Fuzzy graphs and their applications in finding the best route, dominant node and influence index in a network under the hesitant bipolar-valued fuzzy environment

Abstract

This paper introduces the concept of hesitant bipolar-valued fuzzy graph (HBVFG), which captures the two opposing perspectives, namely the positive and negative opinions. The novelty, importance and implications of this concept are illustrated by some results, examples, and graphical representations. There are, respectively, some theoretical terms of graphs such as partial directed hesitant bipolar-valued fuzzy subgraph (HBVFSG), directed HBVFSG, directed spanning HBVFSG, strong directed HBVFG and complete directed HBVFG which are introduced. The operations, such as Cartesian, direct, lexicographical, and strong products, are also defined between two HBVFGs with examples. The mapping relations, such as homomorphism, isomorphism, weak isomorphism, and co-weak isomorphism, are derived with an example. The applications of directed HBVFGs with algorithms for finding the optimal path in a network and the dominant node and influence of index with the self-persistence degree of a node in a social network are presented. For each problem, an algorithm is developed and its effectiveness is demonstrated by examples. The proposed concept is assessed in terms of theory and practice. The benefits of the proposed solution are highlighted and a clear comparison is made with the existing methods.

Introduction

Graph theory plays a key role in different related areas such as decision-making, engineering, computer sciences, physics, sociology, medicine, networking systems, etc. However, the presence of uncertainty in our real-life graph theory is generalized by Rosenfeld [1] using fuzzy set (FS) theory Zadeh [2] named fuzzy graph (FG). Then, it applied various fields to solving decision-making problems [3, 42], telecommunication system [4], computer network [5, 6], image representation and segmentation [7, 43], etc. To handle uncertainty in various practical problems, the number of scholars generalized fuzzy sets at the same time parallelly FG also generalized. For example, intuitionistic fuzzy graph [8] from an intuitionistic fuzzy set (IFS) [9], Picture fuzzy graph [10] from Picture fuzzy set [11], Pythagorean fuzzy graph [12] from Pythagorean fuzzy set [13], hesitant fuzzy graph [14] from the hesitant fuzzy set (HFS) [15], etc. are, respectively, introduced.

Some real-world situations involve contrasting outcomes, such as positive and negative impacts, reliable likelihood and preference, neutrality and indifference. To handle these kinds of scenarios, Zhang [16] proposed a bipolar fuzzy set (BFS). Then, he applied it to graph theory and developed a bipolar fuzzy graph (BFG) [17]. In [18], Mandal and Ranadive presented a new kind of fuzzy set called hesitant bipolar-valued fuzzy set (HBVFS), which expresses a pair of opposite reactions such as positive information is a set of potential values in [0, 1] for what is deemed to be possible and negative information is a set of potential values in $[-1, 0]$ for what is regarded to be impossible. In HBVFS, two concepts like BFS and HFS are considered simultaneously. Inspired by this idea, we introduce a directed hesitant bipolar-valued fuzzy graph (HBVFG) in this paper, which captures more uncertainty in the graph nodes than other generalized FG structures. We define the basic structure such as partial directed hesitant bipolar-valued fuzzy subgraph (HBVFSG), directed HBVFSG, spanning directed HBVFSG, strongly directed HBVFG and completely directed HBVFG. We also explore a series of operational laws such as the Cartesian product, direct product, lexicographical product and strong product of directed HBVFGs in detail. We examine the mapping relations between two directed HBVFGs as well. In terms of connection between two nodes, the best pathfinding has been widely discussed in various generalized fuzzy structures [19,20,21,22,23,24,25,26,27,28]. Therefore, as an application of directed HBVFGs, we propose an algorithm for finding the best path in a network. Another important application of HBVFGs in graph theory is the dominant degree and influence index of nodes with self-persistence, which has been pointed out in [14, 29,30,31,32,33,34]. We design an algorithm for finding the dominant degree and influence index of nodes with self-persistence in a social network like Facebook, WhatsApp, Twitter, etc.

Motivation to introduce HBVFGs

The example of journeys order discussed in [35], we see that authors pointed out "Fast" or "Fast and cheap" is the positive side information, and "Early" or "Early and late" is the negative side information. However, we observe that for computation of the best journey, they only considered the positive side information "Fast" and the negative side information "Early" as a bipolar fuzzy number. The information "Fast and cheap" and “Early and late" is not considered. It is because there are no existing tools in our hands for considering at a time two different types of bipolar fuzzy information. This motivates us to introduce the HBVFGs, which capture at a time two or more bipolar fuzzy information. In the same example, the authors considered each journey path as a node and found the membership values of edges among all nodes using the concepts of BFG. Then, apply the Wiener index of a BFG to find the best node, i.e., the best path. Since, in this example, the journey route or path, i.e., the edges information, is known, we solve this problem using our given algorithm to find the best path in a network.

Structure of this paper

“Preliminaries” recalls basic results related to graph theory and HBVFSs. Directed HBVFGs and their related basic definitions with examples and theories are introduced in “Hesitant bipolar-valued fuzzy graph”. This section also discussed product operations and mapping relations between two directed HBVFGs. The applications of directed HBVFGs, such as finding the shortest path in a network and finding the dominant node in a social network, are discussed in “Application of directed HBVFGs” with a comparative analysis. The influence index with the self-persistence of a node is also introduced in this section. The paper is concluded in “Conclusion with future work”.

Preliminaries

We are reminded in this section of the primary findings of graph theory [36] and HBVFS [18].

Graph theory

From graph theory [36], it is well known that a directed graph is a pair of ordered $G=(V, E)$, where $V=\{x_{1},x_{2}, \cdots , x_{n}\}$ is the set of finite nodes, and E is the set of edges of G. Let $(x_{i},x_{j}) \in E (i \ne j)$ be the directed edge from $x_{i}$ to $x_{j}$, then $(x_{i},x_{j})$ is called the incoming edge of $x_{j}$ and the outgoing edge of $x_{i}$. A subgraph of a directed graph $G_{1}=(V_{1},E_{1})$ is a directed graph $G_{2}=(V_{2},E_{2})$, where $V_{2} \subseteq V_{1}$ and $E_{2} \subseteq E_{1}$.

Hesitant bipolar-valued fuzzy set

Definition 1

[18]. Let X be a reference set. A HBVFS A on X is defined as

$$\begin{aligned} A = \{\langle x, \xi _{A}(x) = \left( \xi _{A}^{P}(x), \xi _{A}^{N}(x) \right) \rangle \mid x \in X\} \end{aligned}$$

(1)

where $\xi _{A}^{P}(x) \in [0,1]$ and $\xi _{A}^{N}(x) \in [-1,0]$ are called hesitant fuzzy positive and negative elements to the set A, respectively. $\xi _{A}(x) =( \xi _{A}^{P}(x), \xi _{A}^{N}(x)) \in [0,1] \times [-1,0]$ is called the hesitant bipolar-valued fuzzy element (HBVFE) to the set A. In this paper, we use $\xi _{A} = (\xi ^{P}_{A}, \xi ^{N}_{A})$ instead of $\xi _{A}(x) =( \xi _{A}^{P}(x), \xi _{A}^{N}(x))$. The set of all HBVFSs on X is denoted by $\Theta (X)$. The algebraic sum between two HBVFEs $\xi _{A} = (\xi ^{P}_{A}, \xi ^{N}_{A})$ and $\xi _{B} = (\xi ^{P}_{B}, \xi ^{N}_{B})$ of $A, B \in \Theta (X)$ is defined in the following way [18]:

$$\begin{aligned} \left( \xi _{A} \oplus \xi _{B}\right)&=\left( \bigcup _{\gamma _{1}^{P} \in \xi ^{P}_{A},\gamma _{2}^{P} \in \xi ^{P}_{B}} \{\gamma _{1}^{P} + \gamma _{2}^{P}-\gamma _{1}^{P}\gamma _{2}^{P}\}, \right. \nonumber \\&\quad \left. \bigcup _{\gamma _{1}^{N} \in \xi ^{N}_{A},\gamma _{2}^{N} \in \xi ^{N}_{B}} \{\gamma _{1}^{N} + \gamma _{2}^{N}+\gamma _{1}^{N}\gamma _{2}^{N}\}\right) . \end{aligned}$$

(2)

Definition 2

[18]. Let $\xi _{A}(x) =( \xi _{A}^{P}(x), \xi _{A}^{N}(x))$ be a corresponding HBVFE of $A \in \Theta (X)$ for all $x \in X$. Then $\delta (\xi _{A}(x))=\frac{1}{2}( \frac{1}{l(\xi _{A}^{P}(x))} \sum _{\gamma ^{P} \in \xi _{A}^{P}(x)}\gamma ^{P}$ $- \frac{1}{l(\xi _{A}^{N}(x))}\sum _{\gamma ^{N} \in \xi _{A}^{N}(x)}\gamma ^{N})$ is called the score function of $\xi _{A}(x)$, where $l(\xi _{A}^{P}(x))$ and $l(\xi _{A}^{N}(x))$ are the numbers of the elements in $\xi _{A}^{P}(x)$ and $\xi _{A}^{N}(x)$, respectively. It is clear that $\delta (\xi _{A}(x))\in [0,1]$ for all $x \in X$.

In the following, we define a subset of two HBVFSs.

Definition 3

Let $\xi _{A} = (\xi ^{P}_{A}, \xi ^{N}_{A})$ and $\xi _{B} = (\xi ^{P}_{B}, \xi ^{N}_{B})$ be the corresponding HBVFEs of $A, B \in \Theta (X)$. If $\delta (\xi _{A}(x)) \le \delta (\xi _{B}(x))$ for all $x \in X$, then the HBVFS A is a subset of HBVFS B and it is denoted by $A \preceq B$.

Example 1

Let $X=\{x_{1}, x_{2}, x_{3}\}$ be a reference set and $A, B \in \Theta (X)$, where $A=\{ \langle x_{1}, (\{0.2,0.4,0.5\}$, $\{-0.3, -0.8\})\rangle$, $\langle x_{2}, (\{0.3,0.4\}, \{-0.9, -0.8, -0.7\})\rangle$, $\langle x_{3}, (\{0.3,0.2,0.5$, $0.6\}, \{-0.7,-0.6,-0.5,-0.2\})\rangle \}$ and $B=\{ \langle x_{1}, (\{0.3,0.5,0.7\}, \{-0.6, -0.4\})\rangle$, $\langle x_{2}$, $(\{0.5, 0.6$, $0.7, 0.8\}, \{-0.8, -0.7, -0.6\})\rangle$, $\langle x_{3}, (\{0.3$, $0.9\}, \{-0.8,-0.5,-0.4\})\rangle \}$. Using Definition 3, we have $\delta (\xi _{A}(x_{1}))=0.4583$, $\delta (\xi _{A}(x_{2}))=0.5750$, $\delta (\xi _{A}(x_{3}))=0.4500$, $\delta (\xi _{B}(x_{1}))=0.5000$, $\delta (\xi _{B}(x_{2}))$ $=0.6750$, $\delta (\xi _{B}(x_{3}))=0.5833$. Therefore, $A \prec B$ according to Definition 4. Since, $\delta (\xi _{A}(x_{1}))=0.4583 < 0.500=\delta (\xi _{B}(x_{1}))$, $\delta (\xi _{A}(x_{2}))=0.5750 < 0.6750=\delta (\xi _{B}(x_{2}))$, $\delta (\xi _{A}(x_{3}))=0.4500 < 0.5833=\delta (\xi _{B}(x_{3}))$.

