Certain models of granular computing based on rough fuzzy approximations

Abstract

An extraction of granular structures using graphs is a powerful mathematical framework in human reasoning and problem solving. The visual representation of a graph and the merits of multilevel or multiview of granular structures suggest the more effective and advantageous techniques of problem solving. In this research study, we apply the combinative theories of rough fuzzy sets and rough fuzzy digraphs to extract granular structures. We discuss the accuracy measures of rough fuzzy approximations and measure the distance between lower and upper approximations. Moreover, we consider the adjacency matrix of a rough fuzzy digraph as an information table and determine certain indiscernible relations. We also discuss some general geometric properties of these indiscernible relations. Further, we discuss the granulation of certain social network models using rough fuzzy digraphs. Finally, we develop and implement some algorithms of our proposed models to granulate these social networks.

Keywords

Rough fuzzy approximations rough fuzzy digraphs information granulation algorithms

1 Introduction

A fundamental mathematical framework to deal with uncertain and vague information is rough set theory (RST), which was originally proposed by Pawlak [22]. In RST, the lower approximation (LA) and the upper approximation (UA) are used to approximate the subset of a universe. The main advantage of RSs to deal uncertainty and vague information is that it does not require any additional information about data. There are many interesting applications of this theory in the fields of cognitive sciences, machine learning, knowledge discovery, expert systems, inductive reasoning and pattern recognition. Rough approximations are widely used in the field of granular computing (GrC) [18]. Granulation of a problem leads to the collection of granules, a granule being a group of elements or objects combined together by similarity or indistinguishability. In RST, the objects are interconnected into equivalence classes, which leads o the formation of elementary blocks or granules. In this way, GrC is inherently similar to the foundation of RST. Pawlak [23] studied the various problems of granularity of knowledge using RSs. The formation of granulation structures by RST and the corresponding approximations were studied by Yao [34]. Wang et al. [30] introduced multi-granularity soft rough sets and multi-granularity soft relative attribute reduction. They studied basic properties of multi-granularity soft rough approximations and illustrated by examples. Furthermore, an algorithm for multi-attribute decision-making problems is proposed. The crisp information granulation fails to reflect the human reasoning and problem solving, when the granules are fuzzy in nature. The granules are fuzzy in the sense that the boundaries of real life phenomena such as small, large, intelligent, etc may not be defined sharply. To overcome the problem of uncertainty in GrC, information granularity of universe based on fuzzy sets (FSs) was proposed by Zadeh [36]. In this granulation, the relationships between clusters, groups or granules are presented using fuzzy sets and fuzzy graphs. Fuzzy information granulation provides a better way of studying the uncertain or indeterminate behaviors of granules. It is worth to note that the RST and FST do not play the same role in the problems of knowledge representation systems. Thus, they are not two rival theories to compare with each other but two different aspects to capture the imperfection of knowledge. Therefore, RSs and FSs can be combined to get an hybrid model of rough fuzzy theory (RFT) to get the better approximations.

To incorporate the advantages of RSs and FSs, Dubious and Prade [14] defined the fuzzy rough sets (FRSs) and rough fuzzy sets (RFSs). FRSs have various applications in many fields, including clustering, attribute reduction and classification [16]. Alsager and Alshehri [7] presented a novel hybrid model called multi Q-dual hesitant fuzzy soft (MkQDHFS) rough set model, by the combination of rough sets and MkQDHFS sets. Shah et al. [27] introduced different kinds of uncertainty measures related to Z-soft rough covering sets and discuss their limitations. Moreover, Zhang et al.[40] investigated a novel technique for multi-attribute decision making applying soft rough sets. They introduced various novel concepts, including soft relative positive regions, dependent degree of decision partition soft sets, soft decision systems, and conditional significance relative to decision partition soft sets. Zhan and Zhu [42] proposed some revised algorithms in decision making and provided certain examples. Finally, they constructed a new decision making method for rough soft sets. Novel decision making methods based on hybrid m-polar fuzzy models were proposed by Akram et al. [2]. They combined the notions of rough m-polar fuzzy sets and m-polar fuzzy soft sets to introduce new hybrid models for soft computing.

Theory of RFSs was applied to digraphs by Zafar and Akram [37]. They proposed a novel decision-making method based on rough fuzzy information. The same authors also defined rough fuzzy digraphs with applications [5]. Wang et al. [31] investigated the multiple attribute decision making (MADM) problem based on the aggregation operators with dual hesitant fuzzy information. Zhang [38] proposed attributes reduction based on intuitionistic fuzzy rough sets. Akram and Zafar [6] generalized the concept of connectivity of fuzzy graphs to rough fuzzy digraphs, introduced the strength of connectedness between vertices, and investigated certain properties. A new approach based on fuzzy rough digraphs for decision-making was proposed by Akram et al. [3]. Zhan et al. [39] introduced the concept of a rough soft hemiring as an extended notion of a rough hemiring and a soft hemiring. Recently, Pythagorean fuzzy soft graphs have been introduced by Shahzadi et al. [28]. They studied the certain properties of the proposed graphs, including perfectly regular and perfectly edge-regular Pythagorean fuzzy soft graphs.

Granulation for simple graphs was firstly introduced by Stell [25]. He demonstrated the different kinds of granulation for simple graphs and discussed the connection of vague graphs with granulation. Chen and Zhong [12] extracted the granular structures using simple graphs. They developed the granular structure models based on three types of elements in a simple graph, i.e., vertices, edges and their combination. In 2016, Chiaselotti et al. [13] studied some interesting concepts of simple graphs, which correspond to the GrC. Micro and macro models of granular computing based on the indiscernibility relation were studied by Bisi et al. [9]. They investigated the indiscernibility structures of knowledge representation system and developed three hypergraphic structures, including the discernibility hypergraph, the reduct and the essential hypergraph. Chen et al. [11] proposed a hypergraph model of GrC and gave an effective way of granular structures and a useful method for problem solving. Furthermore, Bianchi et al. [10] discussed the matching of labeled graphs using the GrC concepts and levels of abstraction. The relational granularity of hypergraphs and the applications of these relations to a theory of granularity were discussed by Stell [26]. Wang and Gong [29] discussed an application of fuzzy hypergraphs in GrC. Information granulation for rough fuzzy hypergraphs was studied by William-West and Singh in [32]. Recently, Luqman et al. [19] have proposed an m-polar fuzzy hypergraph model of GrC.

The construction of granules and grouping the clusters for crisp graphs have been explored many researchers in various domains. But they did not provide a detailed discussion on the construction of granular structures for graphs, use of these structures in problem solving, representation of relationships among granules and transforming the different levels of granularities. Moreover, the crisp approximations of simple graphs lack some information in case of uncertain network models and do not provide an adequate and exact visualization of certain problems. For example, in case of road networking vehicle travel time or vehicle capacity on a road network can not be determined exactly and in such cases, it is necessary to use fuzzy approximations to deal with the uncertainty. These deficiencies of previously proposed models motivate us to combine the hybrid model of RFSs with graph theory to extract the granular structures. We examine the information granulation by applying RFSs and review the corresponding approximations of granules. We discuss the accuracy measures of rough fuzzy sets and measure the distance between lower and upper approximations. The visual representation of RFDGs and merits of multilevel of granularity enable us to adopt the most fruitful techniques for problem solving. In addition. the main focus of our work is to present a brief discussion on the construction of granular structures using RFDGs, grouping these granules and representation of relationships between different levels of granularities. We consider the adjacency matrix of a RFDG as information table and determine some indiscernible relations. Thus, we study how to granulate the certain network models using RFDGs.

The contents of this paper are as follows: In section 2, we review some fundamental concepts of FSs, RSs, RFSs and RFDGs. In section 3, we define accuracy measures and distance between lower and upper approximations. We define RF information tables and set-valued information tables. In section 4, we study the role of RFDGs in information granulation. We discuss some general geometric properties of RFDGs and define certain mappings based on indiscernibility relations. In section 5, we discuss the granulation of certain social networks through RF approximations. We develop and implement some algorithms to granulate our social structures. In section 6, we give a brief comparison of our proposed models to granulate social networks with other existing models of crisp granulation. The last section deals with conclusions and future directions.

2 Preliminaries

Definition 2.1. [35] A fuzzy set η on a non-empty set Z is a mapping η:Z → [0, 1]. A fuzzy relation λ is a fuzzy set on Z × Z. Let η and λ be fuzzy sets on Z and Z × Z, respectively, such that λ (uw) ≤ min {η (u) , η (w)}, for all u, w ∈ Z, then λ is a fuzzy relation on η.

Definition 2.2. [22] Let Z be the set of universe and ξ be an equivalence relation (ER) on Z. Then the pair (Z, ξ) is called an approximation space. Let W^∗ be the non-empty subset of Z and determines the lower approximation (LA) and upper approximation (UA) as follows: $\begin{matrix} \underline{ξ} W^{*} & = & {z \in Z | [z]_{ξ} \subseteq W^{*}}, \\ \bar{ξ} W^{*} & = & {z \in Z | [z]_{ξ} \cap W^{*} \neq \emptyset}, \end{matrix}$ respectively. Here, [z] _ξ represents the equivalence class of an element z ∈ Z. The pair $(\underline{ξ} W^{*}, \bar{ξ} W^{*})$ is called a rough set. Definition 2.3. [14] Let Z be the set of universe and ξ be an ER on Z. Let $F \in F (Z)$ , where $F (Z)$ is the collection of fuzzy subsets of Z. The LA and UA of F are determined by fuzzy sets in Z, denoted by $\underline{ξ} F$ and $\bar{ξ} F$ , respectively. These approximations are defined as: $\begin{matrix} \underline{ξ} F (z) & = & ⋀_{z_{1} \in Z} {max (1 - ξ (z, z_{1}), F (z_{1})}, \\ \bar{ξ} F (z) & = & ⋁_{z_{1} \in Z} {min (ξ (z, z_{1}), F (z_{1}))}, \end{matrix}$ for all z ∈ Z and the pair $(\underline{ξ} F, \bar{ξ} F)$ is called a rough fuzzy set (RFS).

Definition 2.4. [37] Let Z be the set of universe and ξ be an ER on Z. Let F be a fuzzy set on Z and $ξ F = (\underline{ξ} F, \bar{ξ} F)$ be a RFS. Let J^∗ ⊆ Z × Z and E be an ER on J^∗ such that: $(z_{1} z_{2}, z_{1}^{'} z_{2}^{'}) \in E \Leftrightarrow (z_{1}, z_{1}^{'}), (z_{2}, z_{2}^{'}) \in ξ,$ for all $z_{1} z_{2}, z_{1}^{'} z_{2}^{'} \in J^{*}$ . Let J be a fuzzy set on J^∗ such that: $J (z_{1} z_{2}) \leq min {\underline{ξ} F (z_{1}), \underline{ξ} F (z_{2})}, \forall z_{1}, z_{2} \in Z .$ Then, the pair $(\underline{E} J, \bar{E} J)$ represents the LA and UA of J, respectively, such that: $\begin{matrix} \underline{E} J (z_{1} z_{2}) & = & ⋀_{z_{1}^{'} z_{2}^{'} \in J^{*}} {max (1 - E (z_{1} z_{2}, z_{1}^{'} z_{2}^{'}), J (z_{1}^{'} \\ z_{2}^{'}))}, \\ \bar{E} J (z_{1} z_{2}) & = & ⋁_{z_{1}^{'} z_{2}^{'} \in J^{*}} {min (E (z_{1} z_{2}, z_{1}^{'} z_{2}^{'}), J (z_{1}^{'} z_{2}^{'}))}, \end{matrix}$ for all z₁z₂ ∈ J^∗. The pair $EJ = (\underline{E} J, \bar{E} J)$ is called a rough fuzzy relation (RFR).

