Regularity Index of Uncertain Random Graph

Abstract

A graph containing some edges with probability measures and other edges with uncertain measures is referred to as an uncertain random graph. Numerous real-world problems in social networks and transportation networks can be boiled down to optimization problems in uncertain random graphs. Actually, information in optimization problems in uncertain random graphs is always asymmetric. Regularization is a common optimization problem in graph theory, and the regularity index is a fundamentally measurable indicator of graphs. Therefore, this paper investigates the regularity index of an uncertain random graph within the framework of chance theory and information asymmetry theory. The concepts of k-regularity index and regularity index of the uncertain random graph are first presented on the basis of the chance theory. Then, in order to compute the k-regularity index and the regularity index of the uncertain random graph, a simple and straightforward calculating approach is presented and discussed. Furthermore, we discuss the relationship between the regularity index and the k-regularity index of the uncertain random graph. Additionally, an adjacency matrix-based algorithm that can compute the k-regularity index of the uncertain random graph is provided. Some specific examples are given to illustrate the proposed method and algorithm. Finally, we conclude by highlighting some potential applications of uncertain random graphs in social networks and transportation networks, as well as the future vision of its combination with symmetry.

1. Introduction

Symmetry is an important concept in the field of physics. It is a general procedure for consciousness to re-evolute, and it describes immutability under a certain variation (Hu et al. [1]). Symmetry has been widely used in mathematics, information science, and management, especially in the field of graph theory. In today’s social life, more and more people use data, which demands that the ability of data mining and analysis be greatly improved. The relationship between points, that is, graph with edge links can be used to store and quickly schedule these data (Chen et al. [2]). The structure of a graph represents the relationship between entities and facilitates the analysis of that relationship for people. Much data represented by graphs has gathered in various aspects of life due to the quick development of current data-collection technologies.

The study of Euler’s solution to the Konigsberg seven bridges problem in 1736 is widely acknowledged as the beginning of graph theory. Graph theory progressively developed into a mathematical subfield after more than 100 years of growth. Many problems in the fields of natural science and social science may be abstracted into graph problems due to its extensive application history [3]. Until now, some research fields, such as decision science (Akram and Sarwar [4]), operational research (Cheng et al. [5]), and optimization (Chen et al. [6]) rely on graph theory to solve practical problems.

In practical application, real-world problems, such as machine failure, signal transmission instability, and so on, may arise unexpectedly, eventually leading to actual data uncertainty (Akram and Waseem [7]). In the information communication network, due to the presence of subjective consciousness or a lack of appropriate objective facts, the interaction connection between various objects frequently includes inaccurate information (Ni [8]). In the wireless sensor network, it is unclear if the wireless communication link between two sensor nodes actually exists in the wireless sensor network because of energy depletion, physical interference, and other potential effects (Majid et al. [9]). In the transportation network, the traffic flow statistics in the existing network cannot adequately reflect the actual situation due to transmission errors and delays in global positioning system (GPS) data (Peng et al. [10]). In addition, many connections between scientific study and actual life are uncertain for a variety of reasons, including unclear ties influenced by personal privacy protection in social networks, uncertain associations influenced by experimental variables in biological networks, and uncertain links influenced by the environment in mobile peer-to-peer networks.

In terms of theoretical research, more and more individuals are becoming aware of the limitations of traditional graph theory as research questions get more complex. Traditionally, graph theory was studied under the assumption that the vertices and the edges of a graph are deterministic [11]. However, when using graph theory in practice, we frequently run into different kinds of uncertainty. For the regularization problem in a graph, whether the edges exist or not is often uncertain in advance. As a result, the use of traditional approaches to analyze the regularization problem under indeterminacy environment is inappropriate. In order to deal with the indeterminacy factors in the graph theory, Erdos and Rényi [12] initialized the random graph by introducing the probability theory into a graph. Different from the classical graph, a random graph has independent probabilistic edges connecting each pair of vertices. Almost at the same time, the random graphs were also investigated by Gilbert [13]. After that, other researchers in this subject made significant contributions, including Bollobás [14] and Luczak et al. [15].

In several real-life problems, we frequently lack historical data [16]. Liu [17] developed the uncertainty theory in 2007 to create a mathematical framework to deal with the lack of historical data. The uncertainty theory is used to describe the imperfect knowledge from the expert’s empirical data, that is, when there are no historical or experimental data available to use as a reference and researchers must rely solely on empirical data [18]. As a result, it is of tremendous significance to investigate the optimization problem with few or no samples in graph theory.

The uncertain variables in uncertainty theory do not depend on the repeatability of events, but only on the empirical evaluation of the actual situation by industry experts. Such empirical assessment is based on axiomatic systems that satisfy normality, subadditivity, and duality. A specific symmetry of the duality axiom states that the probability of an event plus its complement added together equals one. Up until now, the uncertainty theory has been widely applied in the fields of network optimization (Majumder et al. [19], Mukherjee et al. [20]), supply chain management (Gao and Zhang [21]), production planning problem (Yang et al. [22,23]), etc. Please refer to the book of Liu [24] for further information on the most recent developments in the uncertainty theory.

Uncertain graphs differ from random graphs in that they have different definitions and computation procedures for both measure and confidence (Peng et al. [25]). The term “uncertain graph” was defined in 2013 by Gao and Gao [26]. As the first paper of the basic theory of uncertain graphs, Gao and Gao’s [26] study focused on investigating the connectedness degree of the uncertain graph and proposed an algorithm that can calculate the connectivity index of the uncertain graph. After that, the uncertain graph has been thoroughly investigated and widely developed in recent years. For example, Gao [27] investigated the tree index, Gao et al. [28] studied the $\alpha$-connectedness index, Gao and Qin [29] calculated the edge connectivity degree, and Gao et al. [30] researched the diameter of an uncertain graph. Based on the method of $\alpha$-cut coloring, Rosyida et al. [31] investigated the chromatic number of an uncertain graph. Recently, Wang et al. [32] studied the connectivity issue with generalized uncertain graphs and developed an approach by which to determine the connectivity index of such a graph. The notion of probability core reliability was introduced by Yu et al. [33], which measures the likelihood that a vertex serves as the core vertex in an uncertain graph. Based on the probability core reliability, they proposed a stable structural clustering algorithm to investigate the clustering problem of an uncertain graph.

In the actual world, we can describe some indeterminacy phenomena with historical data by using random variables, while studying others without historical data using uncertain variables (Chen et al. [34]). Therefore, it is logical to assume that a complex system contains both random variables and uncertain variables. In order to describe such a complicated system, Liu [35] initialized the chance theory and gave the definition of an uncertain random variable. Up until now, the chance theory has been applied in various areas, such as statistics (Gao et al. [36], Nowak and Hryniewicz [37]), network optimization (Sheng et al. [38,39]), risk decision (Liu and Ralescu [40], Liu et al. [41], Shi et al. [42]), financial markets (Wang et al. [43], Li et al. [44]), and so on.