The score-based intersection operation is defined in the following way:

Definition 4

Let $\xi _{A} = (\xi ^{P}_{A}, \xi ^{N}_{A})$ and $\xi _{B} = (\xi ^{P}_{B}, \xi ^{N}_{B})$ be the corresponding HBVFEs of $A, B \in \Theta (X)$. Then, the score based intersection between $\xi _{A} = (\xi ^{P}_{A}, \xi ^{N}_{A})$ and $\xi _{B} = (\xi ^{P}_{B}, \xi ^{N}_{B})$ denoted by $\xi _{A} {\tilde{\wedge }} \xi _{B}$ is defined as follows:

$$\begin{aligned} (\xi _{A} {\tilde{\wedge }} \xi _{B})(x)= {\left\{ \begin{array}{ll} \xi _{A}(x), &{} \text {if } \delta (\xi _{A}(x))< \delta (\xi _{B}(x)),\\ \xi _{B}(x), &{} \text {if } \delta (\xi _{B}(x)) < \delta (\xi _{A}(x)),\\ \xi _{A}(x) \text {or } \xi _{B}(x), &{} \text {if } \delta (\xi _{A}(x)) = \delta (\xi _{B}(x)), \end{array}\right. }\nonumber \\ \end{aligned}$$

(3)

for all $x \in X$.

From Definition 3, we have the following proposition.

Proposition 1

Let $\xi _{A} = (\xi ^{P}_{A}, \xi ^{N}_{A})$ and $\xi _{B} = (\xi ^{P}_{B}, \xi ^{N}_{B})$ be the corresponding HBVFEs of $A, B \in \Theta (X)$. If $A \preceq B$, then $\delta (\xi _{A}(x)) \le \delta (\xi _{B}(x))$ for all $x \in X$.

From Definitions 2 and 4, we have the following proposition.

Proposition 2

. Let $\xi _{A} = (\xi ^{P}_{A}, \xi ^{N}_{A})$ and $\xi _{B} = (\xi ^{P}_{B}, \xi ^{N}_{B})$ be the corresponding HBVFEs of $A, B \in \Theta (X)$, then $\delta (\xi _{A}(x) {\tilde{\wedge }} \xi _{B}(x)) = min \left( \delta (\xi _{A}(x)), \delta (\xi _{B}(x))\right)$ for all $x \in X$.

Hesitant bipolar-valued fuzzy relation

Definition 5

Let $\xi _{A} = (\xi ^{P}_{A}, \xi ^{N}_{A})$ and $\xi _{B} = (\xi ^{P}_{B}, \xi ^{N}_{B})$ be the corresponding HBVFEs of $A \in \Theta (X)$ and $B \in \Theta (Y)$, are respectively. Then, $A {\tilde{\times }} B$ represent the Cartesian product of HBVFSs A and B, and it is defined in the following way:

$$\begin{aligned} A {\tilde{\times }} B=\left\{ \langle (x,y), \xi _{A}(x) {\tilde{\wedge }} \xi _{B}(y) \rangle \mid (x,y) \in X \times Y \right\} . \end{aligned}$$

(4)

Definition 6

Let $\xi _{A} = (\xi ^{P}_{A}, \xi ^{N}_{A})$ and $\xi _{B} = (\xi ^{P}_{B}, \xi ^{N}_{B})$ be the corresponding HBVFEs of $A \in \Theta (X)$ and $B \in \Theta (Y)$, are respectively. Then, the hesitant bipolar-valued fuzzy relation (HBVFR) ${\mathcal {R}}$ from the HBVFS A into HBVFS B is a HBVFS on $X \times Y$ such that $\delta (\xi _{{\mathcal {R}}}(x,y)) \le min \left( \delta (\xi _{A}(x)), \delta (\xi _{B}(y))\right)$, for all $(x,y) \in X \times Y$ and $\xi _{{\mathcal {R}}}$ is a corresponding HBVFE of ${\mathcal {R}}$.

If $X=Y$, then ${\mathcal {R}}$ is called a HBVFR on X.

Example 2

Let $X=\{x_{1}, x_{2}, x_{3}\}$ and $Y=\{y_{1}, y_{2}\}$. Let $A \in \Theta (X)$ and $B \in \Theta (Y)$, where $A=\{ \langle x_{1}$, $(\{0.2$, 0.4, $0.8\}$, $\{-0.3$, $-0.8\})\rangle$, $\langle x_{2}$, $(\{0.6$, $0.9\}$, $\{-0.9$, $-0.8$, $-0.2\})\rangle$, $\langle x_{3}$, $(\{0.1$, 0.2, 0.5, $0.6\}$, $\{-0.7$, $-0.6$, $-0.5\})\rangle \}$ and $B=\{ \langle y_{1}$, $(\{0.4$, 0.5, $0.7\}$, $\{-0.6$, $-0.3\})\rangle$, $\langle y_{2}$, $(\{0.2$, 0.5, 0.7, $0.8\}$, $\{-0.9$, $-0.7$, $-0.6\})\rangle$. Using Definition 3, we have $\delta (\xi _{A}(x_{1}))=0.5083$, $\delta (\xi _{A}(x_{2}))=0.6917$, $\delta (\xi _{A}(x_{3}))$ $= 0.4750$, $\delta (\xi _{B}(y_{1}))=0.4917$, $\delta (\xi _{B}(y_{2}))=0.6417$. Then, $A {\tilde{\times }} B=\{\langle (x_{1},y_{1})$, $(\{0.4$, 0.5, $0.7\}$, $\{-0.6$, $-0.3\})\rangle$, $\langle (x_{1}$, $y_{2})$, $(\{0.2$, 0.4, $0.8\}$, $\{-0.3$, $-0.8\})\rangle$, $\langle (x_{2}$, $y_{1})$, $(\{0.4$, 0.5, $0.7\}$, $\{-0.6$, $-0.3\})\rangle$, $\langle (x_{2}$, $y_{2})$, $(\{0.2$, 0.5, 0.7, $0.8\}$, $\{-0.9$, $-0.7$, $-0.6\})\rangle$, $\langle (x_{3}$, $y_{1})$, $(\{0.1$, 0.2, 0.5, $0.6\}$, $\{-0.7$, $-0.6$, $-0.5\})\rangle$, $\langle (x_{3}$, $y_{2})$, $(\{0.1$, 0.2, 0.5, $0.6\}$, $\{-0.7$, $-0.6$, $-0.5\})\rangle \}$ and a HBVFR ${\mathcal {R}}$ can be written as ${\mathcal {R}}= \{\langle (x_{1}$, $y_{1})$, $(\{0.2$, 0.4, 0.5, $0.7\}$, $\{-0.5$, $-0.2$, $-0.1\})\rangle$, $\langle (x_{1}$, $y_{2})$, $(\{0.3$, 0.5, $0.2\}$, $\{-0.8$, $-0.5\})\rangle$, $\langle (x_{2}$, $y_{1})$, $(\{0.1$, 0.3, 0.4, $0.8\}$, $\{-0.4$, $-0.2$, $-0.1\})\rangle$, $\langle (x_{2}$, $y_{2})$, $(\{0.4$, $0.8\}$, $\{-0.8$, $-0.7$, $-0.5\})\rangle$, $\langle (x_{3}$, $y_{1})$, $(\{0.2$, 0.3, 0.4, $0.8\}$, $\{-0.9$, $-0.4$, $-0.1\})\rangle$, $\langle (x_{3}$, $y_{2})$, $(\{0.2$, $0.8\}$, $\{-0.4$, $-0.3$, $-0.2\})\rangle \}$. All elements in ${\mathcal {R}}$ are satisfied $\delta (\xi _{{\mathcal {R}}}(x_{i}, y_{j})) \le min \left( \delta (\xi _{A}(x_{i})), \delta (\xi _{B}(y_{j}))\right)$, $i=1,2,3$ and $j=1,2$. For example, if $i=1, j=1$, then $\delta (\xi _{{\mathcal {R}}}(x_{1}$, $y_{1}))$ $= 0.3583$ $\le min ( \delta (\xi _{A}(x_{1}))$, $\delta (\xi _{B}(y_{1})))$ $= 0.4917$.

Hesitant bipolar-valued fuzzy graph

The concept of FGs is first extended to the HBVFSs background. Then we proposed the definition and the corresponding concepts of HBVFGs.

Definition 7

A directed HBVFG of $G=(V,E)$ is a pair ${\tilde{G}}=({\mathcal {A}}, {\mathcal {G}})$, where ${\mathcal {A}}$ is a HBVFS in V whose corresponding HBVFE $\xi _{{\mathcal {A}}} = (\xi ^{P}_{{\mathcal {A}}}, \xi ^{N}_{{\mathcal {A}}})$ and ${\mathcal {G}}$ is a HBVFS in $V \times V$ whose corresponding HBVFE $\xi _{{\mathcal {G}}} = (\xi ^{P}_{{\mathcal {G}}}, \xi ^{N}_{{\mathcal {G}}})$ such that for all $x_{1}, x_{2}$ in V, $\delta (\xi _{{\mathcal {G}}}(x_{1}, x_{2})) \le min(\delta (\xi _{{\mathcal {A}}}(x_{1})), \delta (\xi _{{\mathcal {A}}}(x_{2})))$.

We call ${\mathcal {A}}$ the hesitant bipolar-valued fuzzy node set (HBVFNS) of ${\tilde{G}}$ and ${\mathcal {G}}$ the hesitant bipolar-valued fuzzy edge set (HBVFES) of ${\tilde{G}}$, respectively. It is observed that ${\mathcal {G}}$ is a symmetric HBVFR on ${\mathcal {A}}$. Here, we do not consider the elements $\xi _{{\mathcal {A}}}(x_{1}) = (\{0\}, \{0\})$.

Following the general definition of directed HBVFGs, we introduce further a special directed HBVFGs named partial directed HBVFSG, directed HBVFSG, spanning directed HBVFSG, strong directed HBVFG and complete directed HBVFG, respectively. In fact, the following definitions are generalizations of the definitions of related fuzzy graph theory results in [39,40,41]. Interested readers, please see [39,40,41] for more explanation of the following definitions. Here, we omitted the explanation of the following definitions because a similar description is also in the following definitions.

Definition 8

Let ${\tilde{G}}=({\mathcal {A}}, {\mathcal {G}})$ and ${\tilde{H}}=({\mathcal {B}}, {\mathcal {H}})$ be directed HBVFGs of $G=(V,E)$. Then, ${\tilde{H}}=({\mathcal {B}}, {\mathcal {H}})$ is called a partial directed HBVFSG of ${\tilde{G}}=({\mathcal {A}}, {\mathcal {G}})$ if $\delta (\xi _{{\mathcal {B}}}(x_{1})) \le \delta (\xi _{{\mathcal {A}}}(x_{1}))$ and $\delta (\xi _{{\mathcal {H}}}(x_{1}, x_{2})) \le \delta (\xi _{{\mathcal {G}}}(x_{1}, x_{2}))$ for all $x_{1},x_{2} \in V$.

