 0.3)} after fuzzing $A$ . Denote MF ${\mu }_2$ as Bill’s aversion to various fruits, and then $A$ ={(apple, 0.1), (banana, 0.4), (orange, 0.7)} is not true in general. That is, ${\mu }_2$ can’t determined by ${\mu }_1$ . It implies that at least two MFs are necessary to describe the uncertainty of different aspects of things.

In order to solve the aforementioned problem in fuzzy graph setting, the concept of bipolar fuzzy graph (BFG) was introduced by Akram [8] in 2011, where two vertex (resp. edge) MFs are used to characterize the positive and negative uncertainty of vertices (resp. edges). In the next 10 years, BFGs have been developed in both theoretical and computer applications, and several important ideas in fuzzy graphs are expanded in the corresponding BFG settings. Akram and Dudek [9] studied the bipolar fuzzy graph in regular settings. Some BFGs with various restrictions are introduced in Akram [10]. Yang et al. [11] corrected some statements in Akram [8], and introduced the generalized BFG. Rashmanlou et al. [12] investigated some characteristics of isomorphism on BFGs. Rashmanlou et al. [13] studied two operations on BFGs. Sarwar and Akram [14] designed algorithms for calculating the strength of competition in BFGs, and applied them in social networks and wireless communication networks. Akram et al. [15] presented a framework for dealing with bipolar fuzzy soft information in terms of bipolar fuzzy soft graphs.

Due to the importance of BFGs, a natural problem to deeply understand it from a theoretical prospect is to classify BFGs, or to calculate the similarity between BFGs. Actually, there is already some literatures discussing the distance between two bipolar fuzzy sets (BFSs), which will elaborated on in detail in Section 0. However, to the best of our knowledge, there is no article introducing the distance between BFGs. It motivates us to explore this field, and the main contributions of this work are to introduce the correspondence, distortion, the distance between two BFGs.

The remainder of the content is organized as follows. The survey is presented in the next section. Then, we give our main definitions and remarks. Finally, we raise an open problem in Section 0.

2. Related Work

In the following context, we summarized the recent contributions in three types: BFG, distance between BFSs, and distance of networks.

2.1. Bipolar Fuzzy Graphs

Let $V$ be a universal set (which is defined as a vertex set), $G=\left(V\left(G\right),E\left(G\right)\right)$ be a crisp graph (without MFs on $V$ and $E$ ). Fuzzy graph is denoted by $G=\left(V,A,B\right)$ , where

$A=\left\{\left(v_1,{\mu }_A\left(v_1\right)\right),\left(v_2,{\mu }_A\left(v_2\right)\right),\cdots ,\left(v_n,{\mu }_A\left(v_n\right)\right)\right\}$

is a fuzzy set on $V$ ,

$B=\left\{\left(v_i,v_j,{\mu }_B\left(v_i,v_j\right)\right):v_i,v_j\in V\right\}$

is a fuzzy set on $V\times V$ , where ${\mu }_A:V\to \left[0,1\right]$ is a MF on vertex set $V$ and ${\mu }_B:V\times V\to \left[0,1\right]$ is a binary MF on $V\times V$ such that ${\mu }_B\left(v_i,v_j\right)\le \text{min}\left\{{\mu }_A\left(v_i\right),{\mu }_A\left(v_j\right)\right\}$ for any $v_i,v_j\in V$ . Moreover, constraint ${\mu }_B\left(v_i,v_j\right)=0$ for any $v_i{v}_j\ne E\left(G\right)$ .

Let $V$ be a universal set, $A=\left\{\left(v,{\mu }_A^P\left(v\right),{\mu }_A^N\left(v\right)\right):v\in V\right\}$ is a BFS on $V$ , where ${\mu }_A^P:V\to \left[0,1\right]$ and ${\mu }_A^N:V\to \left[-1,0\right]$ . Let $B=\left\{\left({\mu }_B^P,{\mu }_B^N\right):V\times V\to \left[0,1\right]\times \left[-1,0\right]\right\}$ be a bipolar fuzzy relation on $V^2$ , where ${\mu }_B^P\left(v,v^{\prime }\right)\in \left[0,1\right]$ , ${\mu }_B^N\left(v,v^{\prime }\right)\in \left[-1,0\right]$ , ${\mu }_B^P\left(v,v^{\prime }\right)\le \text{min}\left\{{\mu }_A^P\left(v\right),{\mu }_A^P\left(v^{\prime }\right)\right\}$ , and ${\mu }_B^N\left(v,v^{\prime }\right)\ge \text{max}\left\{{\mu }_A^N\left(v\right),{\mu }_A^N\left(v^{\prime }\right)\right\}$ hold for any $\left(v,v^{\prime }\right)\in V\times V$ . Furthermore, ${\mu }_B^P\left(v,v^{\prime }\right)={\mu }_B^N\left(v,v^{\prime }\right)=0$ for any $\left(v,v^{\prime }\right)\in {\tilde{V}}^2-E$ . Then, $G=\left(V,A,B\right)$ is a bipolar fuzzy graph (BFG), and $G^{\text{*}}=\left(V,E\right)$ is the corresponding crisp graph.