There may be both uncertainty and randomness in a complicated graph (Liu [18]). Such a graph is called an uncertain random graph (URG). As the first paper of the basic theory of URG, Liu [18] discussed the connectivity degree of URG. After that, Zhang et al. [45] considered the Euler tour in URG. Chen et al. [46] investigated the cycle index of URG and answered the question of what was the likelihood of URG being a cycle. Based on the chance theory, Zhang et al. [47] gave a method by which to compute the matching index for URG. Li and Gao [48] and Li and Zhang [49] investigated the vertex connectivity and the edge connectivity of URG, respectively. Sheng and Mei [50] looked into the shortest path problem under an uncertain random environment. Recently, Li and Gao [51] proposed the concept of edge connectivity in URG and gave an algorithm that could compute the edge connectivity index of URG. Li and Zhang [52] discussed the shortest path problem under uncertain random environments through uncertain random digraphs.

In graph theory, the regularization problem is a common optimization problem, and the regularity index is a fundamental component of graphs. However, in the subject of uncertain random graphs, such a crucial component of a graph has not been explored. Therefore, this paper investigates the regularity index of an uncertain random graph within the framework of chance theory. The three research questions below have a special interest for us. First, what is the likelihood of URG being the k-regular graph? Secondly, how do we compute the k-regularity index and the regularity index of URG? Third, how do the k-regularity index and the regularity index of URG relate to one another?

In order to address the aforementioned research questions, we originally introduced the notions of the k-regularity index and the regularity index of URG. We then give a formula for calculating the k-regularity index and the regularity index of URG. The regularity index of URG is discovered to be the accumulation of all k-regularity indices of the graph. In addition, an adjacency matrix-based algorithm that can compute the k-regularity index of URG is provided. Finally, a few examples are given to examine the proposed method and algorithm.

The rest of this study is described as follows. Section 2 recalls some fundamental preliminaries about the uncertainty theory and the chance theory. In Section 3, the k-regularity index and the regularity index of URG are discussed. Section 4 gives three numerical examples to illustrate the process for calculating the k-regularity index and the regularity index of URG. We put forward summary comments and suggestions about future research in Section 5.

2. Preliminaries

2.1. Uncertainty Theory

Suppose that $\mathsf{\Gamma }$ is a nonempty set, and $\mathcal{L}$ is a $\sigma$-algebra over $\mathsf{\Gamma }$. Each element $\mathsf{\Lambda }\in \mathcal{L}$ is referred to as an event, which is a measurable set (Liu [17]). Following Liu [17], suppose that a number $\mathcal{M}$$\left\{\mathsf{\Lambda }\right\}$ to indicate the belief degree with which we believe an event $\mathsf{\Lambda }\in \mathcal{L}$ will happen. The number $\mathcal{M}$$\left\{\mathsf{\Lambda }\right\},\mathsf{\Lambda }\in \mathcal{L}$ is an uncertain measure (Liu [17]) if the undermentioned axioms are satisfied.

$AxiomI.$$\mathcal{M}\left\{\mathsf{\Gamma }\right\}=1$ for the universal set $\mathsf{\Gamma }$.

$AxiomII.$$\mathcal{M}\left\{\mathsf{\Lambda }\right\}+\mathcal{M}\left\{{\mathsf{\Lambda }}^c\right\}=1$ for any event $\mathsf{\Lambda }$, where ${\mathsf{\Lambda }}^c$ is the complement of $\mathsf{\Lambda }.$

$AxiomIII.$ For every enumerable sequence of events ${\mathsf{\Lambda }}_1,{\mathsf{\Lambda }}_2,\dots ,$ we have

$$\mathcal{M}\left\{\bigcup _{i=1}^{\infty }{\mathsf{\Lambda }}_i\right\}⩽\sum _{i=1}^{\infty }\mathcal{M}\left\{{\mathsf{\Lambda }}_i\right\}.$$

The axioms mentioned above correspond to normality, duality, and subadditivity, respectively. The duality axiom is symmetrical in that the total of the probability of an occurrence and its counterpart equals one. The triple system $(\mathsf{\Gamma },\mathcal{L},\mathcal{M})$ is known as uncertainty space. Uncertain variables are used in uncertainty theory to distinguish variables from random variables used in probability theory. Formally, an uncertain variable is a function $\xi \left(\gamma \right)$ from an uncertainty space $(\mathsf{\Gamma },\mathcal{L},\mathcal{M})$ to the set of real numbers such that $\left\{\xi \right(\gamma )\in B\}$ is a measurable function for any Borel set B (Liu [17]).

An uncertain variable is referred to as a boolean if its value is 0 or 1 with uncertain measure. For instance, the undermentioned an uncertain variable $\xi$ is a boolean uncertain variable,

$$\xi =\left\{\begin{array}{cc}1& \mathrm{with} \mathrm{uncertain} \mathrm{measure} \alpha \hfill \\ 0& \mathrm{with} \mathrm{uncertain} \mathrm{measure} 1-\alpha ,\hfill \end{array}\right.$$

where $\alpha \in [0,1]$.

In order to characterize an uncertain variable $\xi$ analytically, Liu [17] suggested the concept of uncertainty distribution as $\Phi \left(x\right)=\mathcal{M}\left\{\xi ⩽x\right\}$ for any $x\in \Re$.

An uncertain graph is defined as a triple system $G=(V,E,\psi )$ in which V is the vertex set of G, E is the edge set of G and $\psi$: $E\to [0,1]$ is a belief degree function in which $\psi \left(e\right)$ designates the uncertain measure of how likely it is that the edge $e\in E$ exists (Peng et al. [25]).

2.2. Chance Theory

In the real world, we may characterize certain indeterminate events with historical data by using random variables, while studying others without historical data by using uncertain variables. Liu [35] initialized the chance theory to describe this type of complex phenomenon.

Let $(\mathsf{\Gamma },\mathcal{L},\mathcal{M})$ be an uncertainty space, and $(\Omega ,\mathcal{A},\mathrm{Pr})$ be a probability space. The product $(\mathsf{\Gamma },\mathcal{L},\mathcal{M})\times (\Omega ,\mathcal{A},\mathrm{Pr})$ is said to be a chance space (Liu [35]). $\xi$ is an uncertain random variable (Liu [35]) if it is a measurable function from a chance space $(\mathsf{\Gamma },\mathcal{L},\mathcal{M})\times (\Omega ,\mathcal{A},\mathrm{Pr})$ to the set of real numbers.