Definition 2.5. [5, 37] A rough fuzzy digraph (RFDG) on Z is an 4-ordered tuple $\vec{G} = (ξ, ξ F, E, EJ)$ , where,

(i) ξ is an ER on Z.

(ii) E is an ER on J^∗ ⊆ Z × Z.

(iii) $ξ F = (\underline{ξ} F, \bar{ξ} F)$ is RFS on Z.

(iv) $EJ = (\underline{E} J, \bar{E} J)$ is RFR on Z.

(v) (ξF, EJ) is rough fuzzy digraph, where $\underline{\vec{G}} = (\underline{ξ} F, \underline{E} J)$ and $\bar{\vec{G}} = (\bar{ξ} F, \bar{E} J)$ are LA and UA of $\vec{G}$ , respectively, such that for all z₁, z₂ ∈ Z, we have: $\begin{matrix} \underline{E} J (z_{1} z_{2}) & \leq & min {\underline{ξ} F (z_{1}), \underline{ξ} F (z_{2})}, \\ \bar{E} J (z_{1} z_{2}) & \leq & min {\bar{ξ} F (z_{1}), \bar{ξ} F (z_{2})} . \end{matrix}$

Definition 2.6. [9, 15] A knowledge representation system (KRS) or information table (IT) is defined as a structure $I = (Z, \tilde{A_{t}}, g, Val)$ , where Z is the universal set, $\tilde{A_{t}} = {a_{i} | i \in N}$ is a collection of attributes and a_i : Z → Val. The information mapping $g : Z \times \tilde{A_{t}} \to Val$ of I is defined as g (u, a_i) = a_i (u).

fuzzy knowledge representation system or fuzzy information table (FIT), in case of finite set, is defined as a structure $F_{I} = (Z$ , $F_{A_{titsc}}$ , φ, [0, 1]), where $F_{A_{titsc}}$ is the set of fuzzy attributes and φ_k (u, a_k) = η_k (u) ∈ Val_k.

Definition 2.7. [17] Let F₁, $F_{2} \in F (Z)$ . The union, intersection, scalar cardinality and relative cardinality of fuzzy sets are defined as follows:

(i) η_F₁∪F₂ (z) = max {η_{F
₁} (z) , η_{F
₂} (z)} ,

(ii) η_F₁∩F₂ (z) = min {η_{F
₁} (z) , η_{F
₂} (z)} ,

(iii) |F₁| = Ση_{F
₁} (z) ,

(iv) $| | F_{1} | | = \frac{| F_{1} |}{| Z |},$ respectively, for z ∈ Z.

3 Accuracy measures of rough fuzzy approximations

Definition 3.1. Let $ξ F = (\underline{ξ} F, \bar{ξ} F)$ be a RFS on Z. The accuracy of ξF is defined as: $α (ξ F) = \frac{| \underline{ξ} F |}{| \bar{ξ} F |},$ where | . | denotes the scalar cardinality of fuzzy sets.

The accuracy of ξF is also defined as: $α^{*} (ξ F) = \frac{| | \underline{ξ} F | |}{| | \bar{ξ} F | |},$ where || . || denotes the relative cardinality of fuzzy sets.

Theorem 3.1. The accuracy of RFS ξF remains same for scalar cardinality and relative cardinality, i.e., α (ξF) = α^* (ξF).

Proof. Let $α^{*} (ξ F) = \frac{| | \underline{ξ} F | |}{| | \bar{ξ} F | |},$ where || . || denotes the relative cardinality of fuzzy sets. Since, we have: $| | \underline{ξ} F | | = \frac{| \underline{ξ} F |}{| Z |}, | | \bar{ξ} F | | = \frac{| \bar{ξ} F |}{| Z |},$ where | . | denotes the scalar cardinality of fuzzy sets. Then, $\begin{matrix} α^{*} (ξ F) & = & \frac{| | \underline{ξ} F | |}{| | \bar{ξ} F | |} = \frac{| \underline{ξ} F | / | Z |}{| \bar{ξ} F | / | Z |} \\ = & \frac{| \underline{ξ} F | / | Z |}{| \bar{ξ} F | / | Z |} = \frac{| \underline{ξ} F |}{| \bar{ξ} F |} \\ = & α (ξ F) . \end{matrix}$ Hence, the result is proved.

Definition 3.2. The distance between LA and UA of a RFS can be measured using Marczewski-Steinhaus metric or MZ metric and is defined as follows: $\begin{matrix} d (\underline{ξ} F, \bar{ξ} F) & = & 1 - \frac{| \underline{ξ} F \cap \bar{ξ} F |}{| \underline{ξ} F \cup \bar{ξ} F |} \\ = & 1 - \frac{| \underline{ξ} F |}{| \bar{ξ} F |} ∵ \underline{ξ} F \subseteq \bar{ξ} F \\ = & 1 - α (ξ F) . \end{matrix}$ Note that, the distance between LA and UA finds out the accuracy of RFS approximations.

Definition 3.3. Let $I = (Z, \tilde{A_{t}}, g, Val)$ be a KRS and $\tilde{A} \subseteq \tilde{A_{t}}$ . Then the relation defined by: $u ξ_{\tilde{A}} w \Leftrightarrow g (u, a_{i}) = g (w, a_{i}), \forall a_{i} \in \tilde{A},$ is an equivalence relation. The elements u and w are called indiscernible or equivalent elements.

Example 3.1 Let $I = (Z, \tilde{A_{t}}, g, Val)$ be a KRS, where Z = {a, b, c} is the set of three students and $\tilde{A_{t}} = {A_{1}, A_{2}, A_{3}}$ is the set of attributes such that A₁ : academic skills, A₂ : attitude and A₃ : communication skills. The information table of this system is given in Table 1.

Table 1
Information table I

I A ₁ A ₂ A ₃

a High High Good

b Medium Medium Normal

c High High Bad

I	A ₁	A ₂	A ₃
a	High	High	Good
b	Medium	Medium	Normal
c	High	High	Bad

Let us consider the subset of attributes given as $\tilde{A} = {A_{1}, A_{2}}$ . Then, the ER $ξ_{\tilde{A}}$ with respect to {A₁, A₂} is given in Table 2.

Table 2

The ER $ξ_{\tilde{A}}$

$ξ_{\tilde{A}}$	a	b	c
a	1	0	1
b	0	1	0
c	1	0	1

The ER $ξ_{\tilde{A}}$ is given by the following partition: $Z / ξ_{\tilde{A}} = {{a, c}, {b}} .$ Let F = {(a, 0.2), (b, 0.3), (c, 0.5)} be the fuzzy set on Z. The lower and upper approximations of F are given as follows: $\begin{matrix} \underline{ξ} F (a) & = & min {0.2, 1.0, 0.5} = 0.2, \\ \underline{ξ} F (b) & = & min {1.0, 0.3, 1.0} = 0.3, \\ \underline{ξ} F (c) & = & min {0.2, 0.3, 0.5} = 0.2, \end{matrix}$ and $\begin{matrix} \bar{ξ} F (a) & = & max {0.2, 0.0, 0.5} = 0.5, \\ \bar{ξ} F (b) & = & max {0.0, 0.3, 0.0} = 0.3, \\ \bar{ξ} F (c) & = & max {0.2, 0.0, 0.5} = 0.5 . \end{matrix}$ Thus, $\underline{ξ} F = {(a, 0.2), (b, 0.3), (c, 0.2)}$ and $\bar{ξ} F = {(a, 0.5), (b, 0.3), (c, 0.5)}$ are the LA and UA of F, respectively. The accuracy of ξF is given as: $α (ξ F) = \frac{| \underline{ξ} F |}{| \bar{ξ} F |} = \frac{0.7}{1.3} = 0.538 .$ Note that, the accuracy of ξF has the maximum value of 1 and minimum value of 0, i.e., 0 ≤ α (ξF) ≤1.

The distance between LA and UA is given as: $\begin{matrix} d (\underline{ξ} F, \bar{ξ} F) & = & 1 - \frac{| \underline{ξ} F \cap \bar{ξ} F |}{| \underline{ξ} F \cup \bar{ξ} F |} = 1 - \frac{0.7}{1.3} \\ = & 1 - 0.538 = 1 - α (ξ F) = 0.462 . \end{matrix}$ Similarly, different approximations can be determined by choosing different subsets of $\tilde{A_{t}}$ .

Definition 3.4. A set-valued information table (SVIT) is defined as a standard information table $I = (Z, \tilde{A_{t}}, ψ, Val)$ with the only difference that the function ψ is set-valued function, i.e., ψ : Z → 2^Val. For a subset A^* of $\tilde{A_{t}}$ , a relation λ_{A
^*} defined by: $u λ_{A^{*}} w \Leftrightarrow ψ (u) \cap ψ (u) \neq \emptyset, \forall a \in A^{*},$ is a tolerance relation, i.e., symmetric and reflexive. Let F be a fuzzy set on Z. The LA and UA of F are determined as: $\begin{matrix} {\underline{λ}}_{A^{*}} F (z) & = & ⋀_{z_{1} \in Z} {max (1 - λ_{A^{*}} (z, z_{1}), F (z_{1})}, \\ {\bar{λ}}_{A^{*}} F (z) & = & ⋁_{z_{1} \in Z} {min (λ_{A^{*}} (z, z_{1}), F (z_{1}))}, \end{matrix}$ for all z ∈ Z.

We now discuss the rough fuzzy approximations using SVITs and find out their accuracy measures.

Example 3.2. Let $I = (Z, \tilde{A_{t}}, ψ, Val)$ be a SVIT, where Z = {s₁, s₂, s₃, s₄, s₅, s₆} is the set of objects and set of attributes is given as, $\tilde{A_{t}} = {A_{1}, A_{2}, A_{3}, A_{4}}$ . The corresponding SVIT is given in Table 3.