The concepts of BFG have mainly been prominent in the field of fuzzy graph theory and computer network engineering. Poulik and Ghorai [16] yielded a characterization of bipolar fuzzy detour g-eccentric vertex, bipolar fuzzy detour g-boundary vertex and bipolar fuzzy detour g-interior vertex. Poulik and Ghorai [17] extended some topological indices from fuzzy graph to BFG, and derived several properties of these topological indices. Gong and Hua [18] defined edge connectivity in bipolar fuzzy networks and applied it in silo connection designing. Gong and Hua [19] studied the bipolar interval-valued fuzzy set in a hypergraph setting. Nithyanandham et al. [20] introduced the covering, matching and domination concepts in bipolar intuitionistic fuzzy graphs in light of effective edges, and applied these concepts in environment science. Nithyanandham and Augustin [21] demonstrated the applicability of the proposed bipolar fuzzy p-competition graph system in prioritizing COVID-19 vaccines. Almousa and Tchier [22] presented the rejection of bipolar picture fuzzy graphs.

More related contributions in bipolar fuzzy graph setting can be referred to Gong and Hua [23] and Rehman et al. [24].

2.2. Distance between Bipolar Fuzzy Sets

Let $X$ be a universal set, ${\mu }^P:X\to \left[0,1\right]$ be the positive MF on $X$ , and ${\mu }^N:X\to \left[-1,0\right]$ be the negative MF on $X$ . Then we say the set $\mathcal{V}=\left\{V,{\mu }^P,{\mu }^N\right\}=\left\{\left(v,{\mu }^P\left(v\right),{\mu }^N\left(v\right)\right):v\in V\right\}$ is a bipolar fuzzy set on $V$ . The collection of all BFSs is denoted by $BF\left(V\right)$ (on the same crisp set $V$ ). Wei et al. [25] introduced score and accuracy functions for BFSs which are used to compare and determine the ordered corrections between BFSs.

Han et al. [26] defined the metric (or distance measure) between two BFSs on the same universal set in view of the axiom point of view, and introduced two normalized distances to measure the similarity between BFSs. Specifically, assume $\left|V\right|=n$ , for two BFSs ${\mathcal{V}}_1=\left\{\left(v_i,{\mu }_{{\mathcal{V}}_1}^P\left(v_i\right),{\mu }_{{\mathcal{V}}_1}^N\left(v_i\right)\right):v_i\in V\right\}$ and ${\mathcal{V}}_2=\left\{\left(v_i,{\mu }_{{\mathcal{V}}_2}^P\left(v_i\right),{\mu }_{{\mathcal{V}}_2}^N\left(v_i\right)\right):v_i\in V\right\}$ ,

i) Hamming metric is denoted by

$d_H\left({\mathcal{V}}_1,{\mathcal{V}}_2\right)=\frac{1}{2n}{\displaystyle \sum }_{i=1}^n\left(\left|{\mu }_{{\mathcal{V}}_1}^P\left(v_i\right)-{\mu }_{{\mathcal{V}}_2}^P\left(v_i\right)\right|+\left|{\mu }_{{\mathcal{V}}_1}^N\left(v_i\right)-{\mu }_{{\mathcal{V}}_2}^N\left(v_i\right)\right|\right).$

ii) Poles-weighted Hamming metric is formulated by

$d_{PH}\left({\mathcal{V}}_1,{\mathcal{V}}_2\right)=\frac{1}{n}{\displaystyle \sum }_{i=1}^n\left({\xi }^P\left|{\mu }_{{\mathcal{V}}_1}^P\left(v_i\right)-{\mu }_{{\mathcal{V}}_2}^P\left(v_i\right)\right|+{\xi }^N\left|{\mu }_{{\mathcal{V}}_1}^N\left(v_i\right)-{\mu }_{{\mathcal{V}}_2}^N\left(v_i\right)\right|\right),$

where ${\xi }^P+{\xi }^N=1$ .

Alghamdi et al. [27] formulated the normalized Euclidean metric by

$d_E\left({\mathcal{V}}_1,{\mathcal{V}}_2\right)=\frac{1}{2}{\displaystyle \sum }_{i=1}^n{\left({\left|{\mu }_{{\mathcal{V}}_1}^P\left(v_i\right)-{\mu }_{{\mathcal{V}}_2}^P\left(v_i\right)\right|}^2+{\left|{\mu }_{{\mathcal{V}}_1}^N\left(v_i\right)-{\mu }_{{\mathcal{V}}_2}^N\left(v_i\right)\right|}^2\right)}^{\frac{1}{2}}.$

Riaz and Tehrim [28] redefined the BFS in terms of connection number, and the distance metrics were also recast. Specifically, for $\mathcal{V}=\left\{\left(v,{\mu }^P\left(v\right),{\mu }^N\left(v\right)\right):v\in V\right\}$ , the connection number set (CNS) corresponding to $\mathcal{V}$ is formulated by

$\mathcal{V}=\left\{\left(v,x_{\mathcal{V}}\left(v\right),y_{\mathcal{V}}\left(v\right)i,z_{\mathcal{V}}\left(v\right)j\right):v\in V\right\},$

where $x_{\mathcal{V}}\left(v\right)$ , $y_{\mathcal{V}}\left(v\right)$ and $z_{\mathcal{V}}\left(v\right)$ represent “identity degree”, “discrepancy degree” and “contrary degree” respectively denoted by