Any element $\Theta$ in $\mathcal{L}\times \mathcal{A}$ is said to be an event in the chance space. The chance measure of event $\Theta$ was defined by Liu [35] as $\mathrm{Ch}\{\Theta \}={\int }_0^1\mathrm{Pr}\{\omega \in \Omega \mid \mathcal{M}\{\gamma \in \mathsf{\Gamma }\mid (\gamma ,\omega )\in \Theta \}⩾x\}\mathrm{d}x.$ Suppose that f is a measurable function. An uncertain random variable

$$\xi =f({\eta }_1,{\eta }_2,\dots ,{\eta }_m,{\tau }_1,{\tau }_2,\dots ,{\tau }_n)$$

is said to be boolean if ${\eta }_1,{\eta }_2,\dots ,{\eta }_m$ are boolean random variables and ${\tau }_1,{\tau }_2,\dots ,{\tau }_n$ are boolean uncertain variables.

Theorem 1.

(Liu [53]). Let ${\eta }_1,{\eta }_2,\dots ,{\eta }_m$ be independent boolean random variables, i.e.,

$${\eta }_i=\left\{\begin{array}{c}\mathit{1} with probability measure a_i\hfill \\ \mathit{0} with probability measure 1-a_i\hfill \end{array}\right.$$

for $i=1,2,\dots ,m$, and ${\tau }_1,{\tau }_2,\dots ,{\tau }_n$ be independent boolean uncertain variables, i.e.,

$${\tau }_j=\left\{\begin{array}{c}\mathit{1} with uncertain measure b_j\hfill \\ \mathit{0} with uncertain measure 1-b_j\hfill \end{array}\right.$$

for $j=1,2,\dots ,n$. If f is a boolean function, then

$$\xi =f({\eta }_1,{\eta }_2,\dots ,{\eta }_m,{\tau }_1,{\tau }_2,\dots ,{\tau }_n)$$

is a boolean uncertain random variable such that

$${\displaystyle \mathrm{Ch}\{\xi =1\}=\sum _{(x_1,x_2,\dots ,x_m)\in {\{0,1\}}^m}\left(\prod _{i=1}^m{\mu }_i\left(x_i\right)\right)f^{*}(x_1,x_2,\dots ,x_m),}$$

where

$$f^{*}(x_1,x_2,\dots ,x_m)=\left\{\begin{array}{c}{\displaystyle \underset{f(x_1,x_2,\dots ,x_m,y_1,y_2,\dots ,y_n)=1}{sup}\underset{1⩽j⩽n}{min}{\nu }_j\left(y_j\right),}\hfill \\ {\displaystyle if\underset{f(x_1,x_2,\dots ,x_m,y_1,y_2,\dots ,y_n)=1}{sup}\underset{1⩽j⩽n}{min}{\nu }_j\left(y_j\right)<0.5}\hfill \\ {\displaystyle 1-\underset{f(x_1,x_2,\dots ,x_m,y_1,y_2,\dots ,y_n)=0}{sup}\underset{1⩽j⩽n}{min}{\nu }_j\left(y_j\right),}\hfill \\ {\displaystyle if\underset{f(x_1,x_2,\dots ,x_m,y_1,y_2,\dots ,y_n)=1}{sup}\underset{1⩽j⩽n}{min}{\nu }_j\left(y_j\right)⩾0.5,}\hfill \end{array}\right.$$

$${\mu }_i\left(x_i\right)=\left\{\begin{array}{cc}{a}_i,& if x_i=1\hfill \\ 1-a_i,& if x_i=0\hfill \end{array}\right.(i=1,2,\dots ,m),$$

$${\nu }_j\left(y_j\right)=\left\{\begin{array}{cc}{b}_j,& if y_j=1\hfill \\ 1-b_j,& if y_j=0\hfill \end{array}\right.(j=1,2,\dots ,n).$$

2.3. Regular Graph

In this part, we briefly review the formal definition of a graph as well as several fundamental terminologies of graph theory that were utilized throughout this paper.

Definition 1.

(Bondy and Murty [54]). A graph G is defined as a triple system $G=(V,E,\psi )$ where V is a nonempty vertex set of G, E is an edge set of G, and ψ is an incidence function that associates with each edge an unordered pair of vertices of G.

For a graph, the edges that have the same pair of endpoints are called multiple edges. An edge with identical endpoints is called a loop (Shahzadi and Akram [55]). A graph that has no multiple edges or loops is said to be a simple graph (Akram and Sattar [3]). The number of vertices in G is called the order of G, and the number of edges is called its size. A graph G is said to be finite if the order and the size of G are all finite. In this paper, every graph is a simple and finite graph.

The number of edges that incident with v is called the degree of v, and is denoted by degv. If the vertices of G have the same degree, then the graph G is called regular (Akram and Sattar [3]). If degv=k ($0⩽k⩽n-1$) for every vertex v of G, then the graph G is said to be k-regular.

3. Regularity Index

In this section, we begin by defining the notions of the k-regularity index and the regularity index of URG in Section 3.1. Then, in Section 3.2, we provide approaches for computing the k-regularity index and the regularity index of URG, and examine the link between these two indices.

3.1. Foundational Concepts

Suppose that the graph has a collection of vertices $\mathcal{V}=\{v_1,v_2,\dots ,v_n\}.$ We define two collections of edges as follows for the purpose of convenience:

$$\begin{array}{c}\mathcal{U}=\left\{(v_i,v_j)\right|1⩽i<j⩽n \mathrm{and} ( v_i,v_j) \mathrm{are} \mathrm{uncertain} \mathrm{edges}\},\\ \mathcal{R}=\left\{(v_i,v_j)\right|1⩽i<j⩽n \mathrm{and} (v_i,v_j) \mathrm{are} \mathrm{random} \mathrm{edegs}\}.\end{array}$$

We also suppose that a deterministic edge is regarded as a special uncertain edge. Then $\mathcal{U}\cup \mathcal{R}=\left\{(v_i,v_j)\right|1⩽i<j⩽n\}$ that contain all $n(n-1)/2$ edges. We call

$$\mathcal{T}=\left(\begin{array}{cccc}{\alpha }_{11}& {\alpha }_{12}& \dots & {\alpha }_{1n}\\ {\alpha }_{21}& {\alpha }_{22}& \dots & {\alpha }_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ {\alpha }_{n1}& {\alpha }_{n2}& \dots & {\alpha }_{nn}\end{array}\right)$$

an uncertain random adjacency matrix if ${\alpha }_{ij}$ represent the likelihood in probability measure or uncertain measure that there exist an edge between vertices $v_i$ and $v_j$, $i,j=1,2,\dots ,n$, respectively. Note that ${\alpha }_{ii}=0$ and ${\alpha }_{ij}={\alpha }_{ji}$ for $i,j=1,2,\dots ,n,$ respectively. That is, $\mathcal{T}$ is a symmetric matrix.