Definition 9

Let ${\tilde{G}}=({\mathcal {A}}, {\mathcal {G}})$ and ${\tilde{H}}=({\mathcal {B}}, {\mathcal {H}})$ be directed HBVFGs of $G_{1}=(V_{1}, E_{1})$ and $G_{2}=(V_{2}, E_{2})$, respectively. Then, the directed HBVFG ${\tilde{H}}=({\mathcal {B}}, {\mathcal {H}})$ is called a directed HBVFSG of ${\tilde{G}}=({\mathcal {A}}, {\mathcal {G}})$ induced by $V_{2}$ if $V_{2} \subseteq V_{1}$, $\delta (\xi _{{\mathcal {B}}}(x_{1})) = \delta (\xi _{{\mathcal {A}}}(x_{1}))$ for all $x_{1} \in V_{2}$ and $\delta (\xi _{{\mathcal {H}}}(x_{1},x_{2}))= \delta (\xi _{{\mathcal {G}}}(x_{1},x_{2}))$ for all $x_{1},x_{2} \in V_{2}$.

Definition 10

Let ${\tilde{G}}=({\mathcal {A}}, {\mathcal {G}})$ and ${\tilde{H}}=({\mathcal {B}}, {\mathcal {H}})$ be directed HBVFGs of $G=(V,E)$. The span of ${\tilde{G}}$ is a partial directed HBVFSG ${\tilde{H}}=({\mathcal {B}}, {\mathcal {H}})$ if $\delta (\xi _{{\mathcal {G}}}(x_{1})) = \delta (\xi _{{\mathcal {H}}}(x_{1}))$ for all $x_{1} \in V$. In this case, ${\tilde{H}}=({\mathcal {B}}, {\mathcal {H}})$ is called a spanning directed HBVFSG of ${\tilde{G}}=({\mathcal {A}}, {\mathcal {G}})$.

Definition 11

A HBVFG ${\tilde{G}}=({\mathcal {A}}, {\mathcal {G}})$ of $G=(V,E)$ is called a strong directed HBVFG if $\delta (\xi _{{\mathcal {G}}}(x_{1},x_{2}))=min (\delta (\xi _{{\mathcal {A}}}(x_{1})), \delta (\xi _{{\mathcal {A}}}(x_{2})))$ for each edge $(x_{1},x_{2}) \in E$.

Definition 12

A directed HBVFG ${\tilde{G}}=({\mathcal {A}}, {\mathcal {G}})$ of $G=(V,E)$ is called a complete directed HBVFG if $\delta (\xi _{{\mathcal {G}}}(x_{1}, x_{2}))=min (\delta (\xi _{{\mathcal {A}}}(x_{1})), \delta (\xi _{{\mathcal {A}}}(x_{2})))$ for all $x_{1}, x_{2} \in V$.

We present below an illustrative example to elaborate on the above-developed definitions.

Example 3

Let $V=\{x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\}$ be the set of nodes and $E=\{(x_{1}, x_{2})$, $(x_{1}, x_{3})$, $(x_{1}, x_{5})$, $(x_{2}, x_{3})$, $(x_{3}, x_{4})$, $(x_{3}, x_{5})$, $(x_{4}, x_{5})\}$ be the set of edge. We are provided the HBVFSs ${\mathcal {A}}$ and ${\mathcal {G}}$ as the following forms:

$$\begin{aligned} {\mathcal {A}}&=\{\langle x_{1}, (\{0.8,0.6,0.2\},\{-0.7,-0.5\})\rangle , \\&\langle x_{2}, (\{0.9,0.7\},\{-0.8,-0.6,-0.4\})\rangle ,\\&\langle x_{3}, (\{0.8,0.5\},\{-0.9,-0.7\})\rangle , \\&\langle x_{4}, (\{0.7,0.6,0.5,0.4\},\{-0.8\})\rangle ,\\&\langle x_{5}, (\{0.7,0.5,0.3\},\{-0.9,-0.8,-0.4\})\rangle \}, \end{aligned}$$

and we have $\delta (\xi _{{\mathcal {A}}}(x_{1}))=0.5667$, $\delta (\xi _{{\mathcal {A}}}(x_{2}))=0.7000$, $\delta (\xi _{{\mathcal {A}}}(x_{3}))=0.7250$, $\delta (\xi _{{\mathcal {A}}}(x_{4}))=0.6750$, $\delta (\xi _{{\mathcal {A}}}(x_{5}))=0.600$;

$$\begin{aligned} {\mathcal {G}}&=\{\langle (x_{1}, x_{2}), (\{0.7,0.6,0.5\},\{-0.6,-0.4\})\rangle ,\\&\langle (x_{1}, x_{3}), (\{0.8,0.4\},\{-0.7,-0.5,-0.2\})\rangle ,\\&\langle (x_{1}, x_{5}), (\{0.8,0.4,0.3,0.2\},\{-0.9,-0.7,-0.5\})\rangle ,\\&\langle (x_{2}, x_{3}), (\{0.9,0.7\},\{-0.8,-0.6,-0.3\})\rangle ,\\&\langle (x_{3}, x_{4}), (\{0.9,0.5,0.4\},\{-0.8,-0.7,-0.6\})\rangle , \\&\langle (x_{3}, x_{5}), (\{0.6,0.5\},\{-0.8,-0.4\})\rangle ,\\&\langle (x_{4}, x_{5}), (\{0.8,0.6,0.4\},\{-0.8,-0.6,-0.5,-0.4\})\rangle \}, \end{aligned}$$

and we have $\delta (\xi _{{\mathcal {G}}}(x_{1}, x_{2}))=0.5500$, $\delta (\xi _{{\mathcal {G}}}(x_{1}, x_{3}))=0.5333$, $\delta (\xi _{{\mathcal {G}}}(x_{1}, x_{5}))=0.5625$, $\delta (\xi _{{\mathcal {G}}}(x_{2}, x_{3}))=0.6833$, $\delta (\xi _{{\mathcal {G}}}(x_{3}, x_{4}))=0.6500$, $\delta (\xi _{{\mathcal {G}}}(x_{3}, x_{5}))=0.5750$. $\delta (\xi _{{\mathcal {G}}}(x_{4}, x_{5}))=0.5875$.

It’s easy to see through routine calculations ${\tilde{G}}=({\mathcal {A}}, {\mathcal {G}})$ is a directed HBVFG of $G=(V,E)$, shown in Fig. 1.

Fig. 1

The directed HBVFG ${\tilde{G}}=({\mathcal {A}}, {\mathcal {G}})$

Fig. 2

The partial HBVFSG ${\tilde{H}}=({\mathcal {B}}, {\mathcal {H}})$ of ${\tilde{G}}=({\mathcal {A}}, {\mathcal {G}})$

Let ${\tilde{H}}=({\mathcal {B}}, {\mathcal {H}})$ be a directed HBVFG of $G=(V,E)$, where

$$\begin{aligned} {\mathcal {B}}&=\{\langle x_{1}, (\{0.8,0.7,0.4\},\{-0.8,-0.1\})\rangle ,\\&\langle x_{2}, (\{0.7,0.6,0.4,0.3\},\{-0.9,-0.7,-0.4\})\rangle ,\\&\langle x_{3}, (\{0.9,0.7,0.4\},\{-0.9,-0.8,-0.5\})\rangle ,\\&\langle x_{4}, (\{0.6,0.3\},\{-0.8,-0.6,-0.2\})\rangle ,\\&\langle x_{5}, (\{0.7,0.5,0.4,0.2\},\{-0.8,-0.6\})\rangle \}, \end{aligned}$$

and we have $\delta (\xi _{{\mathcal {B}}}(x_{1}))=0.5417$, $\delta (\xi _{{\mathcal {B}}}(x_{2}))=0.5833$, $\delta (\xi _{{\mathcal {B}}}(x_{3}))=0.7000$, $\delta (\xi _{{\mathcal {B}}}(x_{4}))=0.4917$, $\delta (\xi _{{\mathcal {B}}}(x_{5}))=0.5750$;

$$\begin{aligned} {\mathcal {H}}&=\{\langle (x_{1}, x_{2}), (\{0.6,0.4\},\{-0.8,-0.5,-0.3\})\rangle ,\\&\langle (x_{1}, x_{3}), (\{0.8,0.3,0.2\},\{-0.8,-0.5,-0.4\})\rangle ,\\&\langle (x_{1},x_{5}), (\{0.9,0.6,0.2\},\{-0.7,-0.4\})\rangle ,\\&\langle (x_{2},x_{3}), (\{0.8,0.7,0.4,0.2\},\{-0.7,-0.4,-0.1\})\rangle ,\\&\langle (x_{3},x_{4}), (\{0.7,0.5,0.2\},\{-0.4,-0.3\})\rangle , \\&\langle (x_{3},x_{5}), (\{0.8,0.4\},\{-0.7,-0.5,-0.2\})\rangle ,\\&\langle (x_{4},x_{5}), (\{0.6,0.4,0.3\},\{-0.8,-0.7,-0.5\})\rangle \}, \end{aligned}$$

and we have $\delta (\xi _{{\mathcal {H}}}(x_{1},x_{2}))=0.5167$, $\delta (\xi _{{\mathcal {H}}}(x_{1},x_{3}))=0.5000$, $\delta (\xi _{{\mathcal {H}}}(x_{1},x_{5}))=0.5583$, $\delta (\xi _{{\mathcal {H}}}(x_{2},x_{3}))=0.4625$, $\delta (\xi _{{\mathcal {H}}}(x_{3},x_{4}))=0.4083$, $\delta (\xi _{{\mathcal {H}}}(x_{3},x_{5}))=0.5333$, $\delta (\xi _{{\mathcal {H}}}(x_{4},x_{5}))=0.5500$.

It’s easy to see through routine calculations ${\tilde{H}}=({\mathcal {B}}, {\mathcal {H}})$ is a partial directed HBVFSG of ${\tilde{G}}=({\mathcal {A}}, {\mathcal {G}})$ shown in Fig. 2.

Let ${\tilde{K}}=({\mathcal {C}}, {\mathcal {K}})$ be a directed HBVFG of $G_{1}=(V_{1},E_{1})$ where $V_{1}=\{x_{1}$, $x_{2}$, $x_{3}$, $x_{5}\}$ and $E_{1}=\{(x_{1},x_{2})$, $(x_{1},x_{3})$, $(x_{1},x_{5})$, $(x_{2},x_{3})$, $(x_{3},x_{5})\}$. The HBVFSs ${\mathcal {C}}$ and ${\mathcal {K}}$ are defined as

$$\begin{aligned} {\mathcal {C}}=\{&\langle x_{1}, (\{0.8,0.4\},\{-0.7,-0.6,-0.3\})\rangle ,\\&\langle x_{2}, (\{0.8,0.7,0.6\},\{-0.9,-0.5\})\rangle ,\\&\langle x_{3}, (\{0.8,0.7,0.6\},\{-0.8,-0.7\})\rangle ,\\&\langle x_{5}, (\{0.9,0.6,0.5,0.4\},\{-0.8,-0.6,-0.4\})\rangle \}, \end{aligned}$$

and we have $\delta (\xi _{{\mathcal {C}}}(x_{1}))=0.5667$, $\delta (\xi _{{\mathcal {C}}}(x_{2}))=0.7000$, $\delta (\xi _{{\mathcal {C}}}(x_{3}))=0.7250$, $\delta (\xi _{{\mathcal {C}}}(x_{5}))=0.6000$;

$$\begin{aligned} {\mathcal {K}}= \{&\langle (x_{1},x_{2}), (\{0.6,0.4\},\{-0.7,-0.6,-0.5\})\rangle ,\\&\langle (x_{1},x_{3}), (\{0.8,0.6,0.4,0.2\},\{-0.8,-0.6,-0.3\})\rangle ,\\&\langle (x_{1},x_{5}), (\{0.8,0.6,0.4\},\{-0.8,-0.6,-0.4,-0.3\})\rangle ,\\&\langle (x_{2},x_{3}), (\{0.9,0.8,0.7\},\{-0.8,-0.7,-0.2\})\rangle ,\\&\langle (x_{3},x_{5}), (\{0.8,0.6,0.4,0.2\},\{-0.7,-0.6\})\rangle \}, \end{aligned}$$

and we have $\delta (\xi _{{\mathcal {K}}}(x_{1},x_{2}))=0.5500$, $\delta (\xi _{{\mathcal {K}}}(x_{1},x_{3}))=0.5333$, $\delta (\xi _{{\mathcal {K}}}(x_{1},x_{5}))=0.5625$, $\delta (\xi _{{\mathcal {K}}}(x_{2},x_{3}))=0.6833$, $\delta (\xi _{{\mathcal {K}}}(x_{3},x_{5}))=0.5750$.