$x_{\mathcal{V}}\left(v\right)=\frac{{\mu }_{\mathcal{V}}^P\left(v\right)\left(2-\left(-{\mu }_{\mathcal{V}}^N\left(v\right)\right)\right)}{2},$

$y_{\mathcal{V}}\left(v\right)=\frac{2-{\mu }_{\mathcal{V}}^P\left(v\right)\left(2-\left(-{\mu }_{\mathcal{V}}^N\left(v\right)\right)\right)-\left(-{\mu }_{\mathcal{V}}^N\left(v\right)\right)\left(2-{\mu }_{\mathcal{V}}^P\left(v\right)\right)}{2},$

$z_{\mathcal{V}}\left(v\right)=\frac{\left(-{\mu }_{\mathcal{V}}^N\left(v\right)\right)\left(2-{\mu }_{\mathcal{V}}^P\left(v\right)\right)}{2}.$

with the help of this definition, Riaz and Tehrim [28] introduced the following two metrics (the CNSs of ${\mathcal{V}}_1$ and ${\mathcal{V}}_2$ are denoted by ${\mathcal{V}}_1=\left\{\left(v,x_{{\mathcal{V}}_1}\left(v\right),y_{{\mathcal{V}}_1}\left(v\right)i,z_{{\mathcal{V}}_1}\left(v\right)j\right):v\in V\right\}$ and ${\mathcal{V}}_2=\left\{\left(v,x_{{\mathcal{V}}_2}\left(v\right),y_{{\mathcal{V}}_2}\left(v\right)i,z_{{\mathcal{V}}_2}\left(v\right)j\right):v\in V\right\}$ respectively).

i) Connection number-based normalized Humming distance:

$d_1\left({\mathcal{V}}_1,{\mathcal{V}}_2\right)=\frac{1}{3n}{\displaystyle \sum }_{i=1}^n\left(\left|x_{{\mathcal{V}}_1}\left(v_i\right)-x_{{\mathcal{V}}_2}\left(v_i\right)\right|+\left|y_{{\mathcal{V}}_1}\left(v_i\right)-y_{{\mathcal{V}}_2}\left(v_i\right)\right|+\left|z_{{\mathcal{V}}_1}\left(v_i\right)-z_{{\mathcal{V}}_2}\left(v_i\right)\right|\right).$

ii) Connection number-based normalized Euclidean distance:

$d_2\left({\mathcal{V}}_1,{\mathcal{V}}_2\right)={\left(\frac{1}{3n}{\displaystyle \sum }_{i=1}^n\left({\left|x_{{\mathcal{V}}_1}\left(v_i\right)-x_{{\mathcal{V}}_2}\left(v_i\right)\right|}^2+{\left|y_{{\mathcal{V}}_1}\left(v_i\right)-y_{{\mathcal{V}}_2}\left(v_i\right)\right|}^2+{\left|z_{{\mathcal{V}}_1}\left(v_i\right)-z_{{\mathcal{V}}_2}\left(v_i\right)\right|}^2\right)\right)}^{\frac{1}{2}}.$

More metrics on fuzzy sets can be referred to Ali [29] which defined the distance measure between q-rung orthopair fuzzy sets and applied it in multi-criteria group decision-making. However, as far as we know, there is no literature research on the distance between bipolar fuzzy graphs so far, and this field is still open.

2.3. Distance between Networks

The distance between graphs, especially the distance between networks (maybe discussed in a metric space), is an emerging hot issue in mathematics and computer science, which is always closely related to concepts such as the limit of objects, subgraph (motif), and isomorphism (both strong and weak versions). Here, we only summarize and analyze the progress of topology-based network distance, because this branch of the content is pertinent to the main conclusions of this article. The analysis of network distance from other perspectives is not within the scope of this article.

Lovász and Szegedy [30] first pointed out that if we assume dense graphs $G_n$ in a sequence become more and more similar as $n$ tends to infinite, then there is a natural “limit object”. This crucial result concluded a very important fact that the object limit (parameter or measurable function) of the dense graph with a small distance exists. According to the idea of mass transportation, Mémoli [31] proposed Gromov–Wasserstein distance two networks (graphs). Mémoli [32] proved the lower bounds for the Gromov–Hausdorff distance between graphs. Chowdhury and Mémoli [33] presented a series of theories on the Gromov-Hausdorff distance between networks, including equivalence with weak isomorphism, stability, and convergence.

Inspired by Chowdhury and Mémoli [33], we consider the distance of two BFGs with the aid of some extension concepts. Indeed, the ideas (such as distortion, correspondence and weight-preserving map, ect.) in [33] are useful for constructing our theory in BFG setting.

2.4. Some Remarks

We give some notes on these previous works.

Remark. In Han et al. [26], Alghamdi et al. [27] and Riaz and Tehrim [28], the elements in metric space they defined are BFSs, and the metric is actually the distances between two BFSs. While, in Mémoli’s contributions ([31], [32] and [33]), metric space is the vertex set equipped with the discrete topology, elements in this space are vertices in a graph, and the metric is the distance for two vertices. Hence, in Mémoli’s network model, it allows infinite or compact networks (i.e., $V$ is compact) with infinite vertices, and the finite network can be obtained by sampling from an infinite network.