Definition 2.

(Liu [18]). Suppose that $\mathcal{V}$ is the collection of vertices, $\mathcal{U}$ is the collection of uncertain edges, $\mathcal{R}$ is the collection of random edges, and $\mathcal{T}$ is the uncertain random adjacency matrix. Then the quartette $\mathbb{G}=(\mathcal{V},\mathcal{U},\mathcal{R},\mathcal{T})$ is said to be an URG.

Different from classical graph theory in which the vertices and the edges are deterministic, the edge set of URG $\mathbb{G}$ is a set of Boolean uncertain random variable

$$E\left(\mathbb{G}\right)=\{{\xi }_{12},{\xi }_{13},\dots ,{\xi }_{1n},{\xi }_{23},\dots ,{\xi }_{2n},{\xi }_{34},\dots ,{\xi }_{3n},\dots ,{\xi }_{(n-1)n}\},$$

where $\mathrm{Ch}\{{\xi }_{ij}=1\}={\alpha }_{ij}$ and $\mathrm{Ch}\{{\xi }_{ij}=0\}=1-{\alpha }_{ij},$ for $1⩽i<j⩽n.$ For the sake of convenience, we remove the edges ${\xi }_{ij}$ that satisfying $\mathrm{Ch}\{{\xi }_{ij}=1\}=0,$ and denote $E\left(\mathbb{G}\right)=\{{\xi }_1,{\xi }_2,\dots ,{\xi }_m\}$. We also denote $\mathcal{E}=\{(v_i,v_j)\mid 1⩽i<j⩽n,0<{\alpha }_{ij}<1\}$. If the cardinality of $\mathcal{E}$ is m, then the range of $\mathbb{G}$ contains $2^m$ models.

In the following, we give the concept of the k-regularity function of URG $\mathbb{G}$.

Definition 3.

Suppose that $\mathbb{G}$ is an URG with edge set $E\left(\mathbb{G}\right)=\{{\xi }_1,{\xi }_2,\dots ,{\xi }_m\}$. For any $0⩽k⩽n-1$, the k-regularity function of $\mathbb{G}$ is denoted as

$$R_k\left(E\left(\mathbb{G}\right)\right)=\left\{\begin{array}{cc}1,& if graph \mathbb{G} is k-regular\hfill \\ 0,& otherwise.\hfill \end{array}\right.$$

Obviously, $R_k\left(E\left(\mathbb{G}\right)\right)$ is a boolean function. For an URG $\mathbb{G}$, how likely is it to be k-regular? A k-regularity index of the graph $\mathbb{G}$ is given below.

Definition 4.

A k-regularity index of an URG $\mathbb{G}$ is the chance measure that the URG is k-regular, that is,

$${\mu }_k\left(\mathbb{G}\right)=\mathrm{Ch}\{R_k\left(E\left(\mathbb{G}\right)\right)=1\}.$$

Following that, we give the concept of the regularity function of URG $\mathbb{G}$.

Definition 5.

Suppose that $\mathbb{G}$ is an URG with edge set $E\left(\mathbb{G}\right)=\{{\xi }_1,{\xi }_2,\dots ,{\xi }_m\}$. The regularity function of $\mathbb{G}$ is denoted as

$$R\left(E\left(\mathbb{G}\right)\right)=\left\{\begin{array}{cc}1,& if graph \mathbb{G} is regular\hfill \\ 0,& otherwise.\hfill \end{array}\right.$$

Obviously, $R\left(E\left(\mathbb{G}\right)\right)$ is a boolean function. In order to show how likely an URG $\mathbb{G}$ is regular, regularity index is defined below.

Definition 6.

A regularity index of an URG $\mathbb{G}$ is the chance measure that the URG is regular, i.e.,

$$\mu \left(\mathbb{G}\right)=\mathrm{Ch}\{R\left(E\left(\mathbb{G}\right)\right)=1\}.$$

3.2. Main Results

Based on the above foundational concepts of URG in Section 3.1, the following three research questions will be addressed in this part: (1) How likely is it that URG will be the k-regular graph? (2) How does one compute the regularity index of URG? (3) How do the k-regularity index and the regularity index of URG relate to one another?

We next introduce some mathematical notations for URG that are derived from Liu [18]. Suppose that

$$X=\left(\begin{array}{cccc}{x}_{11}& x_{12}& \dots & x_{1n}\\ x_{21}& x_{22}& \dots & x_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ x_{n1}& x_{n2}& \dots & x_{nn}\end{array}\right),$$

(1)

and

$$\mathbb{X}=\left\{X\left|\begin{array}{c}{x}_{ij}=0 or 1,\mathrm{if}(v_i,v_j)\in \mathcal{R}\hfill \\ x_{ij}=0,\mathrm{if}(v_i,v_j)\in \mathcal{U}\hfill \\ x_{ij}=x_{ji},i,j=1,2,\dots ,n\hfill \\ x_{ii}=0,i=1,2,\dots ,n\hfill \end{array}\right.\right\}.$$

(2)

For a given matrix

$$Y=\left(\begin{array}{cccc}{y}_{11}& y_{12}& \dots & y_{1n}\\ y_{21}& y_{22}& \dots & y_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ y_{n1}& y_{n2}& \dots & y_{nn}\end{array}\right),$$

(3)

the extension class of Y is defined by

$$Y^{*}=\left\{X\left|\begin{array}{c}{x}_{ij}=y_{ij},\mathrm{if}(v_i,v_j)\in \mathcal{R}\hfill \\ x_{ij}=0 or 1,\mathrm{if}(v_i,v_j)\in \mathcal{U}\hfill \\ x_{ij}=x_{ji},i,j=1,2,\dots ,n\hfill \\ x_{ii}=0,i=1,2,\dots ,n\hfill \end{array}\right.\right\}.$$

(4)

For a given URG of order n, how may the k-regularity index be calculated? To respond to this question, we provide the subsequent theorem.

Theorem 2.