In the following, we will explore a detailed series of operational laws of directed HBVFGs.

Definition 13

Let ${\tilde{G}}_{1}=({\mathcal {A}}_{1}, {\mathcal {G}}_{1})$ and ${\tilde{G}}_{2}=({\mathcal {A}}_{2}, {\mathcal {G}}_{2})$ be directed HBVFGs of $G_{1}=(V_{1},E_{1})$ and $G_{2}=(V_{2},E_{2})$, respectively. Then, we define the following operations.

• (1) The Cartesian product ${\tilde{G}}_{1} \boxtimes {\tilde{G}}_{2} = ({\mathcal {A}}_{1} {\tilde{\times }} {\mathcal {A}}_{2}, {\mathcal {G}}_{1}$ ${\tilde{\times }}$ ${\mathcal {G}}_{2})$ of ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ is provided as follows:

  • (i) $(\xi _{{\mathcal {A}}_{1}} \times \xi _{{\mathcal {A}}_{2}})(x_{1},x_{2})= \xi _{{\mathcal {A}}_{1}}(x_{1}) {\tilde{\wedge }} \xi _{{\mathcal {A}}_{2}}(x_{2})$, for all $(x_{1},x_{2}) \in V_{1} \times V_{2}$,

  • (ii) $(\xi _{{\mathcal {G}}_{1}} \times \xi _{{\mathcal {G}}_{2}})((x_{1}, x_{2}), (x_{1}, x_{3})) = \xi _{{\mathcal {A}}_{1}}(x_{1}) {\tilde{\wedge }}$ $\xi _{{\mathcal {G}}_{2}}$ $(x_{2}$, $x_{3})$, for all $x_{1} \in V_{1}, (x_{2},x_{3}) \in E_{2}$,

  • (iii) $(\xi _{{\mathcal {G}}_{1}} \times \xi _{{\mathcal {G}}_{2}})((x_{1}, x_{2}), (x_{1}, x_{3})) = \xi _{{\mathcal {G}}_{1}}(x_{1}, x_{2})$ ${\tilde{\wedge }}$ $\xi _{{\mathcal {A}}_{2}} (x_{3})$, for all $(x_{1},x_{2}) \in E_{1}, x_{3} \in V_{2}$.

• (2) The direct product ${\tilde{G}}_{1} *{\tilde{G}}_{2}=({\mathcal {A}}_{1} *{\mathcal {A}}_{2}, {\mathcal {G}}_{1} *{\mathcal {G}}_{2})$ of ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ is provided as follows:

  • (i) $(\xi _{{\mathcal {A}}_{1}} *\xi _{{\mathcal {A}}_{2}}) (x_{1}, x_{2}) = \xi _{{\mathcal {A}}_{1}} (x_{1}) {\tilde{\wedge }} \xi _{{\mathcal {A}}_{2}} (x_{2})$, for all $(x_{1},x_{2}) \in V_{1} \times V_{2}$,

  • (ii) $(\xi _{{\mathcal {G}}_{1}} *\xi _{{\mathcal {G}}_{2}})((x_{1}, x_{2}), (x_{3}, x_{4})) = \xi _{{\mathcal {G}}_{1}} (x_{1}, x_{3})$ ${\tilde{\wedge }}$ $\xi _{{\mathcal {G}}_{2}}$ $(x_{2}$, $x_{4})$, for all $(x_{1},x_{3}) \in E_{1}, (x_{2},x_{4}) \in E_{2}$.

• (3) The lexicographical product ${\tilde{G}}_{1} \bullet {\tilde{G}}_{2}=({\mathcal {A}}_{1} \bullet {\mathcal {A}}_{2}, {\mathcal {G}}_{1} \bullet {\mathcal {G}}_{2})$ of ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ is provided as follows:

  • (i) $(\xi _{{\mathcal {A}}_{1}} \bullet \xi _{{\mathcal {A}}_{2}})(x_{1},x_{2})= \xi _{{\mathcal {A}}_{1}}(x_{1}) {\tilde{\wedge }} \xi _{{\mathcal {A}}_{2}}(x_{2})$, for all $(x_{1},x_{2}) \in V_{1} \times V_{2}$,

  • (ii) $(\xi _{{\mathcal {G}}_{1}} \bullet \xi _{{\mathcal {G}}_{2}})((x_{1}, x_{2}), (x_{1}, x_{3})) = \xi _{{\mathcal {A}}_{1}} (x_{1}) {\tilde{\wedge }}$ $\xi _{{\mathcal {G}}_{2}}$ $(x_{2}$, $x_{3})$, for all $x_{1} \in V_{1}, (x_{2}, x_{3}) \in E_{2}$,

  • (iii) $(\xi _{{\mathcal {G}}_{1}} \bullet \xi _{{\mathcal {G}}_{2}})((x_{1}, x_{2}), (x_{3}, x_{4})) = \xi _{{\mathcal {G}}_{1}}(x_{1}, x_{4})$ ${\tilde{\wedge }}$ $\xi _{{\mathcal {G}}_{2}}$ $(x_{2}$, $x_{4})$, for all $(x_{1}, x_{3}) \in E_{1}, (x_{2}, x_{3}) \in E_{1}$.

• (4) The strong product ${\tilde{G}}_{1} \otimes {\tilde{G}}_{2}=({\mathcal {A}}_{1} \otimes {\mathcal {A}}_{2}, {\mathcal {G}}_{1} \otimes {\mathcal {G}}_{2})$ of ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ is provided as follows:

  • (i) $(\xi _{{\mathcal {A}}_{1}} \otimes \xi _{{\mathcal {A}}_{2}})(x_{1}, x_{2}) = \xi _{{\mathcal {A}}_{1}} (x_{1}) {\tilde{\wedge }} \xi _{{\mathcal {A}}_{2}} (x_{2})$, for all $(x_{1}, x_{2}) \in V_{1} \times V_{2}$,

  • (ii) $(\xi _{{\mathcal {G}}_{1}} \otimes \xi _{{\mathcal {G}}_{2}})((x_{1}, x_{2}), (x_{1}, x_{3})) = \xi _{{\mathcal {A}}_{1}}(x_{1})$ ${\tilde{\wedge }}$ $\xi _{{\mathcal {G}}_{2}}$ $(x_{2}$, $x_{3})$, for all $x_{1} \in V_{1}, (x_{2}, x_{3}) \in E_{2}$,

  • (iii) $(\xi _{{\mathcal {G}}_{1}} \otimes \xi _{{\mathcal {G}}_{2}})((x_{1}, x_{3}), (x_{2}, x_{3})) = \xi _{{\mathcal {G}}_{1}} (x_{1}, x_{2}) {\tilde{\wedge }}$ $\xi _{{\mathcal {A}}_{2}}$ $(x_{3})$, for all $(x_{1}, x_{2}) \in E_{1}, x_{3} \in V_{2}$,

  • (iv) $(\xi _{{\mathcal {G}}_{1}} \otimes \xi _{{\mathcal {G}}_{2}})((x_{1}, x_{3}), (x_{2}, x_{4})) = \xi _{{\mathcal {G}}_{1}}(x_{1}, x_{3})$ ${\tilde{\wedge }}$ $\xi _{{\mathcal {G}}_{2}}$ $(x_{2}$, $x_{4})$, for all $(x_{1}, x_{3}) \in E_{1}, (x_{2}, x_{4}) \in E_{2}$.

Example 4

Let us assume two directed HBVFGs ${\tilde{G}}_{1}=({\mathcal {A}}_{1}, {\mathcal {G}}_{1})$ and ${\tilde{G}}_{2}=({\mathcal {A}}_{2}, {\mathcal {G}}_{2})$, shown in Fig. 3. Then the Cartesian product of two directed HBVFGs ${\tilde{G}}_{1}=({\mathcal {A}}_{1}, {\mathcal {G}}_{1})$ and ${\tilde{G}}_{2}=({\mathcal {A}}_{2}, {\mathcal {G}}_{2})$ is shown in Fig. 4.

Fig. 3

The directed HBVFGs ${\tilde{G}}_{1}=({\mathcal {A}}_{1}, {\mathcal {G}}_{1})$ and ${\tilde{G}}_{2}=({\mathcal {A}}_{2}, {\mathcal {G}}_{2})$

Fig. 4

The directed HBVFGs ${\tilde{G}}_{1} \boxtimes {\tilde{G}}_{2}$

Similarly, we can obtain the direct product, the lexicographical product, and the strong product of the two directed HBVFGs ${\tilde{G}}_{1}=({\mathcal {A}}_{1}, {\mathcal {G}}_{1})$ and ${\tilde{G}}_{2}=({\mathcal {A}}_{2}, {\mathcal {G}}_{2})$. It is a routine calculation. Hence, here we omitted.

Proposition 3

Let ${\tilde{G}}_{1}=({\mathcal {A}}_{1}, {\mathcal {G}}_{1})$ and ${\tilde{G}}_{2}=({\mathcal {A}}_{2}, {\mathcal {G}}_{2})$ be directed HBVFGs of $G_{1}=(V_{1},E_{1})$ and $G_{2}=(V_{2},E_{2})$, respectively. Then,

• (1) ${\tilde{G}}_{1} \boxtimes {\tilde{G}}_{2}=({\mathcal {A}}_{1} {\tilde{\times }} {\mathcal {A}}_{2}, {\mathcal {G}}_{1} {\tilde{\times }} {\mathcal {G}}_{2})$ is a directed HBVFG of $G_{1} \times G_{2}$.

• (2) ${\tilde{G}}_{1} *{\tilde{G}}_{2}=({\mathcal {A}}_{1} *{\mathcal {A}}_{2}, {\mathcal {G}}_{1} *{\mathcal {G}}_{2})$ is a directed HBVFG of $G_{1} \times G_{2}$.

• (3) ${\tilde{G}}_{1} \bullet {\tilde{G}}_{2}=({\mathcal {A}}_{1} \bullet {\mathcal {A}}_{2}, {\mathcal {G}}_{1} \bullet {\mathcal {G}}_{2})$ is a directed HBVFG of $G_{1} \times G_{2}$.

• (4) ${\tilde{G}}_{1} \otimes {\tilde{G}}_{2}=({\mathcal {A}}_{1} \otimes {\mathcal {A}}_{2}, {\mathcal {G}}_{1} \otimes {\mathcal {G}}_{2})$ is a directed HBVFG of $G_{1} \times G_{2}$.