Table 3

The corresponding SVIT I

I	A ₁	A ₂	A ₃	A ₄
s ₁	{0, 1}	{1}	{1}	{0, 1}
s ₂	{1}	{1, 2}	{1}	{0}
s ₃	{1}	{2}	{0, 1}	{0, 1}
s ₄	{1}	{1, 2, 3}	{0, 1}	{1}
s ₅	{0, 1}	{1}	{0}	{0, 1}
s ₆	{0}	{3}	{1}	{1}

We now determine the tolerance classes according to the relation λ_{A
^*} for different subsets of $\tilde{A_{t}}$ . Let $A_{1}^{*} = {\emptyset}$ , then the tolerance relation is given by: $\begin{matrix} ⌈ s_{1} ⌉_{A_{1}^{*}} = {s_{1}}, ⌈ s_{2} ⌉_{A_{1}^{*}} = {s_{2}}, ⌈ s_{3} ⌉_{A_{1}^{*}} = {s_{3}}, \\ ⌈ s_{4} ⌉_{A_{1}^{*}} = {s_{4}}, ⌈ s_{5} ⌉_{A_{1}^{*}} = {s_{5}}, ⌈ s_{6} ⌉_{A_{1}^{*}} = {s_{6}} . \end{matrix}$

Note that, the relation matrix of $A_{1}^{*}$ is an identity matrix. Let $A_{2}^{*} = {A_{1}$ , A₂}, then the tolerance classes are determined as follows: $u λ_{A_{2}^{*}} w \Leftrightarrow ψ (u) \cap ψ (u) \neq \emptyset, \forall a \in A_{2}^{*} .$ By routine calculations, we have: $\begin{matrix} ψ_{A_{1}} (s_{1}) \cap ψ_{A_{1}} (s_{1}) & = & {0, 1} \cap {0, 1} = {0, 1} \neq \emptyset, \\ ψ_{A_{2}} (s_{1}) \cap ψ_{A_{2}} (s_{1}) & = & {1} \cap {1} = {1} \neq \emptyset, \\ ψ_{A_{1}} (s_{1}) \cap ψ_{A_{1}} (s_{2}) & = & {0, 1} \cap {1} = {1} \neq \emptyset, \\ ψ_{A_{2}} (s_{1}) \cap ψ_{A_{2}} (s_{2}) & = & {1} \cap {1, 2} = {1} \neq \emptyset, \\ ψ_{A_{1}} (s_{1}) \cap ψ_{A_{1}} (s_{3}) & = & {0, 1} \cap {1} = {1} \neq \emptyset, \\ ψ_{A_{2}} (s_{1}) \cap ψ_{A_{2}} (s_{3}) & = & {1} \cap {2} = \emptyset, \\ ψ_{A_{1}} (s_{1}) \cap ψ_{A_{1}} (s_{4}) & = & {0, 1} \cap {1} = {1} \neq \emptyset, \end{matrix}$ $\begin{matrix} ψ_{A_{2}} (s_{1}) \cap ψ_{A_{2}} (s_{4}) & = & {1} \cap {1, 2, 3} = {1} \neq \emptyset, \\ ψ_{A_{1}} (s_{1}) \cap ψ_{A_{1}} (s_{5}) & = & {0, 1} \cap {0, 1} = {0, 1} \neq \emptyset, \\ ψ_{A_{2}} (s_{1}) \cap ψ_{A_{2}} (s_{5}) & = & {1} \cap {1} = {1} \neq \emptyset, \\ ψ_{A_{1}} (s_{1}) \cap ψ_{A_{1}} (s_{6}) & = & {0, 1} \cap {0} = {0} \neq \emptyset, \\ ψ_{A_{2}} (s_{1}) \cap ψ_{A_{2}} (s_{6}) & = & {1} \cap {3} = \emptyset . \end{matrix}$ Hence, we have: $\begin{matrix} ⌈ s_{1} ⌉_{A_{2}^{*}} = ⌈ s_{5} ⌉_{A_{2}^{*}} = {s_{1}, s_{2}, s_{4}, s_{5}}, ⌈ s_{2} ⌉_{A_{2}^{*}} \\ = ⌈ s_{4} ⌉_{A_{2}^{*}} = {s_{1}, s_{2}, s_{3}, s_{4}, s_{5}} . \end{matrix}$ Similarly, the tolerance classes of s₃ are given as: $\begin{matrix} ψ_{A_{1}} (s_{3}) \cap ψ_{A_{1}} (s_{1}) & = & {1} \cap {0, 1} = {1} \neq \emptyset, \\ ψ_{A_{2}} (s_{3}) \cap ψ_{A_{2}} (s_{1}) & = & {2} \cap {1} = \emptyset, \\ ψ_{A_{1}} (s_{3}) \cap ψ_{A_{1}} (s_{2}) & = & {1} \cap {1} = {1} \neq \emptyset, \\ ψ_{A_{2}} (s_{3}) \cap ψ_{A_{2}} (s_{2}) & = & {2} \cap {1, 2} = {2} \neq \emptyset, \\ ψ_{A_{1}} (s_{3}) \cap ψ_{A_{1}} (s_{3}) & = & {1} \cap {1} = {1} \neq \emptyset, \\ ψ_{A_{2}} (s_{3}) \cap ψ_{A_{2}} (s_{3}) & = & {2} \cap {2} = {2} \neq \emptyset, \\ ψ_{A_{1}} (s_{3}) \cap ψ_{A_{1}} (s_{4}) & = & {1} \cap {1} = {1} \neq \emptyset, \\ ψ_{A_{2}} (s_{3}) \cap ψ_{A_{2}} (s_{4}) & = & {2} \cap {1, 2, 3} = {2} \neq \emptyset, \\ ψ_{A_{1}} (s_{3}) \cap ψ_{A_{1}} (s_{5}) & = & {1} \cap {0, 1} = {1} \neq \emptyset, \\ ψ_{A_{2}} (s_{3}) \cap ψ_{A_{2}} (s_{5}) & = & {2} \cap {1} = \emptyset, \\ ψ_{A_{1}} (s_{3}) \cap ψ_{A_{1}} (s_{6}) & = & {1} \cap {0} = \emptyset, \\ ψ_{A_{2}} (s_{3}) \cap ψ_{A_{2}} (s_{6}) & = & {2} \cap {3} = \emptyset . \end{matrix}$ Thus, we have: $⌈ s_{3} ⌉_{A_{2}^{*}} = {s_{2}, s_{3}, s_{4}} .$ The tolerance classes of s₆ are given as: $\begin{matrix} ψ_{A_{1}} (s_{6}) \cap ψ_{A_{1}} (s_{1}) & = & {0} \cap {0, 1} = {0} \neq \emptyset, \\ ψ_{A_{2}} (s_{6}) \cap ψ_{A_{2}} (s_{1}) & = & {3} \cap {1} = \emptyset, \\ ψ_{A_{1}} (s_{6}) \cap ψ_{A_{1}} (s_{2}) & = & {0} \cap {1} = \emptyset, \\ ψ_{A_{2}} (s_{6}) \cap ψ_{A_{2}} (s_{2}) & = & {3} \cap {1, 2} = \emptyset, \\ ψ_{A_{1}} (s_{6}) \cap ψ_{A_{1}} (s_{3}) & = & {0} \cap {1} = \emptyset, \\ ψ_{A_{2}} (s_{6}) \cap ψ_{A_{2}} (s_{3}) & = & {3} \cap {2} = \emptyset, \\ ψ_{A_{1}} (s_{6}) \cap ψ_{A_{1}} (s_{4}) & = & {0} \cap {1} = \emptyset, \\ ψ_{A_{2}} (s_{6}) \cap ψ_{A_{2}} (s_{4}) & = & {3} \cap {1, 2, 3} = {3} \neq \emptyset, \end{matrix}$ $\begin{matrix} ψ_{A_{1}} (s_{6}) \cap ψ_{A_{1}} (s_{5}) & = & {0} \cap {0, 1} = {0} \neq \emptyset, \\ ψ_{A_{2}} (s_{6}) \cap ψ_{A_{2}} (s_{5}) & = & {3} \cap {1} = \emptyset, \\ ψ_{A_{1}} (s_{6}) \cap ψ_{A_{1}} (s_{6}) & = & {0} \cap {0} = {0} \neq \emptyset, \\ ψ_{A_{2}} (s_{6}) \cap ψ_{A_{2}} (s_{6}) & = & {3} \cap {3} = {3} \neq \emptyset . \end{matrix}$ Hence, we obtain: $⌈ s_{6} ⌉_{A_{2}^{*}} = {s_{6}} .$ The relation matrix of $A_{2}^{*} = {A_{1}, A_{2}}$ is given in Table 4.

Table 4

The relation matrix $λ_{A_{2}^{*}}$

$λ_{A_{2}^{*}}$	s ₁	s ₂	s ₃	s ₄	s ₅	s ₆
s ₁	1	1	0	1	1	0
s ₂	1	1	1	1	1	0
s ₃	0	1	1	1	0	0
s ₄	1	1	1	1	1	0
s ₅	1	1	0	1	1	0
s ₆	0	0	0	0	0	1

Let F = {(s₁, 0.8) , (s₂, 0.6) , (s₃, 0.7) , (s₄, 0.4) , (s₅, 0.9) , (s₆, 0.3)} be a fuzzy set on Z. The LA and UA of F are determined as: $\begin{matrix} {\underline{λ}}_{A^{*}} F (z) & = & ⋀_{z_{1} \in Z} {max (1 - λ_{A^{*}} (z, z_{1}), F (z_{1})}, \\ {\bar{λ}}_{A^{*}} F (z) & = & ⋁_{z_{1} \in Z} {min (λ_{A^{*}} (z, z_{1}), F (z_{1}))}, \end{matrix}$ for all z ∈ Z. By direct calculations, we have: $\begin{matrix} {\underline{λ}}_{A_{2}^{*}} F (⌈ s_{1} ⌉_{A_{2}^{*}}) = {(s_{1}, 0.7), (s_{2}, 0.6), (s_{4}, 0.4), \\ (s_{5}, 0.4)} = {\underline{λ}}_{A_{2}^{*}} F (⌈ s_{5} ⌉_{A_{2}^{*}}), \\ {\underline{λ}}_{A_{2}^{*}} F (⌈ s_{2} ⌉_{A_{2}^{*}}) = {(s_{1}, 0.7), (s_{2}, 0.6), (s_{3}, 0.7), \\ (s_{4}, 0.4), (s_{5}, 0.4)} = {\underline{λ}}_{A_{2}^{*}} F (⌈ s_{4} ⌉_{A_{2}^{*}}), \\ {\underline{λ}}_{A_{2}^{*}} F (⌈ s_{3} ⌉_{A_{2}^{*}}) = {(s_{2}, 0.6), (s_{3}, 0.7), (s_{4}, 0.4)}, \\ {\underline{λ}}_{A_{2}^{*}} F (⌈ s_{6} ⌉_{A_{2}^{*}}) = {(s_{6}, 0.3)}, \end{matrix}$ and $\begin{matrix} {\bar{λ}}_{A_{2}^{*}} F (⌈ s_{1} ⌉_{A_{2}^{*}}) = {(s_{1}, 0.8), (s_{2}, 0.6), (s_{4}, 0.9), \\ (s_{5}, 0.9)} = {\bar{λ}}_{A_{2}^{*}} F (⌈ s_{5} ⌉_{A_{2}^{*}}), \\ {\bar{λ}}_{A_{2}^{*}} F (⌈ s_{2} ⌉_{A_{2}^{*}}) = {(s_{1}, 0.8), (s_{2}, 0.6), (s_{3}, 0.8), \\ (s_{4}, 0.9), (s_{5}, 0.9)} = {\bar{λ}}_{A_{2}^{*}} F (⌈ s_{4} ⌉_{A_{2}^{*}}), \\ {\bar{λ}}_{A_{2}^{*}} F (⌈ s_{3} ⌉_{A_{2}^{*}}) = {(s_{2}, 0.6), (s_{3}, 0.8), (s_{4}, 0.9)}, \\ {\bar{λ}}_{A_{2}^{*}} F (⌈ s_{6} ⌉_{A_{2}^{*}}) = {(s_{6}, 0.9)}, \end{matrix}$ respectively. These RFSs determine the LA and UA of granules or granular structures. Note that, the elements in same equivalence class or tolerance class correspond to a single unit or granule. Every relation $λ_{A_{i}^{*}}$ gives the i-th level of granularity and different levels can be constructed according to distinct relations $λ_{A_{i}^{*}}$ .