Remark. Kim et al. [34] introduced a series of concepts on BFS involving bipolar fuzzy point, bipolar fuzzy topology (BFT), bipolar fuzzy base, etc. Note that in Mémoli’s theory, the vertex set $V$ in the network always assumed to be a first countable topological space. However, Kim’s contribution [34] in BFT can’t used directly in BFG setting when applying Mémoli’s framework. Because in [34], the elements in BFT are BFS, where the empty set (resp. whole set) in BFT is equipped the bipolar MFs as ${\mu }^P\left(v\right)={\mu }^N\left(v\right)=0$ (resp. ${\mu }^P\left(v\right)=1$ and ${\mu }^N\left(v\right)=-1$ ) for each $v\in V$ . Thus, topology space consists of BFS in vertex set $V$ . While in Mémoli’s setting, the topological space is $V$ and elements in this space is the vertices. Obviously, they are two completely different settings. Furthermore, when we assume ${\mu }_A^P\left(v\right)=1$ and ${\mu }_A^N\left(v\right)=-1$ for all $v\in V$ , then the empty set for BFT in Kim’s setting does not exist.

Remark. By reviewing the literatures on bipolar fuzzy graphs (BFGs), some common points can be summarized. For instance, variables about positive and negative aspects always appear in pairs, such as positive connectivity and negative connectivity, positive cut sets and negative cut sets, ect. Therefore, applying a similar idea to BFG to define the corresponding distance will inevitably produce both a positive distance and a negative distance, where the value of the negative distance is a non-positive number. On the other hand, if you follow Mémoli’s idea and regard the BFG as a metric space, then it only allows one non-negative distance function as metric function. It can be seen that the traditional BFG processing trick cannot be directly applied to the corresponding model of Mémoli’s. To this end, to follow the idea of Mémoli, it is necessary to convert some related concepts into the BFG setting.

The BFGs discussed in this paper are directed and with loops, unless there are special statements to clarify the BFG is symmetry. In what follows, for the convenience of discussion, we assume that ${\mu }_A^P\left(v\right)=1$ and ${\mu }_A^N\left(v\right)=-1$ for all $v\in V$ . In this way, we focus on the edge set of BFG and the bipolar MF defined on the edges. For any matrix $X$ , we denote $X^T$ by the transpose of $X$ .

3. Extended Concepts and Some Remarks

In this part, we always consider the case of $V$ as finite (unless in the special remark). Hence, followed by Chowdhury and Mémoli [33], the collection of BFGs can be written as

$\mathcal{G}:=\left\{\left(V,{\mu }_B^P,{\mu }_B^N\right):\left|V\right|<\infty ,{\mu }_B^P:V\times V\to \left[0,1\right],{\mu }_B^N:V\times V\to \left[-1,0\right]\right\}.$

Definition 1. Let $G=\left(V,A,B\right)$ and $G^{\prime }=\left(V^{\prime },A^{\prime },B^{\prime }\right)$ be two BFGs. A map $\varphi :V\to V^{\prime }$ is positive membership preserving if ${\mu }_B^P\left(v,v^{\prime }\right)={\mu }_{B^{\prime }}^P\left(\varphi \left(v\right),\varphi \left(v^{\prime }\right)\right)$ for any $v,v^{\prime }\in V$ , and is negative membership preserving if ${\mu }_B^N\left(v,v^{\prime }\right)={\mu }_{B^{\prime }}^N\left(\varphi \left(v\right),\varphi \left(v^{\prime }\right)\right)$ for any $v,v^{\prime }\in V$ . We call $\varphi :V\to V^{\prime }$ is bipolar membership preserving if it is both positive membership preserving and negative membership preserving.

Definition 2. (Strong isomorphism) Let $G=\left(V,A,B\right)$ and $G^{\prime }=\left(V^{\prime },A^{\prime },B^{\prime }\right)$ be two BFGs. We say $G$ and $G^{\prime }$ are strongly isomorphic (denoted by $G{\cong }^s{G}^{\prime }$ ) if there is a bipolar membership preserving bijection $\varphi :V\to V^{\prime }$ .

Remark. The strong isomorphism is equivalent to the tripod condition. That is, there is a set $V^{″}$ and bijective maps ${\varphi }_V:V^{″}\to V$ , ${\varphi }_{V^{\prime }}:V^{″}\to V^{\prime }$ such that ${\mu }_B^P\left({\varphi }_V\left(v_1\right),{\varphi }_V\left(v_2\right)\right)={\mu }_{B^{\prime }}^P\left({\varphi }_{V^{\prime }}\left(v_1\right),{\varphi }_{V^{\prime }}\left(v_2\right)\right)$ and ${\mu }_B^N\left({\varphi }_V\left(v_1\right),{\varphi }_V\left(v_2\right)\right)={\mu }_{B^{\prime }}^N\left({\varphi }_{V^{\prime }}\left(v_1\right),{\varphi }_{V^{\prime }}\left(v_2\right)\right)$ for any $v_1,v_2\in V^{″}$ . To see the equivalence, set $V^{″}=\left\{\left(v,\varphi \left(v\right)\right):v\in V\right\}$ and let ${\varphi }_V$ and ${\varphi }_{V^{\prime }}$ be the projection maps on the first and second coordinates, respectively. This equivalence will be used to define the weak isomorphism between two BFGs.