Suppose that $\mathbb{G}=(\mathcal{V},\mathcal{U},\mathcal{R},\mathcal{T})$ is an URG of order n. If all edges are independent, then the k-regularity index of $\mathbb{G}$ is

$${\displaystyle {\mu }_k\left(\mathbb{G}\right)=\sum _{Y\in \mathbb{X}}\left(\prod _{(v_i,v_j)\in \mathcal{R}}{\nu }_{ij}\left(Y\right)\right)R_k^{*}\left(Y\right),}$$

where

$$R_k^{*}\left(Y\right)=\left\{\begin{array}{cc}{\displaystyle \underset{X\in Y^{*},R_k\left(X\right)=1}{sup}\underset{(v_i,v_j)\in \mathcal{U}}{min}{\nu }_{ij}\left(X\right),}& {\displaystyle if\underset{X\in Y^{*},R_k\left(X\right)=1}{sup}\underset{(v_i,v_j)\in \mathcal{U}}{min}{\nu }_{ij}\left(X\right)<0.5}\hfill \\ {\displaystyle 1-\underset{X\in Y^{*},R_k\left(X\right)=0}{sup}\underset{(v_i,v_j)\in \mathcal{U}}{min}{\nu }_{ij}\left(X\right),}& {\displaystyle if\underset{X\in Y^{*},R_k\left(X\right)=1}{sup}\underset{(v_i,v_j)\in \mathcal{U}}{min}{\nu }_{ij}\left(X\right)⩾0.5,}\hfill \end{array}\right.$$

$${\nu }_{ij}\left(X\right)=\left\{\begin{array}{cc}{\alpha }_{ij},& if x_{ij}=1\hfill \\ 1-{\alpha }_{ij},& if x_{ij}=0\hfill \end{array}\right.(v_i,v_j)\in \mathcal{U},$$

$$R_k\left(X\right)=\left\{\begin{array}{cc}1,& {\displaystyle \mathrm{if}\sum _{j=1}^n{x}_{ij}=kfori=1,2,\dots ,n}\hfill \\ 0,& otherwise\hfill \end{array}\right.,0⩽k⩽n-1,$$

where $\mathbb{X}$ is the class of matrices satisfying Equation (2), and $Y^{*}$ is the extension class of Y satisfying Equation (4).

Proof. 

Note that all random edges are independent boolean random variables, and represented by

$${\eta }_{ij}=\left\{\begin{array}{c}1 \mathrm{with} \mathrm{probability} \mathrm{measure} {\alpha }_{ij}\hfill \\ 0 \mathrm{with} \mathrm{probability} \mathrm{measure} 1-{\alpha }_{ij}\hfill \end{array}\right.(v_i,v_j)\in \mathcal{R},$$

and all uncertain edges are independent Boolean uncertain variables which are represented by

$${\tau }_{ij}=\left\{\begin{array}{c}1 \mathrm{with} \mathrm{uncertain} \mathrm{measure} {\alpha }_{ij}\hfill \\ 0 \mathrm{with} \mathrm{uncertain} \mathrm{measure} 1-{\alpha }_{ij}\hfill \end{array}\right.(v_i,v_j)\in \mathcal{U}.$$

Let

$$X=\left(\begin{array}{cccc}{x}_{11}& x_{12}& \dots & x_{1n}\\ x_{21}& x_{22}& \dots & x_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ x_{n1}& x_{n2}& \dots & x_{nn}\end{array}\right),$$

where $x_{ij}\in \{0,1\},i,j=1,2,\dots ,n$, and X is a symmetric matrix with $x_{ii}=0,i=1,2,\dots ,n$.

For a vertex $v_i$, the degree of it is the sum of entries in the row corresponding to vertex $v_i$ in X. It is well known that a graph is k-regular if and only if the degree of all vertices is k. That is,

$$R_k\left(X\right)=1,$$

where

$$R_k\left(X\right)=\left\{\begin{array}{cc}1,& {\displaystyle \mathrm{if}\sum _{j=1}^n{x}_{ij}=kfori=1,2,\dots ,n}\hfill \\ 0,& \mathrm{otherwise}.\hfill \end{array}\right.$$

According to Definition 3, the function $R_k\left(X\right)$ is a boolean function. Then it follows from Theorem 1 that the theorem is verified. □

Remark 1.

If the uncertain factors in URG disappear, then the k-regularity index of $\mathbb{G}$ is

$${\displaystyle {\mu }_k\left(\mathbb{G}\right)=\sum _{X\in \mathbb{X}}\left(\prod _{1⩽i<j⩽n}{\nu }_{ij}\left(X\right)\right)R_k\left(X\right),}$$

where

$$\mathbb{X}=\left\{X\left|\begin{array}{c}{x}_{ij}=0 or 1,i,j=1,2,\dots ,n\hfill \\ x_{ij}=x_{ji},i,j=1,2,\dots ,n\hfill \\ x_{ii}=0,i=1,2,\dots ,n\hfill \end{array}\right.\right\}.$$

Remark 2.

If the random factors in URG disappear, then the k-regularity index of $\mathbb{G}$ is

$${\mu }_k\left(\mathbb{G}\right)=\left\{\begin{array}{cc}{\displaystyle \underset{X\in \mathbb{X},R_k\left(X\right)=1}{sup}\underset{1⩽i<j⩽n}{min}{\nu }_{ij}\left(X\right),}& {\displaystyle if\underset{X\in \mathbb{X},R_k\left(X\right)=1}{sup}\underset{1⩽i<j⩽n}{min}{\nu }_{ij}\left(X\right)<0.5}\hfill \\ {\displaystyle 1-\underset{X\in \mathbb{X},R_k\left(X\right)=0}{sup}\underset{1⩽i<j⩽n}{min}{\nu }_{ij}\left(X\right),}& {\displaystyle if\underset{X\in \mathbb{X},R_k\left(X\right)=1}{sup}\underset{1⩽i<j⩽n}{min}{\nu }_{ij}\left(X\right)⩾0.5,}\hfill \end{array}\right.$$

where

$$\mathbb{X}=\left\{X\left|\begin{array}{c}{x}_{ij}=0 or 1,i,j=1,2,\dots ,n\hfill \\ x_{ij}=x_{ji},i,j=1,2,\dots ,n\hfill \\ x_{ii}=0,i=1,2,\dots ,n\hfill \end{array}\right.\right\}.$$

Theorem 2 is still a pretty abstract formula for actual use, even if we can derive the k-regularity index. Next, we will present a method that can compute the k-regularity index.

Theorem 3.

Suppose that $\mathbb{G}=(\mathcal{V},\mathcal{U},\mathcal{R},\mathcal{T})$ is an URG of order n. If all edges are independent, then the k-regularity index of $\mathbb{G}$ is

$${\displaystyle \mu \left(\mathbb{G}\right)=\sum _{Y\in \mathbb{X}}\left(\prod _{(v_i,v_j)\in \mathcal{R}}{\nu }_{ij}\left(Y\right)\right)R^{*}\left(Y\right),}$$

where

$${\displaystyle R^{*}\left(Y\right)=\underset{X\in Y^{*},R_k\left(X\right)=1}{sup}\underset{(v_i,v_j)\in \mathcal{U}}{min}\left\{{\displaystyle \underset{x_{ij}=1}{min}{\alpha }_{ij},1-\underset{x_{ij}=1}{max}{\alpha }_{ij}}\right\},}$$

and $R_k\left(X\right)$ is the Boolean function defined in Theorem 2.