Definition 3.5. Let ξ₁ ⊆ ξ₂ ⊆ ⋯ ⊆ ξ_m be a nested sequence of equivalence relations on Z such that: $[z]_{ξ_{1}} \subseteq [z]_{ξ_{2}} \subseteq \dots \subseteq [z]_{ξ_{m}} .$ Let F be a fuzzy set on Z, then the sequence of RFS approximations satisfies the following condition: $\begin{matrix} ξ_{1} \subseteq ξ_{2} \subseteq \dots \subseteq ξ_{m} & \Rightarrow & {\underline{ξ}}_{m} F \subseteq {\underline{ξ}}_{m - 1} F \subseteq \dots \subseteq \\ {\underline{ξ}}_{2} F \subseteq {\underline{ξ}}_{1} F \subseteq F (Z), \\ ξ_{1} \subseteq ξ_{2} \subseteq \dots \subseteq ξ_{m} & \Rightarrow & F (Z) \subseteq {\bar{ξ}}_{1} F \subseteq {\bar{ξ}}_{2} F \\ \subseteq \dots \subseteq {\bar{ξ}}_{m - 1} F \subseteq {\bar{ξ}}_{m} F . \end{matrix}$ This implies that: $\begin{matrix} ξ_{1} \subseteq ξ_{2} \subseteq \dots \subseteq ξ_{m} \Rightarrow α_{ξ_{m}} (F) \leq α_{ξ_{m_{1}}} (F) \\ \leq \dots \leq α_{ξ_{1}} (F) . \end{matrix}$

A multi-level partition of the universe is obtained through the nested sequence of equivalence relations and implies to a simple multi-level granulation of Z.

4 Rough fuzzy digraphs in granular computing

In this section, we study a RFDG $\vec{G}$ by considering its adjacency matrix as an information table $J$ . Let us denote by $V (\vec{G})$ , the crisp set of vertices and by $E (\vec{G})$ , the crisp set of edges of $\vec{G}$ .

Definition 4.1. Let $\vec{G} = (ξ F, EJ)$ be a RFDG. The adjacency matrix of $\vec{G}$ is defined as: (b_ij) _n×n

= $(({\underline{b}}_{ij})_{n \times n}$ , $({\bar{b}}_{ij})_{n \times n})$ , where

$({\underline{b}}_{ij})_{n \times n} = {\begin{matrix} \underline{E} J (z_{i} z_{j}), & if z_{i} z_{j} \in J^{*}, \\ 0, & otherwise . \end{matrix}$ $({\bar{b}}_{ij})_{n \times n} = {\begin{matrix} \bar{E} J (z_{i} z_{j}), & if z_{i} z_{j} \in J^{*}, \\ 0, & otherwise . \end{matrix}$

Definition 4.2. Let $\vec{G} = (ξ F, EJ)$ be a RFDG, where ξF is the RFS of vertices and EJ is the RFR, i.e., the rough fuzzy edge set of $\vec{G}$ . Two vertices z_i, z_j of $\vec{G}$ are said to be related vertices, i.e., z_ithicksimz_j ⇔ z_iz_i ∈ EJ and z_jz_i ∈ EJ .

Let z ∈ ξF. Then, the neighborhood of z in $\vec{G}$ , denoted by $δ_{\vec{G}} (z)$ , is defined as: $δ_{\vec{G}} (z) = {w \in ξ F : (z, w) \in EJ} .$

Definition 4.3. An information table of a RFDG $\vec{G}$ , denoted by $J (\vec{G}),$ is fuzzy information table (FIT) (Z, $F_{A_{titsc}}$ , φ, [0, 1]), where the information mapping φ is defined as φ (z_i, z_j) = EJ (z_i, z_j) if z_ithicksimz_j and 0, otherwise. Here, the set of attributes and universe set are both same.

Definition 4.4. Let $J (\vec{G})$ be FIT of $\vec{G}$ and $\tilde{A} \subseteq A_{t}$ . For z₁, z₂ ∈ Z, the relation defined by: $\begin{matrix} z_{1} ξ_{\tilde{A}} z_{2} \Leftrightarrow \underline{φ} (z_{1}, a_{i}) = \underline{φ} (z_{2}, a_{i}) \\ \bar{φ} (z_{1}, a_{i}) = \bar{φ} (z_{2}, a_{i}), \forall a_{i} \in \tilde{A}, \end{matrix}$ is an equivalence relation and called the $\tilde{A} -$ indiscernibility relation on Z.

We denote by $P_{\tilde{A}} (\vec{G})$ , the $\tilde{A} -$ indiscernibility partition of $\vec{G}$ , where $\tilde{A} \subseteq A_{t}$ .

Lemma 4.1. Let $\tilde{A} \subseteq A_{t} = V (\vec{G})$ and $z_{1}, z_{2} \in V (\vec{G})$ . Then the following conditions:

(i) $z_{1} ξ_{\tilde{A}} z_{2}$ ,

(ii) z₁thicksimw ⇔ z₂thicksimw, for $w \in \tilde{A}$ ,

(iii) $δ_{\vec{G}} (z_{1}) \cap \tilde{A} = δ_{\vec{G}} (z_{2}) \cap \tilde{A}$ , are equivalent.

The equivalence between (i) and (iii) implies the following result.

Corollary 4.1. Let $\tilde{A} \subseteq A_{t} = V (\vec{G})$ and $z \in V (\vec{G})$ , then $[z]_{\tilde{A}} = {w : δ_{\vec{G}} (z) \cap \tilde{A} = δ_{\vec{G}} (w) \cap \tilde{A}} .$

Definition 4.5. Let $P_{\tilde{A}} (\vec{G})$ be the $\tilde{A} -$ indiscernibility partition of $\vec{G}$ , where $\tilde{A} \subseteq A_{t}$ . Any equivalence class of $P_{\tilde{A}} (\vec{G})$ is called a symmetry block of $\tilde{A}$ .

The triplet $(\vec{G}, \tilde{A}, P_{\tilde{A}} (\vec{G}))$ is called $\tilde{A} -$ granular information system of $\vec{G}$ having origin $\tilde{A}$ .

Example 4.1. Let Z = {a₁, a₂, a₃, a₄, a₅} be the set of elements and attributes and ξ be an ER on Z as given in Table 5.

Table 5
Equivalence relation ξ

ξ a ₁ a ₂ a ₃ a ₄ a ₅

a ₁ 1 0 1 0 0

a ₂ 0 1 0 0 0

a ₃ 1 0 1 0 0

a ₄ 0 0 0 1 1

a ₅ 0 0 0 1 1

ξ	a ₁	a ₂	a ₃	a ₄	a ₅
a ₁	1	0	1	0	0
a ₂	0	1	0	0	0
a ₃	1	0	1	0	0
a ₄	0	0	0	1	1
a ₅	0	0	0	1	1

Let F = {(a₁, 0.8) , (a₂, 0.6) , (a₃, 0.7) , (a₄, 0.4) , (a₅, 0.9)} be a fuzzy set on Z. Let ξF be a RFS, where the LA and UA are given as: $\begin{matrix} \underline{ξ} F & = & {(a_{1}, 0.7), (a_{2}, 0.6), (a_{3}, 0.7), (a_{4}, 0.4), \\ (a_{5}, 0.4)}, \\ \bar{ξ} F & = & {(a_{1}, 0.8), (a_{2}, 0.6), (a_{3}, 0.8), (a_{4}, 0.9), \\ (a_{5}, 0.9)}, \end{matrix}$ respectively. Let J^* = {a₁a₁, a₁a₃, a₁a₄, a₂a₁, a₂a₅, a₃a₅, a₄a₂, a₄a₃, a₄a₄, a₅a₄} ⊆ Z × Z and E be an ER on J^* as given in Table 6. Let J = {(a₁a₁, 0.6) , (a₁a₃, 0.6) , (a₁a₄, 0.3) , (a₂a₁, 0.5) , (a₂a₅, 0.4) , (a₃a₅, 0.2) , (a₄a₂, 0.2) , (a₄a₃, 0.4) , (a₄a₄, 0.3) , (a₅a₄, 0.1)} be a fuzzy set on J^*. Then, the LA and UA of corresponding RFDG $\vec{G} = (ξ F, EJ)$ are shown in Figure 1.

Table 6

Equivalence relation on J^*

E	a ₁ a ₁	a ₁ a ₃	a ₁ a ₄	a ₂ a ₁	a ₂ a ₅	a ₃ a ₅	a ₄ a ₂	a ₄ a ₃	a ₄ a ₄	a ₅ a ₄
a ₁ a ₁	1	1	0	0	0	0	0	0	0	0
a ₁ a ₃	1	1	0	0	0	0	0	0	0	0
a ₁ a ₄	0	0	1	0	0	1	0	0	0	0
a ₂ a ₁	0	0	0	1	0	0	0	0	0	0
a ₂ a ₅	0	0	0	0	1	0	0	0	0	0
a ₃ a ₅	0	0	1	0	0	1	0	0	0	0
a ₄ a ₂	0	0	0	0	0	0	1	0	0	0
a ₄ a ₃	0	0	0	0	0	0	0	1	0	0
a ₄ a ₄	0	0	0	0	0	0	0	0	1	1
a ₅ a ₄	0	0	0	0	0	0	0	0	1	1

Fig.1

LA and UA of RFDG $\vec{G}$ .

The FIT $J (\vec{G})$ of $\vec{G}$ is given through the adjacency matrix $(b_{ij})_{5 \times 5} = J (\vec{G}) = (\underline{J} (\vec{G}), \bar{J} (\vec{G}))$ , where $\underline{J} (\vec{G})$ and $\bar{J} (\vec{G})$ are LA and UA of $J (\vec{G})$ , respectively. These approximations are given in Table 7.

Table 7

LA and UA of FIT, respectively

$\underline{J} (\vec{G})$	a ₁	a ₂	a ₃	a ₄	a ₅
a ₁	0.6	0	0.6	0.2	0
a ₂	0.5	0	0	0	0.4
a ₃	0	0	0	0	0.2
a ₄	0	0.2	0.4	0.1	0
a ₅	0	0	0	0.1	0

$\bar{J} (\vec{G})$	a ₁	a ₂	a ₃	a ₄	a ₅
a ₁	0.7	0	0.7	0.3	0
a ₂	0.5	0	0	0	0.4
a ₃	0	0	0	0	0.3
a ₄	0	0.2	0.4	0.3	0
a ₅	0	0	0	0.3	0

Let ${\tilde{A}}_{1} = {a_{1}, a_{2}} \subset A_{t}$ . The ${\tilde{A}}_{1} -$ indiscernibility relation $ξ_{{\tilde{A}}_{1}}^{*}$ is given as: $\begin{matrix} [a_{1}]_{{\tilde{A}}_{1}} = {a_{1}}, [a_{2}]_{{\tilde{A}}_{1}} = {a_{2}}, [a_{3}]_{{\tilde{A}}_{1}} = {a_{3}, a_{5}}, \\ [a_{4}]_{{\tilde{A}}_{1}} = {a_{4}}, [a_{5}]_{{\tilde{A}}_{1}} = {a_{3}, a_{5}} . \end{matrix}$ Note that, ${\tilde{A}}_{1} -$ indiscernibility partition $P_{{\tilde{A}}_{1}} (\vec{G})$ of $\vec{G}$ is given as follows: $P_{{\tilde{A}}_{1}} (\vec{G}) = {{a_{1}}, {a_{2}}, {a_{3}, a_{5}}, {a_{4}}},$ where {a₁} , {a₂} , {a₃, a₅} , {a₄} are the symmetry blocks of ${\tilde{A}}_{1}$ and the triplet ({a₁, a₂, a₃, a₄, a₅}, {a₁, a₂}, {{a₁} , {a₂} , {a₃, a₅} , {a₄}}) is the ${\tilde{A}}_{1}$ -granular information system of $\vec{G}$ having origin ${\tilde{A}}_{1}$ . By considering the different subsets of attributes, distinct partitions of Z can be obtained and each partition corresponds to a single level of granularity. The elements containing in same block are equivalent or indiscernible elements.