Using the same fashion as Example 2.1.12 in Chowdhury and Mémoli [33], we have the following truths.

· Suppose both $G$ and $G^{\prime }$ only contain one vertex $v$ and $v^{\prime }$ with a loop respectively. Then $G{\cong }^s{G}^{\prime }$ if and only if ${\mu }_B^P\left(v,v\right)={\mu }_{B^{\prime }}^P\left(v^{\prime },v^{\prime }\right)$ and ${\mu }_B^N\left(v,v\right)={\mu }_{B^{\prime }}^N\left(v^{\prime },v^{\prime }\right)$ .

· Two BFGs $G$ and $G^{\prime }$ with vertex set $V=\left\{v_1,v_2\right\}$ and $V^{\prime }=\left\{{v^{\prime }}_1,{v^{\prime }}_2\right\}$ respectively. Then $G{\cong }^s{G}^{\prime }$ if and only if there is a $2\times 2$ permutation matrix $X^P$ satisfying ${\Omega }_2^P=X^P{\Omega }_1^P{\left(X^P\right)}^{\text{T}}$ and another $2\times 2$ permutation matrix $X^N$ satisfying ${\Omega }_2^N=X^N{\Omega }_1^N{\left(X^N\right)}^{\text{T}}$ , where

${\Omega }_1^P=\left[\begin{array}{ll}{\mu }_B^P\left(v_1,v_1\right)\hfill & {\mu }_B^P\left(v_1,v_2\right)\hfill \\ {\mu }_B^P\left(v_2,v_1\right)\hfill & {\mu }_B^P\left(v_2,v_2\right)\hfill \end{array}\right],$

${\Omega }_2^P=\left[\begin{array}{ll}{\mu }_{B^{\prime }}^P\left({v^{\prime }}_1,{v^{\prime }}_1\right)\hfill & {\mu }_{B^{\prime }}^P\left({v^{\prime }}_1,{v^{\prime }}_2\right)\hfill \\ {\mu }_{B^{\prime }}^P\left({v^{\prime }}_2,{v^{\prime }}_1\right)\hfill & {\mu }_{B^{\prime }}^P\left({v^{\prime }}_2,{v^{\prime }}_2\right)\hfill \end{array}\right],$

${\Omega }_1^N=\left[\begin{array}{ll}{\mu }_B^N\left(v_1,v_1\right)\hfill & {\mu }_B^N\left(v_1,v_2\right)\hfill \\ {\mu }_B^N\left(v_2,v_1\right)\hfill & {\mu }_B^N\left(v_2,v_2\right)\hfill \end{array}\right],$

and

${\Omega }_2^N=\left[\begin{array}{ll}{\mu }_{B^{\prime }}^N\left({v^{\prime }}_1,{v^{\prime }}_1\right)\hfill & {\mu }_{B^{\prime }}^N\left({v^{\prime }}_1,{v^{\prime }}_2\right)\hfill \\ {\mu }_{B^{\prime }}^N\left({v^{\prime }}_2,{v^{\prime }}_1\right)\hfill & {\mu }_{B^{\prime }}^N\left({v^{\prime }}_2,{v^{\prime }}_2\right)\hfill \end{array}\right].$

· From the representation mentioned above, the BFG $G$ with $n$ vertices can be determined by two $n\times n$ matrices denoted by ${\Omega }_G^P$ and ${\Omega }_G^N$ , which record the positive and negative uncertainty information on two BFGs respectively. In detail, set it’s vertex set as $\left\{v_1,\cdots ,v_{\left|V\left(G\right)\right|}\right\}$ , and then

${\Omega }_G^P=\left[\begin{array}{ccc}{\mu }_G^P\left(v_1,v_1\right)& \cdots & {\mu }_G^P\left(v_1,v_{\left|V\left(G\right)\right|}\right)\\ {\mu }_G^P\left(v_2,v_1\right)& \cdots & {\mu }_G^P\left(v_2,v_{\left|V\left(G\right)\right|}\right)\\ ⋮& \ddots & ⋮\\ {\mu }_G^P\left(v_{\left|V\left(G\right)\right|},v_1\right)& \cdots & {\mu }_G^P\left(v_{\left|V\left(G\right)\right|},v_{\left|V\left(G\right)\right|}\right)\end{array}\right],$

${\Omega }_G^N=\left[\begin{array}{ccc}{\mu }_G^N\left(v_1,v_1\right)& \cdots & {\mu }_G^N\left(v_1,v_{\left|V\left(G\right)\right|}\right)\\ {\mu }_G^N\left(v_2,v_1\right)& \cdots & {\mu }_G^N\left(v_2,v_{\left|V\left(G\right)\right|}\right)\\ ⋮& \ddots & ⋮\\ {\mu }_G^N\left(v_{\left|V\left(G\right)\right|},v_1\right)& \cdots & {\mu }_G^N\left(v_{\left|V\left(G\right)\right|},v_{\left|V\left(G\right)\right|}\right)\end{array}\right].$