Proof

(1) For all $(x_{1}, x_{2}) \in V_{1} \times V_{2}$ we have

$$\begin{aligned}&\delta ((\xi _{{\mathcal {A}}_{1}} \times \xi _{{\mathcal {A}}_{2}})(x_{1},x_{2}))\nonumber \\&=\delta ( \xi _{{\mathcal {A}}_{1}}(x_{1}) {\tilde{\wedge }} \xi _{{\mathcal {A}}_{2}}(x_{2}))\nonumber \\&= min (\delta (\xi _{{\mathcal {A}}_{1}}(x_{1})), \delta (\xi _{{\mathcal {A}}_{2}}(x_{2})))~~\nonumber \\&~~~~~[\text {By Proposition 2}]. \end{aligned}$$

(5)

Let $x_{1} \in V_{1}$ and $(x_{2},x_{3}) \in E_{2}$, then we have

$$\begin{aligned}&\delta ((\xi _{{\mathcal {G}}_{1}} \times \xi _{{\mathcal {G}}_{2}})((x_{1},x_{2}),(x_{1},x_{3})))\nonumber \\&=\delta (\xi _{{\mathcal {A}}_{1}}(x_{1}) {\tilde{\wedge }} \xi _{{\mathcal {G}}_{2}}(x_{2},x_{3}))\nonumber \\&= min \{ \delta (\xi _{{\mathcal {A}}_{1}}(x_{1})), \delta (\xi _{{\mathcal {G}}_{2}}(x_{2},x_{3}))\} ~~[\text {By Proposition }] \nonumber \\&\le min \{ \delta (\xi _{{\mathcal {A}}_{1}}(x_{1})), min \{ \delta (\xi _{{\mathcal {A}}_{2}}(x_{2})), \delta (\xi _{{\mathcal {A}}_{2}}(x_{3}))\}\} \nonumber \\&= min \{ min \{ \delta (\xi _{{\mathcal {A}}_{1}}(x_{1})), \delta (\xi _{{\mathcal {A}}_{2}}(x_{2}))\},\nonumber \\&min \{ \delta (\xi _{{\mathcal {A}}_{1}}(x_{1})), \delta (\xi _{{\mathcal {A}}_{2}}(x_{3}))\}\} \nonumber \\&= min \{ \delta (\xi _{{\mathcal {A}}_{1}}(x_{1}) {\tilde{\wedge }} \xi _{{\mathcal {A}}_{2}}(x_{2})), \delta (\xi _{{\mathcal {A}}_{1}}(x_{1}) {\tilde{\wedge }} \xi _{{\mathcal {A}}_{2}}(x_{3}))\}\nonumber \\&~~~~~~~~~~~~~~~[\text {By Proposition }] \nonumber \\&=min \{\delta ((\xi _{{\mathcal {A}}_{1}} \times \xi _{{\mathcal {A}}_{2}})(x_{1},x_{2}),\nonumber \\&\delta ((\xi _{{\mathcal {A}}_{1}} \times \xi _{{\mathcal {A}}_{2}})(x_{1},x_{3}))\}. \end{aligned}$$

(6)

Let $(x_{1},x_{2}) \in E_{1}$ and $x_{3} \in V_{2}$, then we have

$$\begin{aligned}&\delta ((\xi _{{\mathcal {G}}_{1}} \times \xi _{{\mathcal {G}}_{2}})((x_{1},x_{2}),(x_{1},x_{3})))\nonumber \\&=\delta (\xi _{{\mathcal {G}}_{1}}(x_{1},x_{2}) {\tilde{\wedge }} \xi _{{\mathcal {A}}_{2}}(x_{3}))\nonumber \\&= min \{ \delta (\xi _{{\mathcal {G}}_{1}}(x_{1},x_{2})), \delta (\xi _{{\mathcal {A}}_{2}}(x_{3}))\}\nonumber \\&~~~~~~~~~~~[\text {By Proposition }] \nonumber \\&\le min \{min \{ \delta (\xi _{{\mathcal {A}}_{1}}(x_{1})), \delta (\xi _{{\mathcal {A}}_{1}}(x_{2}))\},\nonumber \\&\delta (\xi _{{\mathcal {A}}_{2}}(x_{3}))\} \nonumber \\&= min \{ min \{ \delta (\xi _{{\mathcal {A}}_{1}}(x_{1})), \delta (\xi _{{\mathcal {A}}_{2}}(x_{3}))\},\nonumber \\&min \{ \delta (\xi _{{\mathcal {A}}_{1}}(x_{2})), \delta (\xi _{{\mathcal {A}}_{2}}(x_{3}))\}\} \nonumber \\&= min \{ \delta (\xi _{{\mathcal {A}}_{1}}(x_{1}) {\tilde{\wedge }} \xi _{{\mathcal {A}}_{2}}(x_{3})),\nonumber \\&\delta (\xi _{{\mathcal {A}}_{1}}(x_{2}) {\tilde{\wedge }} \xi _{{\mathcal {A}}_{2}}(x_{3}))\} ~~[\text {By Proposition }] \nonumber \\&=min \{\delta ((\xi _{{\mathcal {A}}_{1}} \times \xi _{{\mathcal {A}}_{2}})(x_{1},x_{2}),\nonumber \\&\delta ((\xi _{{\mathcal {A}}_{1}} \times \xi _{{\mathcal {A}}_{2}})(x_{1},x_{3}))\}. \end{aligned}$$

(7)

Hence, the result (1) is true from equations (5)–(7). The remaining part can be proved in a similar way.

Moreover, the following are discussed with some mapping relationships of the directed HBVFGs.

Definition 14

Let ${\tilde{G}}_{1}=({\mathcal {A}}_{1}, {\mathcal {G}}_{1})$ and ${\tilde{G}}_{2}=({\mathcal {A}}_{2}, {\mathcal {G}}_{2})$ be directed HBVFGs of $G_{1}=(V_{1},E_{1})$ and $G_{2}=(V_{2},E_{2})$, respectively. Then, we define the following mapping relationships of directed HBVFGs.

• 1. The bijective mapping $f: V_{1} \rightarrow V_{2}$ is called a homomorphism between ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$, if f satisfies the following properties:

  • (a) $\delta (\xi _{{\mathcal {A}}_{1}}(x_{1})) \le \delta (\xi _{{\mathcal {A}}_{2}}(f(x_{1})))$ for all $x_{1} \in V_{1}$,

  • (b) $\delta (\xi _{{\mathcal {G}}_{1}}(x_{1}, x_{2})) \le \delta (\xi _{{\mathcal {G}}_{2}}(f(x_{1}),f(x_{2})))$ for all $x_{1}, x_{2} \in V_{1}$.

• 2. The bijective mapping $f: V_{1} \rightarrow V_{2}$ is called an isomorphism between ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$, if f satisfies the following properties:

  • (a) $\delta (\xi _{{\mathcal {A}}_{1}}(x_{1})) = \delta (\xi _{{\mathcal {A}}_{2}}(f(x_{1})))$ for all $x_{1} \in V_{1}$,

  • (b) $\delta (\xi _{{\mathcal {G}}_{1}}(x_{1}, x_{2})) = \delta (\xi _{{\mathcal {G}}_{2}}(f(x_{1}),f(x_{2})))$ for all $x_{1}, x_{2} \in V_{1}$.

  In this case, we write ${\tilde{G}}_{1} \cong {\tilde{G}}_{2}$.

• 3. The bijective mapping $f: V_{1} \rightarrow V_{2}$ is called a weak isomorphism between ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$, if f satisfies the following properties:

  • (a) f is a homomorphism,

  • (b) $\delta (\xi _{{\mathcal {A}}_{1}}(x_{1})) = \delta (\xi _{{\mathcal {A}}_{2}}(f(x_{1})))$ for all $x_{1} \in V_{1}$.

• 4. The bijective mapping $f: V_{1} \rightarrow V_{2}$ is called a co-weak isomorphism between ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$, if f satisfies the following properties:

  • (a) f is a homomorphism,

  • (b) $\delta (\xi _{{\mathcal {G}}_{1}}(x_{1},x_{2})) = \delta (\xi _{{\mathcal {G}}_{2}}(f(x_{1}),f(x_{2})))$ for all $x_{1},x_{2} \in V_{1}$.

In the justification of Definition 14, we have the following example.

Example 5

Let $V_{1}=\{x_{1},x_{2},x_{3}\}$ and $V_{2}=\{y_{1}, y_{2},y_{3}\}$ be the set of nodes and a bijective function $f: V_{1} \rightarrow V_{2}$ is defined as $f(x_{1})=y_{1}$, $f(x_{2})=y_{2}$ and $f(x_{3})=y_{3}$. Let ${\tilde{G}}_{1}=({\mathcal {A}}_{1}, {\mathcal {G}}_{1})$ and ${\tilde{G}}_{2}=({\mathcal {A}}_{2}, {\mathcal {G}}_{2})$ be directed HBVFGs of $G_{1}=(V_{1},E_{1})$ and $G_{2}=(V_{2},E_{2})$, respectively, where,

$$\begin{aligned} {\mathcal {A}}_{1}&=\{\langle x_{1}, (\{0.8,0.4\},\{-0.7,-0.6,-0.3\})\rangle ,\\&\langle x_{2}, (\{0.8,0.7,0.6\},\{-0.9,-0.5\})\rangle ,\\&\langle x_{3}, (\{0.9,0.6,0.5,0.4\},\{-0.8,-0.6,-0.4\})\rangle \},\\ {\mathcal {G}}_{1}&=\{\langle (x_{1}, x_{2}), (\{0.6,0.4\},\{-0.7,-0.6,-0.5\})\rangle ,\\&\langle (x_{1}, x_{3}), (\{0.8,0.6,0.4\},\{-0.8,-0.6,-0.3\})\rangle \},\\ {\mathcal {A}}_{2}&=\{\langle y_{1}, (\{0.8,0.6,0.2\},\{-0.7,-0.5\})\rangle ,\\&\langle y_{2}, (\{0.9,0.7\},\{-0.8,-0.6,-0.4\})\rangle ,\\&\langle y_{3}, (\{0.7,0.5,0.3\},\{-0.9,-0.8,-0.4\})\rangle \}, \\ {\mathcal {G}}_{2}&=\{\langle (y_{1}, y_{2}), (\{0.7,0.6,0.5\},\{-0.6,-0.4\})\rangle ,\\&\langle (x_{1}, x_{3}), (\{0.8,0.4\},\{-0.7,-0.5,-0.2\})\rangle \}. \end{aligned}$$

Here, since $\delta (\xi _{{\mathcal {A}}_{1}}(x_{1})) = \delta (\xi _{{\mathcal {A}}_{2}}(y_{1}))$, $\delta (\xi _{{\mathcal {A}}_{1}}(x_{2})) = \delta (\xi _{{\mathcal {A}}_{2}}(y_{2}))$, $\delta (\xi _{{\mathcal {A}}_{1}}(x_{3})) = \delta (\xi _{{\mathcal {A}}_{2}}(y_{3}))$, $\delta (\xi _{{\mathcal {G}}_{1}}(x_{1},x_{2}))$ $=$ $\delta (\xi _{{\mathcal {G}}_{2}}$ $(y_{1}$, $y_{2}))$ and $\delta (\xi _{{\mathcal {G}}_{1}}(x_{1},x_{3})) = \delta (\xi _{{\mathcal {G}}_{2}}(y_{1},y_{3}))$. Therefore, f is isomorphism between ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$.

Definition 15

Let ${\tilde{G}}=({\mathcal {A}}, {\mathcal {G}})$ be a directed HBVFG of $G=(V,E)$. Then, a sequence of some distinct nodes $x_{i} \in V$ for $i \in \{1,2,\cdots , n\}$ is called a directed path of ${\tilde{G}}$ if $\delta (\xi _{{\mathcal {G}}}(x_{i},x_{j})) > 0$ for some $i,j \in \{1,2,\cdots , n\}$.