4.1 Automorphisms in RFDGs and indiscernibility relation

In this subsection, we discuss some general geometric properties of RFDGs and define certain mappings based on indiscernibility relations.

Definition 4.6.Let $\vec{G} = (Z, ξ F, EJ)$ be a RFDG. A bijective mapping ψ : Z × Z defined as:

$\begin{matrix} (u, w) \in EJ & \Leftrightarrow & (ψ (u), ψ (w)) \in EJ, \end{matrix}$ for all (u, w) ∈ Z × Z, is called an automorphism of $\vec{G}$ .

Definition 4.7. Let $\vec{G} = (Z, ξ F, EJ)$ be a RFDG and let H ⊆ Z. Then, a rough fuzzy directed subgraph generated by H(RFDSG generated by H) is an 5-ordered tuple (H, ξ_H, ξ_HM, E_H, E_HK), denoted by $\vec{G}$ , such that:

(i) H ⊆ Z,

(ii) ξ_H is an ER on H such that z_iz_j ∈ ξ_H ⇔ z_iz_j ∈ ξ,

(iii) ξ_HM is a RFS on H, where M is fuzzy set on H such that M (z_i) ≤ F (z_i), for all z_i ∈ H,

(iv) E_H is an ER on K^* ⊆ H × H such that (z_iz_j, z_kz_l) ∈ E_H ⇔ (z_iz_j, z_kz_l) ∈ E,

(v) E_HK is a RFR on K^*, where K is a fuzzy set on K^* such that K (z_iz_j) ≤ J (z_iz_j).

Then, the ordered pair $\vec{G} [H] = (\underline{\vec{G}} [H], \bar{\vec{G}} [H])$ is RFDSG generated by H, where $\underline{\vec{G}} [H] = (H, {\underline{ξ}}_{H} M, {\underline{E}}_{H} K)$ and $\bar{\vec{G}} [H] = (H, {\bar{ξ}}_{H} M, {\bar{E}}_{H} K)$ are called LA an UA of $\vec{G} [H]$ , respectively.

Definition 4.8. Let $\tilde{A}$ be a fixed subset of vertices of $\vec{G}$ and u, w be two fixed vertices of $\vec{G}$ . Let $H = \tilde{A} \cup {u, w}$ , then we define a mapping $χ_{(u, w)}^{\tilde{A}} : H \to H$ as follows:

$χ_{(u, w)}^{\tilde{A}} (z) = {\begin{matrix} z, & if z \in \tilde{A} \ {u, w}, \\ u, & if z = w, \\ w, & if z = u, \end{matrix}$

Example 4.2. Let $\vec{G} = (Z, ξ F, EJ)$ be a RFDG as shown in Figure 1, where Z = {a₁, a₂, a₃, a₄, a₅}. Let H = {a₁, a₂, a₃} and ξ_H be an ER on H as given in Table 8.

Table 8
ER ξ_H on H

ξ _H	a ₁	a ₂	a ₃
a ₁	1	0	1
a ₂	0	1	0
a ₃	1	0	1

Let M = {(a₁, 0.6) , (a₂, 0.5) , (a₃, 0.6)} be a fuzzy set on H. Then, the LA and UA of ξ_HM are given as: $\begin{matrix} {\underline{ξ}}_{H} M & = & {(a_{1}, 0.6), (a_{2}, 0.5), (a_{3}, 0.6)}, \\ {\underline{ξ}}_{H} M & = & {(a_{1}, 0.6), (a_{2}, 0.5), (a_{3}, 0.6)} . \end{matrix}$

Let K^* = {a₁a₁, a₁a₃, a₂a₁} ⊆ H × H and K = {(a₁a₁, 0.4) , (a₁a₃, 0.5) , (a₂a₁, 0.3)} be a fuzzy set on K^*. The ER E_H on K^* is given in Table 9.

Table 9

ER E_H on H

E _H	a ₁ a ₁	a ₁ a ₃	a ₂ a ₁
a ₁ a ₁	1	1	0
a ₁ a ₃	1	1	0
a ₂ a ₁	0	0	1

Then, the LA and UA of RFR E_HK is given as follows: $\begin{matrix} {\underline{E}}_{H} M & = & {(a_{1} a_{1}, 0.4), (a_{1} a_{3}, 0.4), (a_{2} a_{1}, 0.3)}, \\ {\bar{E}}_{H} M & = & {(a_{1} a_{1}, 0.5), (a_{1} a_{3}, 0.5), (a_{2} a_{1}, 0.3)} . \end{matrix}$

The LA and UA of corresponding RFSDG generated by H are given in Figure 4(a) and 4(b).

Fig.2

LA and UA of $\vec{G}$ .

We now prove that if unotsimw or $\tilde{A} \cap {u, w} = \emptyset$ , then u and w are $\tilde{A} -$ indiscernible if and only if $\tilde{A}$ is symmetry axis.

Theorem 4.1. Let $\vec{G} = (Z, ξ F, EJ)$ be a RFDG such that RFR EJ is symmetric on Z, i.e., z_i ∼ z_j ⇔ z_j ∼ z_i, $\tilde{A} \subseteq Z$ and x₁, x₂ be distinct fixed vertices of $\vec{G}$ . Let $H = \tilde{A} \cup {x_{1}, x_{2}}$ , if x₁notsimx₂ or $\tilde{A} \cap {x_{1}, x_{2}} = \emptyset$ , then we have: $x_{1} ξ_{\tilde{A}} x_{2} \Leftrightarrow χ_{(x_{1}, x_{2})}^{\tilde{A}} \in aut (\vec{G} [H])$ .

Proof.

(i). First we prove that $x_{1} ξ_{\tilde{A}} x_{2} \Rightarrow χ_{(x_{1}, x_{2})}^{\tilde{A}} \in aut (\vec{G} [H])$ . Assume that $x_{1} ξ_{\tilde{A}} x_{2}$ .

From Definition 4.1, we observe that $χ_{(x_{1}, x_{2})}^{\tilde{A}}$ is bijective mapping. Therefore, we have only to prove that $z_{1} \sim z_{2} \Leftrightarrow χ_{(z_{1}, z_{2})}^{\tilde{A}} (z_{1}) \sim χ_{(z_{1}, z_{2})}^{\tilde{A}} (z_{2})$ . Let z₁, z₂ ∈ H. Note that, if $z_{1}, z_{2} \in \tilde{A} \ {z_{1}, z_{2}}$ , then the definition of $χ_{(z_{1}, z_{2})}^{\tilde{A}}$ implies that $χ_{(z_{1}, z_{2})}^{\tilde{A}} (z_{1}) = z_{1}$ and $χ_{(z_{1}, z_{2})}^{\tilde{A}} (z_{2}) = z_{2}$ . Hence, in this case the result is proved. Now, we assume that {z₁, z₂}∩ {x₁, x₂} ≠ ∅. Then, two cases arise here:

z₁ ∈ {x₁, x₂}, z₂ ∉ {x₁, x₂},

z₂ ∈ {x₁, x₂}, z₁ ∉ {x₁, x₂}.

We first suppose that z₁ ∈ {x₁, x₂}, z₂ ∉ {x₁, x₂}, (as the second case z₂ ∈ {x₁, x₂}, z₁ ∉ {x₁, x₂} is similar to prove). WLOG, we suppose that x₁ = z₁. Since, z₂ ∈ H and z₂ ∉ {x₁, x₂}, this implies that

z_{2} \in \tilde{A}

, then condition (ii) of Theorem 4 implies that x₁ ∼ z₂ ⇔ x₂ ∼ z₂. Since, z₂ ∉ {x₁, x₂}, we deduce that

χ_{(z_{1}, z_{2})}^{\tilde{A}} (z_{2}) = z_{2}

and this implies that x₁ ∼ z₂ ⇔ x₂ ∼ z₂ is equivalent to

z_{1} \sim z_{2} \Leftrightarrow χ_{(z_{1}, z_{2})}^{\tilde{A}} (z_{1}) \sim χ_{(z_{1}, z_{2})}^{\tilde{A}} (z_{2})

The other condition, which we have to examine, is {z₁, z₂} = {x₁, x₂}. WLOG, we assume that z₁ = x₁ and z₂ = x₂, the Definition 4.1 implies that $χ_{(z_{1}, z_{2})}^{\tilde{A}} (z_{1}) = x_{2}$ and $χ_{(z_{1}, z_{2})}^{\tilde{A}} (z_{2}) = x_{1}$ . Since, the relation ∼ is symmetric on $\vec{G}$ , we obtain $z_{1} \sim z_{2} \Leftrightarrow χ_{(z_{1}, z_{2})}^{\tilde{A}} (z_{1}) \sim χ_{(z_{1}, z_{2})}^{\tilde{A}} (z_{2})$ . Hence, the result is proved.

(ii). We now suppose that $χ_{(x_{1}, x_{2})}^{\tilde{A}} \in aut (\vec{G} [H])$ and prove that a ∼ x₁ ⇔ a ∼ x₂, for all $a \in \tilde{A}$ . Note that, a ∼ x₁ ⇔ a ∼ x₂ is equivalent to prove that $x_{1} ξ_{\tilde{A}} x_{2}$ . Let $A \in \tilde{A}$ and a ≠ {x₁, x₂}. Then, Definition 4.1 implies that $χ_{(x_{1}, x_{2})}^{\tilde{A}} (a) = a$ . From $χ_{(x_{1}, x_{2})}^{\tilde{A}} (a) = a$ and $χ_{(x_{1}, x_{2})}^{\tilde{A}} \in aut (\vec{G} [H])$ , we deduce that $x_{1} \sim a \Leftrightarrow χ_{(x_{1}, x_{2})}^{\tilde{A}} (x_{1}) \sim χ_{(x_{1}, x_{2})}^{\tilde{A}} (a) \Leftrightarrow x_{2} \sim a$ , which completes the proof.

Now, if $\tilde{A} \cap {x_{1}, x_{2}} = \emptyset$ , since a ∈ A, then a ∉ {x₁, x₂}, the result is proved trivially. Let x₁notsimx₂ and a ∈ {x₁, x₂}. WLOG, suppose that a = x₁, since RFR on $\vec{G}$ is symmetric, then (a = x₁) notsimx₂ and a ∼ x₁ ⇔ a ∼ x₂ is satisfied. Hence, the result is proved. □

Fig.3

LA and UA of $\vec{G}$ .