For any two BFGs $G=\left(V,A,B\right)$ and $G^{\prime }=\left(V^{\prime },A^{\prime },B^{\prime }\right)$ , $G{\cong }^s{G}^{\prime }$ if and only if $\left|A\right|=\left|A^{\prime }\right|$ (i.e., $\left|V\left(G\right)\right|=\left|V\left(G^{\prime }\right)\right|$ , and denote the vertex number by $n$ ) and there is a $n\times n$ permutation matrix $X^P$ satisfying ${\Omega }_{G^{\prime }}^P=X^P{\Omega }_G^P{\left(X^P\right)}^{\text{T}}$ and a $n\times n$ permutation matrix $X^N$ satisfying ${\Omega }_{G^{\prime }}^N=X^N{\Omega }_G^N{\left(X^N\right)}^{\text{T}}$ .

Remark. The motif sets can be extended to BFG setting in terms of the similar fashion. For each $n\in N$ and let $G$ be a BFG, define ${\Psi }_G^{n,P},{\Psi }_G^{n,N}:V^n\to R^{n\times n}$ to be the maps:

${\Psi }_G^{n,P}:\left\{v_1,\cdots ,v_n\right\}\mapsto {\left[{\mu }_G^P\left(v_i,v_j\right)\right]}_{i,j=1}^n,$

${\Psi }_G^{n,N}:\left\{v_1,\cdots ,v_n\right\}\mapsto {\left[{\mu }_G^N\left(v_i,v_j\right)\right]}_{i,j=1}^n.$

Note that ${\Psi }_G^{n,P}$ and ${\Psi }_G^{n,N}$ are the maps which map each vertex subset with $n$ vertices to its corresponding positive and negative MF matrices. Denote $\mathcal{C}\left(R^{n\times n}\right)$ by the closed subsets of $R^{n\times n}$ , and $M_n^P$ and $M_n^P$ are two maps defined by

$\left(V,A,B\right)\mapsto \left\{{\text{Ψ}}_G^{n,P}\left(v_1,\cdots ,v_n\right):\left\{v_1,\cdots ,v_n\right\}\subseteq V\left(G\right)\right\},$

$\left(V,A,B\right)\mapsto \left\{{\text{Ψ}}_G^{n,N}\left(v_1,\cdots ,v_n\right):\left\{v_1,\cdots ,v_n\right\}\subseteq V\left(G\right)\right\}.$

The following definitions are revised by Chowdhury and Mémoli [33].

Definition 3. (Correspondence) Let $G=\left(V,A,B\right)$ and $G^{\prime }=\left(V^{\prime },A^{\prime },B^{\prime }\right)$ be two BFGs. A correspondence between $V$ and $V^{\prime }$ is a relation $R\subseteq V\times V^{\prime }$ such that ${\pi }_V\left(R\right)=V$ and ${\pi }_{V^{\prime }}\left(R\right)=V^{\prime }$ , where ${\pi }_V$ and ${\pi }_{V^{\prime }}$ are the canonical projections of $V\times V^{\prime }$ onto $V$ and $V^{\prime }$ , respectively. The collection of all correspondences between $V$ and $V^{\prime }$ is denoted by $ℛ\left(V,V^{\prime }\right)$ , or simply denoted by $ℛ$ .

Definition 4. (Distortion of a correspondence) Let $G=\left(V,A,B\right)$ and $G^{\prime }=\left(V^{\prime },A^{\prime },B^{\prime }\right)$ be two BFGs, and $R\in ℛ\left(V,V^{\prime }\right)$ . The distortion of $R$ is defined by

$\text{dis}\left(R\right)=\underset{\left(v_1,v_2\right),\left({v^{\prime }}_1,{v^{\prime }}_2\right)\in R}{\text{sup}}\left\{\left|{\mu }_B^P\left(v_1,{v^{\prime }}_1\right)-{\mu }_{B^{\prime }}^P\left(v_2,{v^{\prime }}_2\right)\right|+\left|{\mu }_B^N\left(v_1,{v^{\prime }}_1\right)-{\mu }_{B^{\prime }}^N\left(v_2,{v^{\prime }}_2\right)\right|\right\}.$

In addition, the positive distortion and negative distortion of $R$ are defined by

${\text{dis}}^P\left(R\right)=\underset{\left(v_1,v_2\right),\left({v^{\prime }}_1,{v^{\prime }}_2\right)\in R}{\text{sup}}\left\{\left|{\mu }_B^P\left(v_1,{v^{\prime }}_1\right)-{\mu }_{B^{\prime }}^P\left(v_2,{v^{\prime }}_2\right)\right|\right\}$

and

${\text{dis}}^N\left(R\right)=\underset{\left(v_1,v_2\right),\left({v^{\prime }}_1,{v^{\prime }}_2\right)\in R}{\text{sup}}\left\{\left|{\mu }_B^N\left(v_1,{v^{\prime }}_1\right)-{\mu }_{B^{\prime }}^N\left(v_2,{v^{\prime }}_2\right)\right|\right\}$

respectively. If $\text{dis}\left(R\right)={\text{dis}}^P\left(R\right)+{\text{dis}}^N\left(R\right)$ , then we call $G$ and $G^{\prime }$ YinYang consistency with regard to $R$ .