Proof. 

Similar to the results provided by Zhang et al. [45], the proof of this theorem is also based on a set of premises. We omit it.

For a given URG of order n, what is the likelihood of an URG being a regular one? This question will be resolved by the subsequent theorem. □

Theorem 4.

Suppose that $\mathbb{G}=(\mathcal{V},\mathcal{U},\mathcal{R},\mathcal{T})$ is an URG of order n. If all edges are independent, then the regularity index of $\mathbb{G}$ is

$${\displaystyle \mu \left(\mathbb{G}\right)=\sum _{Y\in \mathbb{X}}\left(\prod _{(v_i,v_j)\in \mathcal{R}}{\nu }_{ij}\left(Y\right)\right)R^{*}\left(Y\right),}$$

where

$$R^{*}\left(Y\right)=\left\{\begin{array}{cc}{\displaystyle \underset{X\in Y^{*},R\left(X\right)=1}{sup}\underset{(v_i,v_j)\in \mathcal{U}}{min}{\nu }_{ij}\left(X\right),}& {\displaystyle if\underset{X\in Y^{*},R\left(X\right)=1}{sup}\underset{(v_i,v_j)\in \mathcal{U}}{min}{\nu }_{ij}\left(X\right)<0.5}\hfill \\ {\displaystyle 1-\underset{X\in Y^{*},R\left(X\right)=0}{sup}\underset{(v_i,v_j)\in \mathcal{U}}{min}{\nu }_{ij}\left(X\right),}& {\displaystyle if\underset{X\in Y^{*},R\left(X\right)=1}{sup}\underset{(v_i,v_j)\in \mathcal{U}}{min}{\nu }_{ij}\left(X\right)⩾0.5,}\hfill \end{array}\right.$$

$${\nu }_{ij}\left(X\right)=\left\{\begin{array}{cc}{\alpha }_{ij},& if x_{ij}=1\hfill \\ 1-{\alpha }_{ij},& if x_{ij}=0\hfill \end{array}\right.(v_i,v_j)\in \mathcal{U},$$

$$R\left(X\right)=\left\{\begin{array}{cc}1,& {\displaystyle if\sum _{l=1}^n{x}_{il}=\sum _{l=1}^n{x}_{jl}fori,j=1,2,\dots ,n}\hfill \\ 0,& otherwise,\hfill \end{array},\right.$$

where $\mathbb{X}$ is the class of matrices satisfying Equation (2), and $Y^{*}$ is the extension class of Y satisfying Equation (4).

Proof. 

Keep in mind that every random edge is a separate boolean random variable represented by

$${\eta }_{ij}=\left\{\begin{array}{c}1 \mathrm{with} \mathrm{probability} \mathrm{measure} {\alpha }_{ij}\hfill \\ 0 \mathrm{with} \mathrm{probability} \mathrm{measure} 1-{\alpha }_{ij}\hfill \end{array}\right.(v_i,v_j)\in \mathcal{R},$$

and every uncertain edge is a separate Boolean uncertain variable, denoted by

$${\tau }_{ij}=\left\{\begin{array}{c}1 \mathrm{with} \mathrm{uncertain} \mathrm{measure} {\alpha }_{ij}\hfill \\ 0 \mathrm{with} \mathrm{uncertain} \mathrm{measure} 1-{\alpha }_{ij}\hfill \end{array}\right.(v_i,v_j)\in \mathcal{U}.$$

Let

$$X=\left(\begin{array}{cccc}{x}_{11}& x_{12}& \dots & x_{1n}\\ x_{21}& x_{22}& \dots & x_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ x_{n1}& x_{n2}& \dots & x_{nn}\end{array}\right),$$

where $x_{ij}\in \{0,1\},i,j=1,2,\dots ,n$, and X is a symmetric matrix with $x_{ii}=0,i=1,2,\dots ,n,$ respectively. □

The degree of a vertex $v_i$ is the sum of entries in the row corresponding to vertex $v_i$ in X. As is well known, a graph is regular one if and only if each vertex’s degree is equal, that is,

$$R\left(X\right)=1,$$

where

$$R\left(X\right)=\left\{\begin{array}{cc}1,& {\displaystyle \mathrm{if}\sum _{l=1}^n{x}_{il}=\sum _{l=1}^n{x}_{jl}\mathrm{for}i,j=1,2,\dots ,n}\hfill \\ 0,& \mathrm{otherwise}.\hfill \end{array}.\right.$$

According to Definition 5, the function $R\left(X\right)$ is a Boolean function. The conclusion that the theorem is verified arises from Theorem 1.

Remark 3.

If the uncertain factors in URG disappear, then the regularity index of $\mathbb{G}$ is

$${\displaystyle \mu \left(\mathbb{G}\right)=\sum _{X\in \mathbb{X}}\left(\prod _{1⩽i<j⩽n}{\nu }_{ij}\left(X\right)\right)R\left(X\right),}$$

where

$$\mathbb{X}=\left\{X\left|\begin{array}{c}{x}_{ij}=0 or 1,i,j=1,2,\dots ,n\hfill \\ x_{ij}=x_{ji},i,j=1,2,\dots ,n\hfill \\ x_{ii}=0,i=1,2,\dots ,n\hfill \end{array}\right.\right\}.$$

Remark 4.

If the random factors in URG disappear, then the regularity index of $\mathbb{G}$ is

$$\mu \left(\mathbb{G}\right)=\left\{\begin{array}{cc}{\displaystyle \underset{X\in \mathbb{X},R\left(X\right)=1}{sup}\underset{1⩽i<j⩽n}{min}{\nu }_{ij}\left(X\right),}& {\displaystyle if\underset{X\in \mathbb{X},R\left(X\right)=1}{sup}\underset{1⩽i<j⩽n}{min}{\nu }_{ij}\left(X\right)<0.5}\hfill \\ {\displaystyle 1-\underset{X\in \mathbb{X},R\left(X\right)=0}{sup}\underset{1⩽i<j⩽n}{min}{\nu }_{ij}\left(X\right),}& {\displaystyle if\underset{X\in \mathbb{X},R\left(X\right)=1}{sup}\underset{1⩽i<j⩽n}{min}{\nu }_{ij}\left(X\right)⩾0.5,}\hfill \end{array}\right.$$

where

$$\mathbb{X}=\left\{X\left|\begin{array}{c}{x}_{ij}=0 or 1,i,j=1,2,\dots ,n\hfill \\ x_{ij}=x_{ji},i,j=1,2,\dots ,n\hfill \\ x_{ii}=0,i=1,2,\dots ,n\hfill \end{array}\right.\right\}.$$

Theorem 5.