Example 4.3. Let $\vec{G} = (Z, ξ F, EJ)$ be a RFDG as shown in Figure 3 such that RFR EJ is symmetric on Z = {x₁, x₂, x₃, x₄}. Let $\tilde{A} = {x_{2}, x_{4}} \subseteq Z$ such that x₂ ≠ x₄. Let x₁, x₃ be distinct fixed vertices of $\vec{G}$ such that $H = {x_{2}, x_{4}} \cup {x_{1}, x_{3}} = \tilde{A} \cup {x_{1}, x_{3}}$ . Since, $\tilde{A} \cap {x_{1}, x_{3}} = \emptyset$ . Note that, $\underline{φ} (x_{1}, a_{i}) = \underline{φ} (x_{3}, a_{i})$ and $\bar{φ} (x_{1}, a_{i}) = \bar{φ} (x_{3}, a_{i})$ , ∀ a_i ∈ Z, i.e., $\begin{matrix} \underline{φ} (x_{1}, x_{1}) = \underline{φ} (x_{3}, x_{1}) = 0.3, \\ \underline{φ} (x_{1}, x_{2}) = \underline{φ} (x_{3}, x_{2}) = 0, \\ \underline{φ} (x_{1}, x_{3}) = \underline{φ} (x_{3}, x_{3}) = 0.3, \\ \underline{φ} (x_{1}, x_{4}) = \underline{φ} (x_{3}, x_{4}) = 0, \\ \bar{φ} (x_{1}, x_{1}) = \bar{φ} (x_{3}, x_{1}) = 0.4, \\ \bar{φ} (x_{1}, x_{2}) = \bar{φ} (x_{3}, x_{2}) = 0, \\ \bar{φ} (x_{1}, x_{3}) = \bar{φ} (x_{3}, x_{3}) = 0.4, \\ \bar{φ} (x_{1}, x_{4}) = \bar{φ} (x_{3}, x_{4}) = 0 . \end{matrix}$

Thus, we have x₁ is equivalent to x₂ with respect to $ξ_{\tilde{A}}$ , i.e., $x_{1} ξ_{\tilde{A}} x_{3}$ . Let {ψ₁, ψ₂} be two distinct automorphisms, as defined in Definition 4.6, of $\vec{G} [H] = \vec{G}$ such that, $\begin{matrix} ψ_{1} (x_{1}) = x_{3}, ψ_{1} (x_{2}) = x_{4}, ψ_{1} (x_{3}) = x_{1}, \\ ψ_{1} (x_{4}) = x_{2}, \end{matrix}$ $\begin{matrix} ψ_{2} (x_{1}) = x_{3}, ψ_{2} (x_{2}) = x_{2}, ψ_{2} (x_{3}) = x_{1}, \\ ψ_{2} (x_{4}) = x_{4} . \end{matrix}$

We now determine the mapping $χ_{(x_{1}, x_{3})}^{\tilde{A}} : H \to H$ following the Definition 4.8, which is given as:

$\begin{matrix} χ_{(x_{1}, x_{3})}^{\tilde{A}} (x_{1}) = x_{3}, χ_{(x_{1}, x_{3})}^{\tilde{A}} (x_{2}) = x_{2}, \\ χ_{(x_{1}, x_{3})}^{\tilde{A}} (x_{3}) = x_{1}, χ_{(x_{1}, x_{3})}^{\tilde{A}} (x_{4}) = x_{4} . \end{matrix}$

Thus, we have $χ_{(x_{1}, x_{3})}^{\tilde{A}} = ψ_{2}$ , i.e., $χ_{(x_{1}, x_{3})}^{\tilde{A}} \in aut (\vec{G} [H])$ . Hence, the result of Theorem 4 is proved.

Theorem 4.2. Let $\vec{G} = (Z, ξ F, EJ)$ be a RFDG such that RFR EJ is symmetric on Z, i.e., z_i ∼ z_j ⇔ z_j ∼ z_i and $(\vec{G}$ , $\tilde{A}$ , $P_{\tilde{A}} (\vec{G}))$ be the $\tilde{A} -$ granular information system of $\vec{G}$ having origin $\tilde{A}$ . If $ω \in aut (\vec{G})$ and $P_{\tilde{A}} (\vec{G}) = {S_{1}, S_{2}, \dots, S_{m}},$ then $P_{ω (\tilde{A})} (\vec{G}) = {ω (S_{1}), ω (S_{2}), \dots, ω (S_{m})} .$

Proof. Let P = {ω (S₁) , ω (S₂) , ⋯ , ω (S_m)} and $P_{ω (\tilde{A})} (\vec{G}) = {D_{1}, D_{2}, \dots, D_{m}} .$ Note that, the set {ω (S₁) , ω (S₂) , ⋯ , ω (S_m)} defines a partition of Z because ω is bijective and {S₁, S₂, ⋯ , S_m} is a partition of Z. We now prove that {ω (S₁) , ω (S₂) , ⋯ , ω (S_m)} = {D₁, D₂, ⋯ , D_m} . Since, P and $P_{ω (\tilde{A})} (\vec{G})$ both define partitions on Z, thus to prove our result, we have to show that for all D_k there exists some S_l such that ω (S_l) = D_k. Let D_k be a general symmetric block of $P_{ω (\tilde{A})} (\vec{G})$ and d be a fixed element of D_k. Thus, we have $[d]_{ω (\tilde{A})} = D_{k}$ . Let $g \in V (\vec{G}) = Z$ such that ω (g) = d. Then, there must exist a symmetry block S_l such that g ∈ S_l, because {S₁, S₂, ⋯ , S_m} , is a partition of $V (\vec{G})$ . Now, we prove that $[d]_{ω (\tilde{A})} = ω (S_{l})$ .

The condition (ii) of Theorem 4.1 implies that, for all $d^{*} \in V (\vec{G})$ , we have: $d ξ_{ω (\tilde{A})} d^{*} \Leftrightarrow d \sim a \Leftrightarrow d^{*} \sim a,$ for all $a \in ω (\tilde{A})$ . Since, $ω \in aut (\vec{G})$ , then the condition $d ξ_{ω (\tilde{A})} d^{*} \Leftrightarrow d \sim a \Leftrightarrow d^{*} \sim a$ is equivalent to the following condition $d ξ_{ω (\tilde{A})} d^{*} \Leftrightarrow ω^{- 1} (d) \sim b \Leftrightarrow ω^{- 1} (d^{*}) \sim b,$ for all $b \in \tilde{A}$ . Thus, we have $d ξ_{ω (\tilde{A})} d^{*} \Leftrightarrow ω^{- 1} (d) ξ_{ω (\tilde{A})} ω^{- 1} (d^{*}) .$

Let $d^{*} \in [d]_{ω (\tilde{A})}$ and $g^{*} \in V (\vec{G})$ such that ω (g^*) = d^*. Since, $d ξ_{ω (\tilde{A})} d^{*}$ , therefore we have $g = ω^{- 1} (d) ξ_{ω (\tilde{A})} ω^{- 1} (d^{*}) = g^{*}$ . But g ∈ S_l and S_l is an indiscernibility block of $ξ_{\tilde{A}}$ , therefore, we also have g^* ∈ S_l. Hence, d^* = ω (g^*) ∈ ω (S_l) and $[d]_{ω (\tilde{A})} \subseteq ω (S_{l}) .$

To prove the other side, let d^* ∈ ω (S_l). Then, there is an element g^* ∈ S_l such that ω (g^*) = d^*. Since, g, g^* ∈ S_l, we have $ω^{- 1} (d) = g ξ_{\tilde{A}} ω^{- 1} (d^{*}) = g^{*}$ , which implies that $d ξ_{ω (\tilde{A})} d^{*}$ , i.e., $d^{*} \in [d]_{ω (\tilde{A})} .$ This implies that $ω (S_{l}) \subseteq [d]_{ω (\tilde{A})}$ , which proves the result. □

Example 4.4. Let $\vec{G} = (Z, ξ F, EJ)$ be a RFDG as shown in Figure 1 such that RFR EJ is symmetric on Z = {x₁, x₂, x₃, x₄}.

Fig.4

LA and UA of $\vec{G}$ .

Let $\tilde{A} = {x_{1}, x_{3}}$ .

We now determine the $\tilde{A} -$ granular information system of $\vec{G}$ . Note that:

$\begin{matrix} \underline{φ} (x_{1}, x_{1}) = \underline{φ} (x_{2}, x_{1}) = 0.3, \\ \underline{φ} (x_{1}, x_{3}) = \underline{φ} (x_{2}, x_{3}) = 0 . \end{matrix}$

That is $x_{2} \in [x_{1}]_{\tilde{A}}$ . Similarly, we have:

$\begin{matrix} \underline{φ} (x_{1}, x_{1}) \neq \underline{φ} (x_{3}, x_{1}), \bar{φ} (x_{1}, x_{1}) \neq \bar{φ} (x_{3}, x_{1}), \\ \underline{φ} (x_{1}, x_{3}) \neq \underline{φ} (x_{3}, x_{3}), \bar{φ} (x_{1}, x_{3}) \neq \bar{φ} (x_{3}, x_{3}), \\ \underline{φ} (x_{1}, x_{1}) = \underline{φ} (x_{4}, x_{1}) = 0.3, \\ \bar{φ} (x_{1}, x_{1}) = \bar{φ} (x_{4}, x_{1}) = 0.4, \\ \underline{φ} (x_{1}, x_{3}) = \underline{φ} (x_{4}, x_{3}) = 0, \\ \bar{φ} (x_{1}, x_{3}) = \bar{φ} (x_{4}, x_{3}) = 0, \\ \underline{φ} (x_{2}, x_{1}) = \underline{φ} (x_{1}, x_{1}) = 0.3, \\ \bar{φ} (x_{2}, x_{1}) = \bar{φ} (x_{1}, x_{1}) = 0.4 \\ \underline{φ} (x_{2}, x_{3}) = \underline{φ} (x_{1}, x_{3}) = 0, \\ \bar{φ} (x_{2}, x_{3}) = \bar{φ} (x_{1}, x_{3}) = 0, \\ \underline{φ} (x_{2}, x_{1}) \neq \underline{φ} (x_{3}, x_{1}), \bar{φ} (x_{2}, x_{1}) \neq \bar{φ} (x_{3}, x_{1}), \\ \underline{φ} (x_{2}, x_{3}) = \underline{φ} (x_{3}, x_{3}) = 0, \\ \bar{φ} (x_{2}, x_{3}) = \bar{φ} (x_{3}, x_{3}) = 0, \\ \underline{φ} (x_{2}, x_{1}) = \underline{φ} (x_{4}, x_{1}) = 0.3, \\ \bar{φ} (x_{2}, x_{1}) = \bar{φ} (x_{4}, x_{1}) = 0.4, \\ \underline{φ} (x_{2}, x_{3}) = \underline{φ} (x_{4}, x_{3}) = 0, \\ \bar{φ} (x_{2}, x_{3}) = \bar{φ} (x_{4}, x_{3}) = 0, \end{matrix}$

$\begin{matrix} \underline{φ} (x_{3}, x_{1}) \neq \underline{φ} (x_{1}, x_{1}), \bar{φ} (x_{3}, x_{1}) \neq \bar{φ} (x_{1}, x_{1}), \\ \underline{φ} (x_{3}, x_{3}) = \underline{φ} (x_{1}, x_{3}) = 0, \\ \bar{φ} (x_{3}, x_{3}) = \bar{φ} (x_{1}, x_{3}) = 0, \\ \underline{φ} (x_{3}, x_{1}) \neq \underline{φ} (x_{2}, x_{1}), \bar{φ} (x_{3}, x_{1}) \neq \bar{φ} (x_{2}, x_{1}), \\ \underline{φ} (x_{3}, x_{3}) = \underline{φ} (x_{2}, x_{1}) = 0, \\ \bar{φ} (x_{3}, x_{3}) = \bar{φ} (x_{2}, x_{1}) = 0, \\ \underline{φ} (x_{3}, x_{1}) \neq \underline{φ} (x_{4}, x_{1}), \bar{φ} (x_{3}, x_{1}) \neq \bar{φ} (x_{4}, x_{1}), \\ \underline{φ} (x_{3}, x_{3}) = \underline{φ} (x_{4}, x_{3}) = 0, \\ \bar{φ} (x_{3}, x_{3}) = \bar{φ} (x_{4}, x_{3}) = 0 . \end{matrix}$