Definition 16

Let $a=x_{1}, x_{2}, \cdots , x_{n+1}=b$ $(n > 0)$ be a directed path in a directed HBVFG ${\tilde{G}}=({\mathcal {A}}, {\mathcal {G}})$ of $G=(V,E)$. Then, its length is n. This path is called a cycle, if $x_{1}=x_{n+1}$ for $(n \ge 3)$. Moreover, the degree of a directed path from the starting node a to the destination node b is defined in the following way: for $i,j \in \{1,2,\cdots , n\}$

$$\begin{aligned} {\mathcal {D}}_{{\tilde{G}}}(a,b)=\sum _{x_{i},x_{j} \in V, i \ne j}&\delta (\xi _{{\mathcal {G}}}(x_{i},x_{j})) \end{aligned}$$

(8)

In justifying Definitions 15 and 16, we have the following example.

Example 6

Let $V=\{x_{1},x_{2},x_{3},x_{4}\}$ and E be the set of edges illustrated by Fig. 5. Here $a=x_{1}$, $x_{2}$, $x_{3}$, $x_{4}=b$ is a directed path and is length 3. Moreover, $x_{1}$, $x_{2}$, $x_{3}$, $x_{4}$, $x_{1}$ form a directed cycle. The degree of directed path from the starting node a to the destination node b is ${\mathcal {D}}_{{\tilde{G}}}(a,b)= \delta (\xi _{{\mathcal {G}}}(x_{1},x_{2}))$ $+$ $\delta (\xi _{{\mathcal {G}}}(x_{2},x_{3}))$ $+$ $\delta (\xi _{{\mathcal {G}}}(x_{3},x_{4})) =0.5167+0.4625+0.5583 =1.5375$.

Fig. 5

The directed HBVFG

Application of directed HBVFGs

Finding best route in a network

Finding the best route is the most common problem in graph theory. Every fuzzy structure [19,20,21,22,23,24] has been rigorously studied with a relatively simple algorithm, and because we achieve the best-predicted results, as in [19] in those times. We discussed in this section finding the best path with a numerical example for directed HBVFGs.

Best route finding Algorithm

The algorithm introduced by Dijkstra is the large-handed, commonly used method for computing a network’s best path. Few modifications have been made to the Dijkstra algorithm from time to time, and multiple formulas are used to obtain optimal performance. These methods contain distance measurements, similarity and aggregation operators, etc. In this respect, some valuable work has been done [20, 23, 25]. Let ${\tilde{G}}=({\mathcal {A}}, {\mathcal {G}})$ be a directed hesitant bipolar-valued fuzzy network of $G=(V,E)$. Let $V=\{x_{1}, x_{2}, \cdots , x_{n}\}$ be the set of nodes. We find the best path of ${\tilde{G}}$, where the directed edge, i.e., the directed path between two nodes, is provided in the form of HBVFEs. The steps of the algorithm are discussed in the following, where the starting node is $x_{1}$ and the destination node is $x_{n}$.

• (1) First, we assume that the path between every node and itself is zero, i.e., $\xi _{{\mathcal {G}}}(x_{i}, x_{i})=(\{0\}, \{0\})$ for all $i \in \{1, 2, \cdots , n\}$.

• (2) Set $i=1$.

• (3) Find j for $\xi _{{\mathcal {G}}}(x_{1}, x_{j}) = \xi _{{\mathcal {G}}} (x_{1}, x_{1}) \oplus {\tilde{\wedge }}_{k \in {\mathcal {N}}(1)}$ $\xi _{{\mathcal {G}}}$ $(x_{1}$, $x_{k})$, where $\oplus$ and ${\tilde{\wedge }}$ obtained by Eqs. (2) and (3), and ${\mathcal {N}}(i)$ indicate the set of all nodes having relation with outgoing edges from i.

• (4) Put $i=j$.

• (5) Obtain k for $\xi _{{\mathcal {G}}}(x_{j}, x_{k}) = \xi _{{\mathcal {G}}} (x_{1}, x_{j}) \oplus {\tilde{\wedge }}_{h \in {\mathcal {N}}(j)}$ $\xi _{{\mathcal {G}}}$ $(x_{j}$, $x_{h})$ and determine $\xi _{{\mathcal {G}}}(x_{j}, x_{k})$.

• (6) It is necessary to continue this process until the goal node is reached.

• (7) The algorithm will be stopped if the goal node is reached.

For validation of our proposed algorithm, we have the following numerical example.

Example 7

Let ${\tilde{G}}=({\mathcal {A}}, {\mathcal {G}})$ be the directed hesitant bipolar-valued fuzzy network of $G=(V,E)$ display in Fig. 6, where $V=\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6}\}$ and the HBVFE form is given for the guided path of each of the two nodes. The algorithm is continued accordingly.

Fig. 6

The directed hesitant bipolar-valued fuzzy network ${\tilde{G}}=({\mathcal {A}}, {\mathcal {G}})$ of $G=(V,E)$

In the graph shown in Fig. 6, the source node is $x_{1} (=a, \text {say})$ and the destination node is $x_{6} (=b, \text {say})$. Thus, for $n=6$ and the first step $\xi _{{\mathcal {G}}}(x_{1},x_{1})=(\{0\}, \{0\})$. Assuming $i = 1$ is needed to obtained j using the Eq. (9).

$$\begin{aligned} \xi _{{\mathcal {G}}}(x_{1},x_{j})&=\xi _{{\mathcal {G}}}(x_{1},x_{1}) \oplus {\tilde{\wedge }}_{k \in {\mathcal {N}}(1)} \xi _{{\mathcal {G}}}(x_{1},x_{k}) \nonumber \\&=\xi _{{\mathcal {G}}}(x_{1},x_{1}) \oplus {\tilde{\wedge }}_{k \in \{2,3\}} \xi _{{\mathcal {G}}}(x_{1},x_{k}) \nonumber \\&=\xi _{{\mathcal {G}}}(x_{1},x_{1})\oplus (\xi _{{\mathcal {G}}}(x_{1},x_{2}) {\tilde{\wedge }} \xi _{{\mathcal {G}}}(x_{1},x_{3})). \end{aligned}$$

(9)

Since, $\delta (\xi _{{\mathcal {G}}}(x_{1},x_{2})=0.5167 > 0.5=(\xi _{{\mathcal {G}}}(x_{1},x_{3})$. Therefore, $\xi _{{\mathcal {G}}}(x_{1},x_{j})=\xi _{{\mathcal {G}}}(x_{1},x_{1})\oplus (\xi _{{\mathcal {G}}}(x_{1},x_{3}) =(\{0\}, \{0\}) \oplus (\{0.8,0.3,0.2\},\{-0.8,-0.5,-0.4\})$. Using Eq. (2), we have $\xi _{{\mathcal {G}}}(x_{1},x_{j})=(\{0.8,0.3,0.2\}$, $\{-0.8,-0.5,-0.4\})$. This implies that $j=3$ and $\xi _{{\mathcal {G}}}(x_{1},x_{3})=(\{0.8,0.3,0.2\},\{-0.8,-0.5,-0.4\})$. For the next step $j=3$, to find k, we have the Eq. (10).

$$\begin{aligned} \xi _{{\mathcal {G}}}(x_{3},x_{k})&=\xi _{{\mathcal {G}}}(x_{1},x_{3}) \oplus {\tilde{\wedge }}_{h \in {\mathcal {N}}(3)} \xi _{{\mathcal {G}}}(x_{3},x_{h}) \nonumber \\&=\xi _{{\mathcal {G}}}(x_{1},x_{3}) \oplus {\tilde{\wedge }}_{k \in \{4,5\}} \xi _{{\mathcal {G}}}(x_{3},x_{h}) \nonumber \\&=\xi _{{\mathcal {G}}}(x_{1},x_{3})\oplus (\xi _{{\mathcal {G}}}(x_{3},x_{4}) {\tilde{\wedge }} \xi _{{\mathcal {G}}}(x_{3},x_{5})). \end{aligned}$$

(10)

In the similar way we obtain $\xi _{{\mathcal {G}}}(x_{3},x_{k})=(\{0.94$, 0.9, 0.84, 0.79, 0.65, 0.44, 0.76, 0.6, $0.36\}$, $\{-0.88$, $-0.7$, $-0.64$, $-0.86$, $-0.65$, $-0.58\})$. This implies that $k=5$ and $\xi _{{\mathcal {G}}}(x_{1},x_{5})=(\{0.94$, 0.9, 0.84, 0.79, 0.65, 0.44, 0.76, 0.6, $0.36\}$,$\{-0.88$, $-0.7$, $-0.64$, $-0.86$, $-0.65$, $-0.58\})$.

Again, for $j = 5$, we have the Eq. (11) for finding k.

$$\begin{aligned} \xi _{{\mathcal {G}}}(x_{5},x_{k})&=\xi _{{\mathcal {G}}}(x_{1},x_{5}) \oplus {\tilde{\wedge }}_{h \in {\mathcal {N}}(5)} \xi _{{\mathcal {G}}}(x_{5},x_{h}) \nonumber \\&=\xi _{{\mathcal {G}}}(x_{1},x_{5}) \oplus {\tilde{\wedge }}_{k \in \{6\}} \xi _{{\mathcal {G}}}(x_{5},x_{h})\nonumber \\&=\xi _{{\mathcal {G}}}(x_{1},x_{3})\oplus \xi _{{\mathcal {G}}}(x_{5},x_{6}). \end{aligned}$$

(11)

In the similar way we obtain $\xi _{{\mathcal {G}}}(x_{5},x_{k})=(\{0.994$, 0.97, 0.964, 0.99, 0.95, 0.94, 0.984, 0.92, 0.904, 0.979, 0.895, 0.874, 0.965, 0.825, 0.79, 0.944, 0.72, 0.664, 0.976, 0.88, 0.856, 0.96, 0.8, 0.76, 0.936, 0.68, $0.616\}$, $\{-0.976$, $-0.964$, $-0.952$, $-0.94$, $-0.91$, $-0.88$, $-0.928$, $-0.892$, $-0.856$, $-0.972$, $-0.958$, $-0.944$, $-0.93$, $-0.895$, $-0.86$, $-0.916$, $-0.874$, $-0.832\})$. This implies that $k=6$. So the best directed path in Fig. 6 from source node a to destination node b is $x_{1} \rightarrow x_{3} \rightarrow x_{5} \rightarrow x_{6}$. From Eq. (8), the degree of this path is 1.5125.

Finding dominant node in a social network

To date, it is well known that a number of people interact with each other using social network platforms. Therefore, naturally, they impact each other more or less. In this section, we can investigate a person’s influence on others with respect to his/her positive and negative thinking.

Using Eq. (2) and the aggregation operators defined in [18], we have the following aggregation operators for the collection of HBVFEs $\xi _{j}=(\xi _{j}^{P}, \xi _{j}^{N})$ $(j \in 1,2, \cdots , n)$ are:

$$\begin{aligned}&\xi _{1} \oplus \xi _{2} \oplus \cdots \oplus \xi _{n} = \oplus _{j = 1}^{n} \xi _{i} \nonumber \\&=\left( \bigcup _{\gamma _{1}^{P} \in \xi _{1}^{P}, \gamma _{2}^{P} \in \xi _{2}^{P}, \cdots , \gamma _{n}^{P} \in \xi _{n}^{P}} \left\{ 1 - \prod _{j=1}^{n} (1-\gamma _{j}^{P}) \right\} ,\right. \nonumber \\&~~~~~\left. \bigcup _{\gamma _{1}^{N} \in \xi _{1}^{N}, \gamma _{2}^{N} \in \xi _{2}^{N}, \cdots , \gamma _{n}^{N} \in \xi _{n}^{N}} \left\{ -1 + \prod _{j=1}^{n} (1+\gamma _{j}^{N}) \right\} \right) . \end{aligned}$$

(12)

Using Eq. (12), we have the following concepts of the indegree and outdegree of a node.