Remark. (Composition of correspondences) Let $G_1=\left(V_1,A_1,B_1\right)$ , $G_2=\left(V_2,A_2,B_2\right)$ and $G_3=\left(V_3,A_3,B_3\right)$ be three BFGs, $R\in ℛ\left(V_1,V_2\right)$ , and $S\in ℛ\left(V_2,V_3\right)$ . Then

$R\circ S=\left\{\left(v_1,v_3\right)\in V_1\times V_3:\exists v_2\in V_2,\left(v_1,v_2\right)\in R,\left(v_2,v_3\right)\in S\right\}$

is the composition of $R$ and $S$ . Clearly, $R\circ S\in ℛ\left(V_1,V_3\right)$ , and $\text{dis}\left(R\circ S\right)\le \text{dis}\left(R\right)+\text{dis}\left(S\right)$ .

Now, the distance between two BFGs is defined as follows.

Definition 5. Let $G=\left(V,A,B\right)$ and $G^{\prime }=\left(V^{\prime },A^{\prime },B^{\prime }\right)$ be two BFGs. The distance between $G$ and $G^{\prime }$ is formulated by

$d\left(G,G^{\prime }\right)=\frac{1}{2}\underset{R\in ℛ}{\text{inf}}\text{dis}\left(R\right).$

In addition, the positive distance and negative distance are defined by

$d^P\left(G,G^{\prime }\right)=\frac{1}{2}\underset{R\in ℛ}{\text{inf}}{\text{dis}}^P\left(R\right)$

and

$d^N\left(G,G^{\prime }\right)=\frac{1}{2}\underset{R\in ℛ}{\text{inf}}{\text{dis}}^N\left(R\right)$

respectively.

From this definition, the optimal correspondence between $V$ and $V^{\prime }$ is further formulated by

${ℛ}^{\text{opt}}=\left\{R\in ℛ:\text{dis}\left(R\right)=2\text{dis}\left(G,G^{\prime }\right)\right\}.$

In addition, the positive optimal correspondences and negative optimal correspondences are formalized by

${ℛ}^{\text{Popt}}=\left\{R\in ℛ:{\text{dis}}^P\left(R\right)=2{\text{dis}}^P\left(G,G^{\prime }\right)\right\}$

and

${ℛ}^{\text{Nopt}}=\left\{R\in ℛ:{\text{dis}}^N\left(R\right)=2{\text{dis}}^N\left(G,G^{\prime }\right)\right\}$

respectively. Since BFG is finite, ${ℛ}^{\text{opt}}$ , ${ℛ}^{\text{Popt}}$ and ${ℛ}^{\text{Popt}}$ are not empty. Furthermore, in the symmetry setting (undirected BFG), the distance defined in Definition 5 is equivalent to Gromov-Hausdorff distance between the corresponding set.

Remark. Note that in Definition 4 and Definition 5, $l^{\infty }$ -norm are acted as a standard measure. In general, let $p\in \left[1,\infty \right)$ and the vertex number in BFG be infinity (we don’t discuss in this case in our paper), then the $L_p$ -norm based distortion of $R$ and the distance between BFG can be defined, and $L_{\infty }$ -norm is the special case when $p$ tends to infinite. Actually, in the discrete case, $l_{\infty }$ -norm is exactly the maximum operator between two MFs (with the same vertex set $V$ ),

i.e., ${\Vert {\mu }_B^P-{\mu }_{B^{\prime }}^P\Vert }_{l_{\infty }\left(V\times V\right)}={\max }_{v,v^{\prime }\in V}\left|{\mu }_B^P\left(v,v^{\prime }\right)-{\mu }_{B^{\prime }}^P\left(v,v^{\prime }\right)\right|$ and ${\Vert {\mu }_B^N-{\mu }_{B^{\prime }}^N\Vert }_{l_{\infty }\left(V\times V\right)}={\max }_{v,v^{\prime }\in V}\left|{\mu }_B^N\left(v,v^{\prime }\right)-{\mu }_{B^{\prime }}^N\left(v,v^{\prime }\right)\right|$ .

Suppose $G$ and $G^{\prime }$ are strong isomorphic BFGs, then $\text{dis}\left(R\right)={\text{dis}}^P\left(R\right)={\text{dis}}^N\left(R\right)=0$ . Unfortunately, the reverse of this proposition does not hold true, see Chowdhury and Mémoli [33] for more details. To overcome this deficiency, the concept of weak isomorphism between BFGs is defined below, which is a relaxation of the strong isomorphism version.