Suppose that $\mathbb{G}=(\mathcal{V},\mathcal{U},\mathcal{R},\mathcal{T})$ is an URG of order n. If all edges are independent, then the regularity index of $\mathbb{G}$ is

$${\displaystyle \mu \left(\mathbb{G}\right)=\sum _{Y\in \mathbb{X}}\left(\prod _{(v_i,v_j)\in \mathcal{R}}{\nu }_{ij}\left(Y\right)\right)R^{*}\left(Y\right),}$$

where

$${\displaystyle R^{*}\left(Y\right)=\underset{X\in Y^{*},R\left(X\right)=1}{sup}\underset{(v_i,v_j)\in \mathcal{U}}{min}\left\{{\displaystyle \underset{x_{ij}=1}{min}{\alpha }_{ij},1-\underset{x_{ij}=1}{max}{\alpha }_{ij}}\right\},}$$

(5)

and

$$R\left(X\right)=\left\{\begin{array}{cc}1,& {\displaystyle \mathrm{if}\sum _{l=1}^n{x}_{il}=\sum _{l=1}^n{x}_{jl}fori,j=1,2,\dots ,n}\hfill \\ 0,& otherwise.\hfill \end{array}\right.$$

Proof. 

The findings in Theorem 2 indicate that we just need to establish Equation (5) verifies for every given Y. On the basis of the findings presented in Gao [56], we may immediately derive Equation (5).

In addition, we can find the following relationship between the k-regularity index and the regularity index of an URG. □

Corollary 1.

The regularity index of URG is the sum of all the k-regularity indexes of the graph.

On the basis of the abovementioned theorems, we construct an adjacency matrix-based algorithm to find the k-regularity index of URG which is described in Algorithm 1.

Algorithm 1. Adjacency matrix-based algorithm
Steps	Commands	Interpretation
1	Input matrix $\tau ={\left[t(i,j)\right]}_{m\times n}$,	Input uncertain random adjacency matrix of $\mathbb{G}$
2	Set $k=0$
3	Input matrix $A_k={\left[a(i,j)\right]}_{m\times n}$
4	For each $A_k$, define
	$U=\left[u(i,j)\right],u(i,j)=a(i,j)=1\in A_k$ and
	$\mathcal{R}=\left[r(k,l)\right],r(k,l)=a(k,l)=1\in A_k$
5	if $r(k,l)=1\in \mathcal{R}$
6	$y(k,l)=1$
7	else
8	$y(k,l)=0$
9	end
10	Set matrix $Y={\left[y(k,l)\right]}_{m\times n}$	Define matrix Y related to elements in $\mathcal{R}$
11	if $y(i,j)=1\in Y$
12	$\nu (i,j)=t(i,j)$	Determine ${\nu }_{ij}\left(Y\right)$
13	else
14	$\nu (i,j)=1-t(i,j)$
15	end
16	$\nu ={\prod }_{r(i,j)=1\in \mathcal{R}}\nu (i,j)$
17	if $a(i,j)=1\in U$
	$R^{*}(i,j)=min\left\{t(i,j)\right|a(i,j)=1\in U\}$
18	else
19	$R^{*}(i,j)=1-\{maxt(i,j)|a(i,j)=0\notin U\}$
20	end
21	$R^{*}=min\left\{R^{*}(i,j)\right|a(i,j)\in U\cup a(i,j)=0\notin U\}$
22	${\mu }_k\left(\mathbb{G}\right)=\nu *R^{*}$	Determine k-regularity index for each matrix $A_k$
23	$k=k+1$
24	If there exist other realization matrix $A_k$
	such that $\mathbb{G}$ is k-regular, go to Step 3, otherwise
25	Stop, and calculate ${\displaystyle \mu \left(\mathbb{G}\right)=\sum _k{\mu }_k\left(\mathbb{G}\right)}$	Determine k-regularity index for $\mathbb{G}$

4. Examples

In order to demonstrate how to compute the k-regularity index and the regularity index of URG as well as the relationship between the two, this section provides three examples.

Example 1.

Let $\mathbb{G}=(\mathcal{V},\mathcal{U},\mathcal{R},\mathcal{T})$ be an URG of order 3 and size 3, where

$$\mathcal{U}=\{(v_1,v_3),(v_2,v_3)\},\mathcal{R}=\left\{(v_1,v_2)\right\},$$

and the uncertain random adjacency matrix is

$$\mathcal{T}=\left(\begin{array}{ccc}0& 0.8& 0.6\\ 0.8& 0& 0.3\\ 0.6& 0.3& 0\end{array}\right).$$

Because the size of URG $\mathbb{G}$ is 3, and the cardinality of $\mathcal{E}$ is 3, thus it contains 8 models. If a model has the following adjacency matrix

$$A_1=\left(\begin{array}{ccc}0& 1& 1\\ 1& 0& 1\\ 1& 1& 0\end{array}\right),$$

then the model is 2-regular, and if a model has the following adjacency matrix,

$$A_2=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right),$$

then the model is 0-regular. If a model has one of the other 6 adjacency matrixes, then it is irregular.

When Y has a following form corresponding to the matrix $A_1$,

$$Y=\left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right),$$

then we obtain that

$$\prod _{(v_i,v_j)\in \mathcal{R}}{\nu }_{ij}\left(Y\right)=0.8,R^{*}\left(Y\right)=0.3.$$

Thus the 2-regularity index of graph $\mathbb{G}$ is

$${\mu }_2\left(\mathbb{G}\right)=0.8\times 0.3=0.24.$$

When Y has a following form corresponding to the matrix $A_2$,

$$Y=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right),$$

then we obtain that

$$\prod _{(v_i,v_j)\in \mathcal{R}}{\nu }_{ij}\left(Y\right)=0.2,R^{*}\left(Y\right)=0.4.$$

Hence the 0-regularity index of graph $\mathbb{G}$ is

$${\mu }_0\left(\mathbb{G}\right)=0.2\times 0.4=0.08.$$

It follows from Corollary 1 that the regularity index is

$$\mu \left(\mathbb{G}\right)={\mu }_0\left(\mathbb{G}\right)+{\mu }_2\left(\mathbb{G}\right)=0.08+0.24=0.32.$$

Example 2.