Definition 17

Let ${\tilde{G}}=({\mathcal {A}}, {\mathcal {G}})$ be the directed HBVFGs of $G=(V,E)$ and $x_{r}$ $(r \in 1,2, \cdots , n)$ be adjacent hesitant bipolar-valued fuzzy nodes of $x_{k} \in V$ $(k \in 1,2, \cdots , n)$. The indegree and outdegree of $x_{k}$ are respectively denoted $\mathcal{I}\mathcal{D}(x_{k})$ and $\mathcal{O}\mathcal{D}(x_{k})$. The indegree is defined as the algebraic sum of the incoming edges of $x_{k}$, which as given in Eq. (13), where $\mathcal{I}\mathcal{N}(x_{k})$ denote the set of all nodes having relation with incoming edges to $x_{k}$.

$$\begin{aligned} \mathcal{I}\mathcal{D}(x_{k})&=\left( \bigcup _{\gamma _{i}^{P} \in \xi _{{\mathcal {G}}}(x_{r},x_{k}), x_{r} \in \mathcal{I}\mathcal{N}(x_{k}), i=1,2,\cdots ,|\mathcal{I}\mathcal{N}(x_{k})|} \left\{ 1 - \prod _{j=1}^{|\mathcal{I}\mathcal{N}(x_{k})|}(1-\gamma _{j}^{P})\right\} , \right. \nonumber \\&\quad \left. \bigcup _{\gamma _{i}^{N} \in \xi _{{\mathcal {G}}}(x_{r},x_{k}), x_{r} \in \mathcal{I}\mathcal{N}(x_{k}), i=1,2,\cdots ,|\mathcal{I}\mathcal{N}(x_{k})|}\left\{ -1 + \prod _{j=1}^{|\mathcal{I}\mathcal{N}(x_{k})|}(1+\gamma _{j}^{N})\right\} \right) . \end{aligned}$$

(13)

Similarly, the outdegree is defined as the algebraic sum of the outgoing edges of $x_{k}$, which as given in Eq. (14), where $\mathcal{O}\mathcal{N}(x_{k})$ denote the set of all nodes having relation with outgoing edges from $x_{k}$.

$$\begin{aligned} \mathcal{O}\mathcal{D}(x_{k})&=\left( \bigcup _{\gamma _{i}^{P} \in \xi _{{\mathcal {G}}}(x_{k},x_{r}), x_{r} \in \mathcal{O}\mathcal{N}(x_{k}), i=1,2,\cdots ,|\mathcal{O}\mathcal{N}(x_{k})|} \left\{ 1 - \prod _{j=1}^{|\mathcal{O}\mathcal{N}(x_{k})|}(1-\gamma _{j}^{P})\right\} ,\right. \nonumber \\&\quad \left. \bigcup _{\gamma _{i}^{N} \in \xi _{{\mathcal {G}}}(x_{k},x_{r}), x_{r} \in \mathcal{O}\mathcal{N}(x_{k}), i=1,2,\cdots ,|\mathcal{O}\mathcal{N}(x_{k})|} \left\{ -1 + \prod _{j=1}^{|\mathcal{O}\mathcal{N}(x_{k})|}(1+\gamma _{j}^{N})\right\} \right) . \end{aligned}$$

(14)

Based on Definition 17, we have the following definition for obtaining the dominant degree of a node.

Definition 18

Let ${\tilde{G}}=({\mathcal {A}}, {\mathcal {G}})$ be the directed HBVFGs of $G=(V,E)$ and $x_{r}$ $(r \in 1,2, \cdots , n)$ be adjacent hesitant bipolar-valued fuzzy nodes of $x_{k} \in V$ $(k \in 1,2, \cdots , n)$. Then the dominant degree of a node $x_{k}$ is denoted by $\mathcal{D}\mathcal{G}(x_{k})=\delta (\mathcal{O}\mathcal{D}(x_{k})) - \delta (\mathcal{I}\mathcal{D}(x_{k}))$.

Ding et al. [29] pointed out that in real life, each person would have a certain degree of self-confidence regarding their own view and would simultaneously be influenced by others linked to them in a social network. In this connection, Zhou et al. [30] introduced the concepts of the self-persistence degree of a person for their viewing. Based on this concept in the following, we define the influence index of a node.

Definition 19

The influence index of a node $x_{k}$ $(k=1,2, \cdots , n)$ is defined as $\mathcal {INF}(x_{k})=(\alpha _{k} + \delta (\mathcal{I}\mathcal{D}(x_{k}) + \delta (\mathcal{O}\mathcal{D}(x_{k}))/3$, where $\alpha _{k} \in [0,1]$ $(k \in 1,2, \cdots , n)$ be the self-persistence degree of $x_{k}$.

Table 1 The HBVFS ${\mathcal {A}}$ on V

Algorithm for finding the dominant node with influence index

Let ${\tilde{G}}=({\mathcal {A}}, {\mathcal {G}})$ be a directed hesitant bipolar-valued fuzzy network of $G=(V,E)$. Let $V=\{x_{1}, x_{2}, \cdots , x_{n}\}$ be the set of nodes. We find the dominant node with influence degree, where nodes are provided in the form of HBVFEs. In Algorithm 1, we discussed the proposed technique.

Algorithm 1 Finding the dominant node with influence index in a network.

Input: The uncertainty information of each node according to HBVFEs with the self-persistence $\alpha _{k} \in [0,1]$ $(k \in \{1,2, \cdots , n))$.

Output: The dominant node and influence index of each node.

• 1. Obtain the HBVFEs, i.e., membership degree of each connecting edge maintaining relation (15), where $i \ne j~~ i,j \in \{1,2, \cdots , n\}$.

  $$\begin{aligned}&\delta (\xi _{{\mathcal {G}}}(x_{i}, x_{j})) \le min(\delta (\xi _{{\mathcal {A}}}(x_{i})), \delta (\xi _{{\mathcal {A}}}(x_{j}))). \end{aligned}$$

  (15)

• 2. Obtain indegree and outdegree of every node by equations (13) and (14).

• 3. Compute the dominant degree of every node using Definition 18.

• 4. Rank the nodes according to a maximum dominant degree.

• 5. Compute the influence index of every node using Definition 19.

Remark 1

It is noted that in Step 2 of Algorithm 1, the membership degree of each connecting edge is not unique. It mainly depends on the nature of the network.

For validation of our proposed algorithm, we have the following numerical example.

For validation of our proposed algorithm, we have the following numerical example.

Example 8

We adopt an application discussed in [14, 31, 32]. There are seven persons who influence each other in a social group on Whatsapp. Let $V=\{x_{1}$, $x_{2}$, $x_{3}$, $x_{4}$, $x_{5}$, $x_{6}$, $x_{7}\}$ be the set of seven people in a social group on Whatsapp. Let ${\mathcal {A}}$ be the HBVFS on V, which given in Table 1. The connection among the seven persons is shown in Fig. 7.

Fig. 7

The directed hesitant bipolar-valued fuzzy network of seven persons in V

From Fig. 7, the relations among the seven persons is a set of edges. The set of edges denoted by E and $E=\{(x_{1},x_{3})$, $(x_{1},x_{6})$, $(x_{1},x_{7})$, $(x_{2},x_{1})$, $(x_{2},x_{3})$, $(x_{3},x_{4})$, $(x_{3},x_{7})$, $(x_{4},x_{7})$, $(x_{5},x_{2})$, $(x_{5},x_{4})$, $(x_{6},x_{2})$, $(x_{6},x_{3})$, $(x_{6},x_{4})$, $(x_{6},x_{7})$, $(x_{7},x_{5})\}$.

Let ${\mathcal {G}}$ be the HBVFS on E obtained from Eq. (15), which is shown in Table 2.

Table 2 The HBVFS ${\mathcal {G}}$ on E

Using Eqs. (12) and (13) to obtain the indegree and outdegree of all nodes $x_{k}$ $(k=1,2,\cdots ,7)$. For example take a node $x_{2}$, we have