From above calculations, we deduce that $[x_{1}]_{\tilde{A}} = {x_{1}, x_{2}, x_{4}}$ , $[x_{2}]_{\tilde{A}} = {x_{1}, x_{2}, x_{4}}$ , $[x_{3}]_{\tilde{A}} = {x_{3}}$ , $[x_{4}]_{\tilde{A}} = {x_{1}, x_{2}, x_{4}}$ and

$\begin{matrix} P_{\tilde{A}} (\vec{G}) & = & {S_{1}, S_{2}, S_{3}, S_{4}} \\ = & {{x_{1}, x_{2}, x_{4}}, {x_{1}, x_{2}, x_{4}}, {x_{3}}, \\ {x_{1}, x_{2}, x_{4}}} . \end{matrix}$

Suppose that $ω \in aut (\vec{G})$ defined by:

$ω (x_{1}) = x_{1}, ω (x_{2}) = x_{2}, ω (x_{3}) = x_{3}, ω (x_{4}) = x_{4} .$

Then, we have $ω (\tilde{A}) = {x_{1}, x_{3}}$ . Also, we obtain

$\begin{matrix} P_{ω (\tilde{A})} (\vec{G}) & = & {{x_{1}, x_{2}, x_{4}}, {x_{1}, x_{2}, x_{4}}, {x_{3}}, \\ {x_{1}, x_{2}, x_{4}}}, \end{matrix}$

where ω (S₁) = {x₁, x₂, x₄}, ω (S₂) = {x₁, x₂, x₄}, ω (S₃) = {x₃}, and ω (S₄) = {x₁, x₂, x₄}. Thus, we have

$P_{ω (\tilde{A})} (\vec{G}) = {ω (S_{1}), ω (S_{2}), ω (S_{3}), ω (S_{4})} .$ Hence, the result of Theorem 4.2 is proved.

5 Rough fuzzy information granulation of certain social networks

In the granulation of social networks, social structures are discussed through graphs (or digraphs). The vertices of a graph represent the elements or members of some network and these vertices are connected through an edge if they have some relationship between them. In this section, we study the granulation and discuss some indiscernible relations of certain social networks using RFDGs.

5.1 Granulation of social network of elite families

In the field of social network analysis, we study the social structures through graphs or digraphs. In this subsection, we study the social network concerning the marriage ties of certain noble families using RFDG $\vec{G}$ . This network is briefly studied through simple graphs in [13], we generalize the granulation of these families through rough fuzzy approximations. Let $\vec{G} = (\underline{\vec{G}}$ , $\bar{\vec{G}})$ be a RFDG, where $\underline{\vec{G}} = (Z, \underline{ξ} F, \underline{E} J)$ and $\bar{\vec{G}} = (Z, \bar{ξ} F, \bar{E} J)$ are the LA and UA of $\vec{G}$ , respectively. Let Z = {n₁, n₂, n₃, n₄, n₅, n₆, n₇, n₈, n₉, n₁₀, n₁₁, n₁₂, n₁₃, n₁₄, n₁₅, n₁₆} be the set of crisp vertices of $\vec{G}$ representing the noble families of Florentine of 15th century. These two vertices or families are linked together through an edge if they both have a marriage tie between them. The LA and UA of the corresponding social network are shown in Figures 2 and 3, respectively. Here, n₁: Acciaiuoli, n₂: Albizzi, n₃: Barbadori, n₄: Bischeri, n₅: Castellani, n₆: Ginori, n₇: Guadagni, n₈: Lamberteschi, n₉: Medici, n₁₀: Pazzi, n₁₁: Peruzzi, n₁₂: Pucci, n₁₃: Ridolfi, n₁₄: Salviati, n₁₅: Strozzi and n₁₆: Tornabuoni are the elite families of Florence in 15th century. Let $\tilde{A}$ ={Medici, Ridolfi, Tornabuoni}, i.e., {n₉, n₁₃, n₁₆}. Then, the equivalence relation $ξ_{\tilde{A}}$ is given as:

Fig.5

LA of social network $\vec{G}$ .

Fig.6

UA of social network $\vec{G}$ .

$\begin{matrix} [n_{1}]_{ξ_{\tilde{A}}} = [n_{2}]_{ξ_{\tilde{A}}} = [n_{3}]_{ξ_{\tilde{A}}} = [n_{14}]_{ξ_{\tilde{A}}} = {n_{1}, n_{2}, \\ n_{3}, n_{14}}, \\ [n_{4}]_{ξ_{\tilde{A}}} = [n_{5}]_{ξ_{\tilde{A}}} = [n_{6}]_{ξ_{\tilde{A}}} = [n_{8}]_{ξ_{\tilde{A}}} = [n_{10}]_{ξ_{\tilde{A}}} = \\ [n_{11}]_{ξ_{\tilde{A}}} = [n_{12}]_{ξ_{\tilde{A}}} = {n_{4}, n_{5}, n_{6}, n_{8}, n_{10}, n_{11}, \\ n_{12}}, \\ [n_{7}]_{ξ_{\tilde{A}}} = {n_{7}}, [n_{9}]_{ξ_{\tilde{A}}} = {n_{9}}, [n_{13}]_{ξ_{\tilde{A}}} = {n_{13}}, \\ [n_{15}]_{ξ_{\tilde{A}}} = {n_{15}}, [n_{16}]_{ξ_{\tilde{A}}} = {n_{16}} . \end{matrix}$ Since, FIT of $\vec{G}$ is a matrix of order 16 × 16, therefore we are unable to show these information in tabular form. This relation can be determined directly from Figures 2 and 3. Hence, $\tilde{A} -$ indiscernibility partition of $\vec{G}$ is given as: $P_{\tilde{A}} (\vec{G}) = {{n_{1}, n_{2}, n_{3}, n_{14}}, {n_{4}, n_{5}, n_{6}, n_{8}, n_{10}, n_{11}, n_{12}}, {n_{7}}, {n_{9}}, {n_{13}}, {n_{15}}, {n_{16}}} .$ The symmetry block {n₁, n₂, n₃, n₁₄} shows that these are the families, which have a marriage tie with Medici family but no relationship with any other member of Ridolfi or Tornabuoni. Similarly, {n₄, n₅, n₆, n₈, n₁₀, n₁₁, n₁₂} is the block of those families, which have no marriage tie with any of the families of Medici, Ridolfi or Tornabuoni. All other blocks also possess the symmetric properties with respect to the attribute set $\tilde{A}$ . The LA of RFDG describes the minimum strength and UA shows the maximum strength of the ties of corresponding families. Thus, by considering the RFDG representation of this social network, we not only see the interactions between families but also the strength of their relationships. The procedure adopted in this granulation of social networking is given in Algorithm 1.

Algorithm 1

Granulation of social network and indiscernibility relations

Input: A RFDG

\vec{G}

representing the relations between elite families.

Output: Granulation of social network and indiscernibility relations between these families.

Begin

1 . Input the set Z of families n₁, n₂, ⋯ , u_m.

2 . Input an equivalence relation ξ on Z.

3 . Input the set S of relations s₁, s₂, ⋯ , s_r, where s_k = n_in_j, for i, j ∈ {1, 2, ⋯ , m} .

4 . Input an ER E on S ⊆ Z × Z.

5 . dol from 1 to m

6 . read*, F (n_l)

7 . end do

8 . dol from 1 to m

9 .

\bar{ξ} F (n_{l}) = 0

10 .

\underline{ξ} F (n_{l}) = 0

11 . dop from 1 to m

12 .

\underline{U} = max {1 - ξ (n_{l}, n_{p}), F (n_{l})}

13 .

\bar{U} = min {ξ (n_{l}, n_{p}), F (n_{l})}

14 .

\underline{ξ} F (n_{l}) = min {\underline{ξ} F (n_{l}), \underline{U}}

15 .

\bar{ξ} F (n_{l}) = max {\bar{ξ} F (n_{l}), \bar{U}}

16 . end do

17 . end do

18 dol from 1 to r

19 . read*, J (s_l)

20 . end do

21 . dol from 1 to r

22 .

\underline{E} J (s_{l})

= 0

23 .

\bar{E} J (s_{l})

= 0

24 . dop from 1 to r

25 .

\underline{T} = max {1 - E (s_{l}, s_{p}), J (s_{l})}

26 .

\bar{T} = min {E (s_{l}, s_{p}), F (s_{l})}

27 .

\underline{E} J (s_{l}) = min {\underline{E} J (s_{l}), \underline{T}}

28 .

\bar{E} J (s_{l}) = max {\bar{E} J (s_{l}), \bar{T}}

29 . end do

30 . end do

31 . Determine the FIT

J (\vec{G})

\vec{G}

as:

32 . dol from 1 to m

33 . dop from 1 to m

34 . if

(n_{l} n_{p} \in \underline{E} J, n_{l} n_{p} \in \bar{E} J)

then

35 . n_lthicksimn_p

36 . end if

37 . if (n_lthicksimn_p) then

38 .

\underline{φ} (n_{l} n_{p}) = \underline{E} J (n_{l}, n_{p})

39 .

\bar{φ} (n_{l} n_{p}) = \bar{E} J (n_{l}, n_{p})

40 . print*,

φ (n_{l} n_{p}) = (\underline{φ} (n_{l} n_{p}), \bar{φ} (n_{l} n_{p}))

41 . else

42 . print*, φ (n_ln_p) =0

43 . end if

44 . print*, [φ] _m×m is FIT

J (\vec{G})

\vec{G}

45 . Fix a subset

\tilde{A}

of attributes a₁, a₂, ⋯ , a_m such that

\tilde{A} \subseteq Z

46 . doq from 1 to m

47 . if

(\underline{φ} (n_{l}, a_{q}) = \underline{φ} (n_{p}, a_{q}), \bar{φ} (n_{l}, a_{q}) = \bar{φ} (n_{p}, a_{q}))

then

48 .

n_{l} ξ_{\tilde{A}} n_{p}

49 . print*,

ξ_{\tilde{A}}

is an

\tilde{A} -

indiscernibility relation on Z.

50 . end if

51 . end do

52 . end do

53 . end do

54 . The symmetry blocks obtained from

\tilde{A} -

indiscernibility relation

ξ_{\tilde{A}}

contain the families,

which have the identical relationships with the families of

\tilde{A}

End

5.2 Indiscernible relations in an organization management structure

In an organization structure, there are different stakeholder, including directors, managers, officers and other employees, who lead a company to fulfill its business objectives. Here, we consider the management structure of a business organization and represent this structure through the LA and UA of a RFDG $\vec{G} = (Z, ξ F, EJ)$ . The corresponding approximations are shown in Figures 5 and 6. Note that, RFR is not symmetric in this case as if v_iv_j ∈ E, it is not necessary that v_jv_i ∈ E. The directed edge shows the influence of an employee on his staff member and the membership degrees of vertices represent the contribution of corresponding worker to fulfil the objectives of organization. The minimum contribution and overall influence is given by LA and maximum contribution and overall influence is given by UA. We now wish to determine the relationships between some fixed members of this organization rather than the whole staff. For this purpose, we fix the set of those elements among which we want to see the relations and then generate a subgraph of $\vec{G}$ .