Definition 6. Let $G=\left(V,A,B\right)$ and $G^{\prime }=\left(V^{\prime },A^{\prime },B^{\prime }\right)$ be two BFGs. We say $G$ and $G^{\prime }$ are weakly isomorphic (denoted by $G{\cong }^w{G}^{\prime }$ ) if there is a set $V^{″}$ and surjective maps ${\varphi }_V:V^{″}\to V$ , ${\varphi }_{V^{\prime }}:V^{″}\to V^{\prime }$ such that ${\mu }_B^P\left({\varphi }_V\left(v_1\right),{\varphi }_V\left(v_2\right)\right)={\mu }_{B^{\prime }}^P\left({\varphi }_{V^{\prime }}\left(v_1\right),{\varphi }_{V^{\prime }}\left(v_2\right)\right)$ and ${\mu }_B^N\left({\varphi }_V\left(v_1\right),{\varphi }_V\left(v_2\right)\right)={\mu }_{B^{\prime }}^N\left({\varphi }_{V^{\prime }}\left(v_1\right),{\varphi }_{V^{\prime }}\left(v_2\right)\right)$ for any $v_1,v_2\in V^{″}$ .

Remark. By Definition 2, we know that the above definition on weak isomorphism can be written as the following equivalent statement. Let $G=\left(V,A,B\right)$ and $G^{\prime }=\left(V^{\prime },A^{\prime },B^{\prime }\right)$ be two BFGs. Then $G$ and $G^{\prime }$ are weakly isomorphic if there is a bipolar membership preserving surjection $\varphi :V\to V^{\prime }$ , i.e., ${\mu }_B^P\left(v_1,v_2\right)={\mu }_{B^{\prime }}^P\left(\varphi \left(v_1\right),\varphi \left(v_2\right)\right)$ and ${\mu }_B^N\left(v_1,v_2\right)={\mu }_{B^{\prime }}^N\left(\varphi \left(v_1\right),\varphi \left(v_2\right)\right)$ for all $v_1,v_2\in V$ .

Remark. If $\left|V\right|=\infty$ , then we have the following revised version of weak isomorphism by imitating Chowdhury and Mémoli [33]’s idea. Let $G=\left(V,A,B\right)$ and $G^{\prime }=\left(V^{\prime },A^{\prime },B^{\prime }\right)$ be two BFGs. We say $G$ and $G^{\prime }$ are type II weakly isomorphic (denoted by $G{\cong }_{II}^w{G}^{\prime }$ ) if for each $\epsilon >0$ , there are two sets $V_{\epsilon }^P$ and $V_{\epsilon }^N$ , surjective maps ${\varphi }_V^{\epsilon ,P}:V_{\epsilon }^P\to V$ , ${\varphi }_{V^{\prime }}^{\epsilon ,P}:V_{\epsilon }^P\to V^{\prime }$ , ${\varphi }_V^{\epsilon ,N}:V_{\epsilon }^N\to V$ and ${\varphi }_{V^{\prime }}^{\epsilon ,N}:V_{\epsilon }^N\to V^{\prime }$ such that

$\left|{\mu }_B^P\left({\varphi }_V^{\epsilon ,P}\left(v_1\right),{\varphi }_V^{\epsilon ,P}\left(v_2\right)\right)-{\mu }_{B^{\prime }}^P\left({\varphi }_{V^{\prime }}^{\epsilon ,P}\left(v_1\right),{\varphi }_{V^{\prime }}^{\epsilon ,P}\left(v_2\right)\right)\right|<\epsilon$

and

$\left|{\mu }_B^N\left({\varphi }_V^{\epsilon ,N}\left(v_3\right),{\varphi }_V^{\epsilon ,N}\left(v_4\right)\right)-{\mu }_{B^{\prime }}^N\left({\varphi }_{V^{\prime }}^{\epsilon ,N}\left(v_3\right),{\varphi }_{V^{\prime }}^{\epsilon ,N}\left(v_4\right)\right)\right|<\epsilon$

for any $v_1,v_2\in V_{\epsilon }^P$ and $v_3,v_4\in V_{\epsilon }^N$ .

By the explanation in Chowdhury and Mémoli [33], in the infinite case, ${ℛ}^{\text{opt}}$ , ${ℛ}^{\text{Popt}}$ and ${ℛ}^{\text{Popt}}$ may become $\varnothing$ , and the above revised definition is well-defined in the infinite situation. Moreover, two kinds of weak isomorphism induce two kinds of equivalence classes on the set of BFGs.

4. Conclusions and Future Work

The article addresses a gap in the existing literature by introducing the concept of distance between BFGs. Building upon the concept of Gromov-Hausdorff distance for networks, it proposes definitions for distortion, correspondence, and distance between BFGs, along with variations like positive and negative distances. It also discusses strong and weak isomorphism in the context of BFGs, culminating in a proposal for a new type of weak isomorphism for infinite BFGs.

The following issues require further in-depth study in the future.

· In the setting with constraints ${\mu }_A^P\left(v\right)=1$ and ${\mu }_A^N\left(v\right)=-1$ , we almost ignore the uncertainty of the vertices. However, in most application circumstances, the bipolar uncertainty of vertices is the first thing to be considered, and the bipolar uncertainty of edges is thus constrained. Therefore, it is necessary to relax the restrictions on the vertex bipolar MF and discuss the general situation.

· The theoretical implications and properties of the proposed distance measures should be explored in sufficient depth in the future.

· The concrete examples or case studies to demonstrate the practical utility of the proposed concepts in potential applications will be provided in the future.

Conflicts of Interest

The author declares no conflicts of interest.

Conflicts of Interest

The author declares no conflicts of interest.

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