$$\begin{aligned}&\mathcal{I}\mathcal{D}(x_{2}) \\&=\left( \bigcup _{\gamma _{i}^{P} \in \xi _{{\mathcal {G}}}(x_{r},x_{2}), x_{r} \in \mathcal{I}\mathcal{N}(2), i=1,2,\cdots ,|\mathcal{I}\mathcal{N}(2)|} \left\{ 1 - \prod _{j=1}^{|\mathcal{I}\mathcal{N}(2)|}(1-\gamma _{j}^{P})\right\} , \right. \\&\left. \bigcup _{\gamma _{i}^{N} \in \xi _{{\mathcal {G}}}(x_{r},x_{2}), x_{r} \in \mathcal{I}\mathcal{N}(2), i=1,2,\cdots ,|\mathcal{I}\mathcal{N}(2)|} \left\{ -1 + \prod _{j=1}^{|\mathcal{I}\mathcal{N}(2)|}(1+\gamma _{j}^{N})\right\} \right) \\&=\left( \bigcup _{\gamma _{i}^{P} \in \xi _{{\mathcal {G}}}(x_{r},x_{2}), x_{r} \in \{x_{1},x_{3}\}, i=1,2} \left\{ 1 - \prod _{j=1}^{2}(1-\gamma _{j}^{P})\right\} , \right. \\&\left. \bigcup _{\gamma _{i}^{N} \in \xi _{{\mathcal {G}}}(x_{r},x_{2}), x_{r} \in \{x_{5},x_{6}\}, i=1,2} \left\{ -1 + \prod _{j=1}^{2}(1+\gamma _{j}^{N})\right\} \right) \\&=\left( \bigcup _{\gamma _{1}^{P} \in \xi _{{\mathcal {G}}}(x_{5},x_{2}), \gamma _{2}^{P} \in \xi _{{\mathcal {G}}}(x_{6},x_{2})} \left\{ 1 - \prod _{j=1}^{2}(1-\gamma _{j}^{P})\right\} , \right. \\&\left. \bigcup _{\gamma _{1}^{N} \in \xi _{{\mathcal {G}}}(x_{5},x_{2}),\gamma _{2}^{N} \in \xi _{{\mathcal {G}}}(x_{6},x_{2})} \left\{ -1 + \prod _{j=1}^{2}(1+\gamma _{j}^{N})\right\} \right) \\&=\left( \bigcup _{\gamma _{1}^{P} \in \{0.6,0.5\},\gamma _{2}^{P} \in \{0.9,0.8,0.7\}} \left\{ 1 - \prod _{j=1}^{2}(1-\gamma _{j}^{P})\right\} , \right. \\&\left. \bigcup _{\gamma _{1}^{N} \in \{-0.8,-0.4\},\gamma _{2}^{N} \in \{-0.8,-0.7,-0.2\}} \left\{ -1 + \prod _{j=1}^{2}(1+\gamma _{j}^{N})\right\} \right) \\&=\bigg (\{1-(1-0.6) \times (1-0.9), 1-(1-0.6) \times (1-0.8), \\&1-(1-0.6) \times (1-0.7), 1-(1-0.5) \times (1-0.9), \\&1-(1-0.5) \times (1-0.5), 1-(1-0.5) \times (1-0.7)\}, \\&\{-1+(1-0.8) \times (1-0.8), -1+(1-0.8) \times (1-0.7), \\&-1+(1-0.8) \times (1-0.2), -1+(1-0.4) \times (1-0.8), \\&-1+(1-0.4) \times (1-0.7), -1+(1-0.4) \times (1-0.2)\} \bigg ) \\&=\bigg (\{0.96,0.92,0.88,0.95,0.9,0.85\}, \\&\{-0.96,-0.94,-0.84,-0.88,-0.82,-0.52\} \bigg ) \\&\mathcal{O}\mathcal{D}(x_{2}) \\&=\left( \bigcup _{\gamma _{i}^{P} \in \xi _{{\mathcal {G}}}(x_{2},x_{r}), x_{r} \in \mathcal{O}\mathcal{N}(2), i=1,2,\cdots ,|\mathcal{I}\mathcal{N}(2)|} \right. \\&\left. \left\{ 1 - \prod _{j=1}^{|\mathcal{O}\mathcal{N}(2)|}(1-\gamma _{j}^{P})\right\} , \right. \\&\left. \bigcup _{\gamma _{i}^{N} \in \xi _{{\mathcal {G}}}(x_{2},x_{r}), x_{r} \in \mathcal{O}\mathcal{N}(2), i=1,2,\cdots ,|\mathcal{I}\mathcal{N}(2)|} \right. \\&\left. \left\{ -1 + \prod _{j=1}^{|\mathcal{O}\mathcal{N}(2)|}(1+\gamma _{j}^{N})\right\} \right) \\&=\left( \bigcup _{\gamma _{i}^{P} \in \xi _{{\mathcal {G}}}(x_{2},x_{r}), x_{r} \in \{x_{1},x_{3}\}, i=1,2} \left\{ 1 - \prod _{j=1}^{2}(1-\gamma _{j}^{P})\right\} , \right. \\&\left. \bigcup _{\gamma _{i}^{N} \in \xi _{{\mathcal {G}}}(x_{2},x_{r}), x_{r} \in \{x_{1},x_{3}\},i=1,2} \left\{ -1 + \prod _{j=1}^{2}(1+\gamma _{j}^{N})\right\} \right) \\&=\left( \bigcup _{\gamma _{1}^{P} \in \xi _{{\mathcal {G}}}(x_{2},x_{1}), \gamma _{2}^{P} \in \xi _{{\mathcal {G}}}(x_{2},x_{3})} \left\{ 1 - \prod _{j=1}^{2}(1-\gamma _{j}^{P})\right\} , \right. \\&\left. \bigcup _{\gamma _{1}^{N} \in \xi _{{\mathcal {G}}}(x_{2},x_{1}), \gamma _{2}^{N} \in \xi _{{\mathcal {G}}(x_{2},x_{3})}} \left\{ -1 + \prod _{j=1}^{2}(1+\gamma _{j}^{N})\right\} \right) \\&=\left( \bigcup _{\gamma _{1}^{P} \in \{0.6,0.4\}, \gamma _{2}^{P} \in \{0.9,0.7\}} \left\{ 1 - \prod _{j=1}^{2}(1-\gamma _{j}^{P})\right\} , \right. \\&\left. \bigcup _{\gamma _{1}^{N} \in \{-0.8,-0.5,-0.3\}, \gamma _{2}^{N} \in \{-0.8,-0.6,-0.3\}} \left\{ -1 + \prod _{j=1}^{2}(1+\gamma _{j}^{N})\right\} \right) \\&=\bigg (\{0.96,0.88,0.94,0.82\},\{-0.96,-0.92,-0.86, \\&-0.9,-0.8,-0.65,-0.86,-0.72,-0.51\} \bigg ) \end{aligned}$$

Similarly, we can also obtain the indegree and outdegree of all nodes; here, we will not list them for vast amounts of data. The dominant degree of all nodes and the influence of nodes are shown in Table 3 for self-persistence of each node is 0.

Table 3 The dominant degree and influence index of each node

According to the dominant degree of nodes listed in Table 3, we have $x_{6}> x_{1}> x_{5}> x_{2}> x_{3}> x_{4} > x_{7}$. But if we consider self-persistence, the ranking result is $x_{3}> x_{2}> x_{7}> x_{4}> x_{6}> x_{1} > x_{5}$. Therefore, according to the dominant degree, $x_{6}$ is a dominant person in this Whatsapp group, but if we consider self-persistence the person is $x_{3}$.

Comparative analysis with discussion

In this section, we compare our proposed best route finding Algorithm, provided in Sect. “Best route finding Algorithm” and the dominant node with influence index finding Algorithm, provided in Sect. “Algorithm for finding the dominant node with influence index”, with existing related literature.

Comparison of best route finding Algorithm with related literature

We consider the example discussed in Sect. 6 [35] for data comparison. We see that the different types of journey information from the city of Paris to Brest are shown in Table 4, where "Fast" and "Fast and cheap" are positive side information, and "Early" and "Early and late" are negative side information. The different types of journeys are shown in Fig. 8, using data present in Table 4.

Table 4 Order of journeys [35]

Fig. 8

The journey route or path

In Fig. [8], we see that every path information is a hesitant bipolar-valued fuzzy information, which mainly depends on the journey time, departure time, cost and mode like plain, train, and bus. Then, using the algorithm given in Sect. “Best route finding Algorithm”, we have the descending order of the journeys are 14, 11, 12, 15, 16, 13, 10. In [35], the authors considered "Fast" as a positive side information and "Early" as a negative side information. The information "Fast and cheap" and "Early and late" are avoided. They obtained the descending order of the journeys are 16, 15, 13, 14, 12, 11, and 10. Therefore, the best journey way of our proposed Algorithm and the method in [35] is 10. However, the journey order is changed in our method because we can not avoid the information of "Fast and cheap" and "Early and late".

Fig. 9

Detailed comparison results with proposed Algorithm provided in section 4.2.1 versus the method in [14]

In the following, we discussed the merits of the proposed shortest path algorithm based on HBVFGs with respect to the existing literature in [19, 20, 23, 25,26,27].

• (1) The best/shortest path processing method in [19] of Chuang and Kung embroils the similarity measures of fuzzy set and is thus unable to work with HBVF data. It also takes more time than the strategy used in our proposed algorithm to find the shortest path using a similarity measure.

• (2) The same technique as our proposed technique was used for trapezoidal fuzzy numbers [26], interval-valued Pythagorean fuzzy numbers [27] and T-spherical fuzzy numbers [25]. But our proposed shortest path Algorithm in Sect. “Best route finding Algorithm”, based on hesitant bipolar-valued fuzzy information, is more comfortable than other existing shortest path Algorithm, based on other extensions of fuzzy information like trapezoidal fuzzy numbers [26], interval-valued Pythagorean fuzzy numbers [27] and T-spherical fuzzy numbers [25].

• (3) A new algorithm for finding the best route in the fuzzy triangular setting was proposed by Gani and Jabarulla [20]. To obtain the node’s distance, they used distance measurements. These methods take more time, and the dependence on distance measurements has often been a concern.

• (4) Mukherjee modified the algorithm of Dijkstra [23] by adding the idea of aggregating operators of intuitionistic fuzzy information for the shortest route, which is an unstable way of calculating a route.

We mentioned a few benefits of the proposed algorithm below.

• (1) It is very straightforward to use the proposed algorithm, and the finding results are much quicker than the current algorithms since it contains the formulas found in equations (2) and (3). The proposed Algorithm for the shortest path in Sect. “Best route finding Algorithm” is more simpler than the one proposed in [28, 37, 38].

• (2) The shortest path results in our proposed Algorithm are more powerful than the results in [39,40,41].

Our proposed shortest path Algorithm in Sect. “Best route finding Algorithm” fails for incomplete or partial path information, which is a disadvantage of this Algorithm.

Comparison of the dominant node with influence index finding Algorithm with related literature

For data comparison with the proposed Algorithm provided in Sect. 4.2 versus the method in [14], we consider Example 8. Then, using two methods, we calculate the score of indegree, score of outdegree, dominant degree and the influence index of each node. The detailed results of both methods are shown in Fig. 9.

Some advantages of the dominant node with an influence index finding Algorithm are:

• (1) It captures more information than the studies in [14, 40].

• (2) In [29] pointed out the importance of self-persistence degree of nodes for computing the dominant node with an influence index. However, in [14, 31, 32, 40] it neglected. It is considered in our proposed dominant node with an influence index finding Algorithm.

Our proposed dominant node with influence index finding Algorithm in Sect. “Algorithm for finding the dominant node with influence index” fails for incomplete or partial vertices and edges information, which is a disadvantage of this Algorithm.

Discussion

Today, any person or people to bring themselves into the light of propaganda using a social network like Facebook, WhatsApp, Twitter, etc. Numbers of politicians also use these platforms for their colonial powers. Therefore, any persons or politicians want to dominate each other using social networks. How do we measure how much each other is affected? It is discussed according to Dynamic topology in [38] and according to DeGroot model in [29, 30, 34] by crisp classical data. In Algorithm in Sect. “Algorithm for finding the dominant node with influence index”, we proposed a model to measure a person or politician dominant each other using social network. In this algorithm, we considered two types of information for each person: positive information, i.e., possibly affected, and negative information, i.e., possibly unaffected. Then, obtain the outdegree of a person, i.e., how much do affect and indegree of a person, i.e., how much has been affected. In Example 8, we see that the rank of persons shown in Table 5 according to the dominant degree and influence index are different if we count only positive, negative, and both information. The detailed comparison result is shown in Fig. 10.

Table 5 Rank of persons by different information using $\mathcal{D}\mathcal{G}$ and $\mathcal {INF}$

Fig. 10

Detailed comparison analysis of dominant degree and influence index, where we considered three types of information such as positive, negative and both information

Conclusion with future work

This paper introduces the concept of directed HBVFG, which can capture more uncertainty in the nodes or edges of a graph than other generalized FGs such as BFGs and HFGs. The paper analyses the novelty of directed HBVFGs and their significance. It also discusses some basic definitions, mapping relations and product operations related to directed HBVFGs. The paper adapts the well-known Dijkstra algorithm for directed HBVFGs and solves the shortest path problem in a network of directed HBVFGs. It also addresses the dominant degree and influence index of a person in a social network. The paper suggests that directed HBVFGs can be very useful for some applications in network optimization, road signal problems and other engineering and computer science problems. In the following we mention the merits with benefit of our study:

• (1) To find the shortest path in a network using our proposed algorithm takes less time than existing literature in [19, 20, 23, 25,26,27].

• (2) The proposed Algorithm for the shortest path in Sect. “Best route finding Algorithm” is simpler than the one proposed in [28, 37, 38].

• (3) The shortest path results in our proposed Algorithm are more powerful than the results in [39,40,41].

• (4) In our proposed dominant node with an influence index finding Algorithm captures more information than the studies in [14, 40].

• (5) In proposed dominant node with an influence index finding Algorithm, we consider self-persistence degree of nodes.

The disadvantages with limitations of our study are listed in the following:

• (1) In this paper we cannot consider incomplete information of vertices and edges. Thus, our proposed shortest path, and dominant node with an influence index finding algorithms are fail for incomplete information of vertices and edges.

• (2) In hybrid type’s data, our proposed approach not applicable.

In future, we shall work the interesting results like fuzzy coloring graph [44], graph centrality measure [41], competition graph [45], etc. based on HBVFG and the two limitations, mentioned above.