Fig.7

Minimum influence of employees on their staff.

Fig.8

Maximum influence of employees on their staff.

Here, the set of universe Z = {m₁, m₂, m₃, ⋯ , m₂₂} represent the members of a business organization and these members are connected through directed edges according to the relations between them. Different designations are shown by distinct shapes of vertices, i.e., the oval vertices represent the CEOs, circular vertices show officers under CEOs, rectangles represent the managers of the company and square vertices represent the teams working under managers. We skip to show the FIT $J (\vec{G})$ of $\vec{G}$ due to the lengthy matrix of order 22 × 22. Let H = {m₁, m₄, m₃, m₇, m₉, m₁₃, m₁₄, m₁₇, m₁₉, m₂₂} be a fixed set of vertices of $\vec{G}$ . Let ξ_H be an ER on H and $\begin{matrix} M & = & {(m_{1}, 0.4), (m_{4}, 0.6), (m_{3}, 0.5), (m_{7}, 0.5), \\ (m_{9}, 0.6), (m_{13}, 0.7), (m_{14}, 0.6), (m_{17}, 0.5), \\ (m_{19}, 0.6), (m_{22}, 0.6)} \end{matrix}$ be a fuzzy set on H. Let K^* = {m₁m₄, m₁m₃, m₄m₉, m₇m₁₃, m₉m₁₇, m₁₃m₁₉} ⊆ H × H such that m_im_j ∈ K^* ⇔ m_im_j ∈ J^* and E_H be an ER on K^*. Then, the RFSDG generated by H, $\vec{G}$ is given in Figures 7 and 8.

Fig.9

LA of $\vec{G} [H]$ .

Note that, m₂₂ and m₁₄ possess no relationships with any member of the fixed set. Similarly, the UA is given in the following figure.

Fig.10

UA of $\vec{G} [H]$ .

Similarly, by fixing any other set of vertices, we can determine the relationships between the corresponding members of an organization. The procedure adopted in the granulation of our network is described in Algorithm 2.

Algorithm 2

Generation of subgraphs

Input: A RFDG

\vec{G}

representing the management structure of a business organization.

Output: Generated subgraph of

\vec{G}

to represent the relations among some fixed members

of management team.

Begin

1 . Input the set Z of employees m₁, m₂, ⋯ , m_n.

2 . Input an equivalence relation ξ on Z.

3 . Input the set R of relations r₁, r₂, ⋯ , r_s, where r_k = m_im_j, for i, j ∈ {1, 2, ⋯ , n} .

4 . Input an ER E on R ⊆ Z × Z.

5 . dol from 1 to n

6 . read*, F (n_l)

7 . end do

8 . dol from 1 to n

9 .

\bar{ξ} F (m_{l}) = 0

10 .

\underline{ξ} F (m_{l}) = 0

11 . dop from 1 to n

12 .

\underline{U} = max {1 - ξ (m_{l}, m_{p}), F (m_{l})}

13 .

\bar{U} = min {ξ (m_{l}, m_{p}), F (m_{l})}

14 .

\underline{ξ} F (m_{l}) = min {\underline{ξ} F (m_{l}), \underline{U}}

15 .

\bar{ξ} F (m_{l}) = max {\bar{ξ} F (m_{l}), \bar{U}}

16 . end do

17 . end do

18 dol from 1 to s

19 . read*, J (r_l)

20 . end do

21 . dol from 1 to s

22 .

\underline{E} J (r_{l})

= 0

23 .

\bar{E} J (r_{l})

= 0

24 . dop from 1 to s

25 .

\underline{T} = max {1 - E (r_{l}, r_{p}), J (r_{l})}

26 .

\bar{T} = min {E (r_{l}, r_{p}), F (r_{l})}

27 .

\underline{E} J (r_{l}) = min {\underline{E} J (r_{l}), \underline{T}}

28 .

\bar{E} J (r_{l}) = max {\bar{E} J (r_{l}), \bar{T}}

29 . end do

30 . end do

31 . Fix a set H of vertices m₁, m₂,⋯, m_k, where k ≤ n.

32 . doi from 1 to k

33 . doj from 1 to k

34 . if (m_im_j ∈ ξ) then

35 . m_im_j ∈ ξ_H

36 . print*, ξ_H is an ER on H.

37 . end if

38 . if (M (m_i) ≤ F (m_i)) then

39 . M is a fuzzy set on H.

40 . end if

41 . Fix a subset K^* of H × H.

42 . dop from 1 to k

43 . doq from 1 to k

44 . if ((m_im_j, m_pm_q) ∈ E) then

45 . (m_im_j, m_pm_q) ∈ E_H

46 . print*, E_His an ER on K^*.

47 . end if

48 . if (K (m_im_j) ≤ J (m_im_j)) then

49 . K is fuzzy set on K^*.

50 . end if

51 . end do

52 . end do

53 . end do

54 . end do

55 . RFS and RFR on H are determined by following the steps from 5 to 30.

55 . The generated subgraph

\vec{G}

\vec{G}

represent the relationships between

fixed members of an organization.

End

6 Comparative analysis

Theory of FSs and RSs are two different approaches to handle uncertainty but not opposite to each other. To combine both theories to obtain more generalized outcomes, Dubois and Prade [14] investigated the roughness of FSs and proposed a more general mathematical tool to deal with vagueness named as RFSs. In GrC, many researchers have constructed the granular structures and multilevels of granularities using simple crisp graphs and rough approximations. The crisp information granulation fails to reflect the human reasoning and problem solving, when the parameters or attributes are fuzzy in nature. To overcome the problem of uncertainty in construction of granules, we apply the hybrid model of RFDG to construct granular structures. A RFS and RFDG model of GrC handles the uncertain relationships between elements of universe and describes the lower and upper approximations of these relationships at different levels of granularity. Thus, our proposed model helps to granulate the complex uncertain networks more effectively and practically by considering the LA and UA. A very used example of constructing the granules in social networks using simple graphs is discussed by Chiaselotti et al. [13]. They considered the marriage ties of noble families in Florence in 15th century. The vertices of a simple undirected graph were considered as the 16 elite families and these vertices were connected by an edge if the corresponding families were linked by a marriage tie. The corresponding crisp graph is shown in Figure 11.

By considering this graph, they studied the symmetry blocks and corresponding partition of set of universe. We generalized the above example using RFDGs as given in 5.1 and the LA and UA of these families are given in Figures 2 and 3, respectively. We studied the indiscernible relations between these families using fuzzy information tables. Our proposed approach is more effective and practical in the way that it does not only provide the information about marriage ties between elite families (as described by crisp approach) but also illustrate the minimum and maximum strengths of these ties. For example, the LA of RFDG describes the minimum strength and UA shows the maximum strength of the ties of corresponding families.

Similarly, we consider another example of social network, i.e., face-to-face interaction networks. The data set information are derived from real situations and is given in Table 12.

Fig.11

The elite families in Florence.

Table 12

Dataset information

Dataset statistics
Number of networks	62
Number of nodes	451
Number of edges	3, 126, 993
Average number of edges per network	50, 435
Total temporal length	142, 005 seconds

The dynamic face-to-face interaction networks represent the interactions that happen during discussions between a group of participants playing the Resistance game. In these networks, each vertex represents a participant. A directed edge from node u to v is is drawn if u looks at participant v or the laptop. We consider a small part of these complex networks to compare our model with crisp models. A crisp directed network of face-to-face interaction is given in Figure 12.

Fig.12

Face to face interaction network.

Now, if we want to obtain the symmetry blocks and indiscernibility partition of these participants according to the relation that how many maximum and minimum times a participant looks at other in some certain interval of time. Then, this crisp network model can not effectively deal with our problem. Consider the LA and UA of the above network as shown in Figure 13. Here, the degrees of vertices (participants) in LA represent the average minimum number of times of interaction and the degrees in UA represent the average maximum number of times of interaction to each other in a specific interval of time. The membership values of edges in LA represent the minimum number of looks between corresponding participants and membership values of edges in UA represent the maximum number of looks between corresponding participants. By considering the fuzzy information tables of these approximations, we can determine the symmetry blocks and partition (as we have described in 5.1) corresponding to the relation how many times a participant looks the other one.

Fig.13

Lower and upper approximations of face-to-face interaction networks.

Hence, by considering the RFDG representation of social network models, we not only see the interactions between different elements of universe but also the maximum and minimum strengths of their relationships. Thus, the proposed models of GrC are more practical and applicable to approximate the social network models in which uncertainty and vagueness occur.

7 Conclusions and future directions

Rough sets and fuzzy sets are two different approaches to deal with uncertainty and inexact information but both theories are capable to handle vagueness occurring in real life. In this paper, we have applied the combined theory of RFSs to deal with uncertainty in information granulation of social networks. Rough fuzzy information granulation provides a multi-level visualization of problems containing uncertain information. We have presented a formal approach of GrC based on RFSs and RFDGS. This approach turns out to design a fruitful connection between modeling of granular structures and rough fuzzy approximations. We have defined the accuracy measures between the LA and UA of RFSs and calculated the distance using these measures. We have defined the FIT and SVIT to determine the indiscernibility relations between elements. Moreover, we have discussed some general geometric properties of indiscernibility relations and RFDGs. Further, we have presented certain social networks through RFDGs and granulated these networks using indiscernibility relations based on FITs. We have developed and implemented some algorithms to granulate the social networks. Finally, we have given a detailed comparative analysis of certain social networks occurring in real situation by considering the public data sets. We have proved the applicability of our proposed models by giving a brief comparison with other existing models of crisp graphs.

We aim to broaden our studies to (1) Fuzzy rough hypergraph models of granular computing, (2) Models of granular structures based on hesitant m-polar fuzzy graphs, (3) Fuzzy rough soft models of granular computing, (4) Hesitant fuzzy hypergraphs in granular computing and (5) Relational granularity of fuzzy hypergraphs.

Conflicts of interest

The authors declare no conflict of interest.

Acknowledgment

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. (DF-168- 135-1441). The authors, therefore, gratefully acknowledge DSR technical and financial support.

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Certain models of granular computing based on rough fuzzy approximations

Abstract

Keywords

1 Introduction

2 Preliminaries

3 Accuracy measures of rough fuzzy approximations

Table 1 Information table I I A 1 A 2 A 3 a High High Good b Medium Medium Normal c High High Bad

Table 5 Equivalence relation ξ ξ a 1 a 2 a 3 a 4 a 5 a 1 1 0 1 0 0 a 2 0 1 0 0 0 a 3 1 0 1 0 0 a 4 0 0 0 1 1 a 5 0 0 0 1 1

Table 8 ER ξ H on H

5.1 Granulation of social network of elite families

Conflicts of interest

Acknowledgment

References

Table 1
Information table I

I A ₁ A ₂ A ₃

a High High Good

b Medium Medium Normal

c High High Bad

Table 5
Equivalence relation ξ

ξ a ₁ a ₂ a ₃ a ₄ a ₅

a ₁ 1 0 1 0 0

a ₂ 0 1 0 0 0

a ₃ 1 0 1 0 0

a ₄ 0 0 0 1 1

a ₅ 0 0 0 1 1

Table 8
ER ξ_H on H