Let $\mathbb{G}=(\mathcal{V},\mathcal{U},\mathcal{R},\mathcal{T})$ be an URG of order 4 and size 6, where

$$\mathcal{U}=\{(v_1,v_4),(v_2,v_3),(v_2,v_4),(v_3,v_4)\},\mathcal{R}=\{(v_1,v_2),(v_1,v_3)\},$$

and the uncertain random adjacency matrix is

$$\mathcal{T}=\left(\begin{array}{cccc}0& 0.7& 0.6& 1\\ 0.7& 0& 1& 0.4\\ 0.6& 1& 0& 0.8\\ 1& 0.4& 0.8& 0\end{array}\right).$$

Because the size of URG $\mathbb{G}$ is 6, but the cardinality of $\mathcal{E}$ is 4, thus it contains 16 models. Suppose that adjacency matrix is

$$A_1=\left(\begin{array}{cccc}0& 0& 0& 1\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ 1& 0& 0& 0\end{array}\right),$$

then the model is 1-regular and the matrix Y has the following form

$$Y=\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right).$$

Thus, we obtain that

$$\prod _{(v_i,v_j)\in \mathcal{R}}{\nu }_{ij}\left(Y\right)=0.12,R^{*}\left(Y\right)=0.2.$$

It follows from Theorem 2 that the 1-regularity index is

$${\mu }_1\left(\mathbb{G}\right)=0.12\times 0.2=0.024.$$

If a model has one of the other two adjacency matrixes,

$$A_2=\left(\begin{array}{cccc}0& 1& 0& 1\\ 1& 0& 1& 0\\ 0& 1& 0& 1\\ 1& 0& 1& 0\end{array}\right),A_3=\left(\begin{array}{cccc}0& 0& 1& 1\\ 0& 0& 1& 1\\ 1& 1& 0& 0\\ 1& 1& 0& 0\end{array}\right),$$

then the model is 2-regular.

When

$$Y=\left(\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right),$$

we obtain that

$$\prod _{(v_i,v_j)\in \mathcal{R}}{\nu }_{ij}\left(Y\right)=0.28,R^{*}\left(Y\right)=0.6.$$

When

$$Y=\left(\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right),$$

we obtain that

$$\prod _{(v_i,v_j)\in \mathcal{R}}{\nu }_{ij}\left(Y\right)=0.18,R^{*}\left(Y\right)=0.4.$$

It follows from Theorem 2 that the 2-regularity index is

$${\mu }_2\left(\mathbb{G}\right)=0.28\times 0.6+0.18\times 0.4=0.24.$$

If a model has the following adjacency matrix,

$$A_4=\left(\begin{array}{cccc}0& 1& 1& 1\\ 1& 0& 1& 1\\ 1& 1& 0& 1\\ 1& 1& 1& 0\end{array}\right),$$

then the model is 3-regular. Thus the matrix Y is

$$Y=\left(\begin{array}{cccc}0& 1& 1& 0\\ 1& 0& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right).$$

We obtain that

$$\prod _{(v_i,v_j)\in \mathcal{R}}{\nu }_{ij}\left(Y\right)=0.42,R^{*}\left(Y\right)=0.4.$$

It follows from Theorem 2 that the 3-regularity index is

$${\mu }_3\left(\mathbb{G}\right)=0.42\times 0.4=0.168.$$

A model is considered irregular if it contains one of the other 12 adjacency matrices. As a result of Corollary 1, the regularity index is

$$\mu \left(\mathbb{G}\right)={\mu }_1\left(\mathbb{G}\right)+{\mu }_2\left(\mathbb{G}\right)+{\mu }_3\left(\mathbb{G}\right)=0.024+0.24+0.168=0.432.$$

Example 3.

Let $\mathbb{G}=(\mathcal{V},\mathcal{U},\mathcal{R},\mathcal{T})$ be an URG of order 4 and size 6, where

$$\mathcal{U}=\{(v_2,v_3),(v_2,v_4),(v_3,v_4)\},\mathcal{R}=\{(v_1,v_2),(v_1,v_3),(v_1,v_4)\}$$

and the uncertain random adjacency matrix is

$$\mathcal{T}=\left(\begin{array}{cccc}0& 0.7& 0.6& 0.1\\ 0.7& 0& 0.3& 0.7\\ 0.6& 0.3& 0& 0.2\\ 0.1& 0.7& 0.2& 0\end{array}\right).$$

Because the size of URG $\mathbb{G}$ is 6, and the cardinality of $\mathcal{E}$ is 4, thus it contains 16 models. Suppose that adjacency matrix is

$$A_1=\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right).$$

Then the model is 0-regular, and the matrix Y has the following form

$$Y=\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right).$$

We obtain that

$$\prod _{(v_i,v_j)\in \mathcal{R}}{\nu }_{ij}\left(Y\right)=0.108,R^{*}\left(Y\right)=0.3.$$

It follows from Theorem 2 that the 0-regularity index is

$${\mu }_0\left(\mathbb{G}\right)=0.108\times 0.3=0.0324.$$

If a model has one of the other 3 adjacency matrices,

$$A_2=\left(\begin{array}{cccc}0& 0& 0& 1\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ 1& 0& 0& 0\end{array}\right),A_3=\left(\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right),A_4=\left(\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ 1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right),$$

then the model is 1-regular.

When

$$Y=\left(\begin{array}{cccc}0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 1& 0& 0& 0\end{array}\right),$$

we obtain that

$$\prod _{(v_i,v_j)\in \mathcal{R}}{\nu }_{ij}\left(Y\right)=0.012,R^{*}\left(Y\right)=0.3.$$

When

$$Y=\left(\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right),$$

we obtain that

$$\prod _{(v_i,v_j)\in \mathcal{R}}{\nu }_{ij}\left(Y\right)=0.252,R^{*}\left(Y\right)=0.2.$$

When

$$Y=\left(\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right),$$

we obtain that

$$\prod _{(v_i,v_j)\in \mathcal{R}}{\nu }_{ij}\left(Y\right)=0.162,R^{*}\left(Y\right)=0.3.$$

It follows from Theorem 2 that the 1-regularity index is

$${\mu }_1\left(\mathbb{G}\right)=0.012\times 0.3+0.252\times 0.2+0.162\times 0.3=0.1026.$$

If a model has one of the other 3 adjacency matrices,

$$A_5=\left(\begin{array}{cccc}0& 0& 1& 1\\ 0& 0& 1& 1\\ 1& 1& 0& 0\\ 1& 1& 0& 0\end{array}\right),A_6=\left(\begin{array}{cccc}0& 1& 0& 1\\ 1& 0& 1& 0\\ 0& 1& 0& 1\\ 1& 0& 1& 0\end{array}\right),A_7=\left(\begin{array}{cccc}0& 1& 1& 0\\ 1& 0& 0& 1\\ 1& 0& 0& 1\\ 0& 1& 1& 0\end{array}\right),$$

then the model is 2-regular.

When

$$Y=\left(\begin{array}{cccc}0& 0& 1& 1\\ 0& 0& 0& 0\\ 1& 0& 0& 0\\ 1& 0& 0& 0\end{array}\right),$$

we obtain that

$$\prod _{(v_i,v_j)\in \mathcal{R}}{\nu }_{ij}\left(Y\right)=0.018,R^{*}\left(Y\right)=0.3.$$

When

$$Y=\left(\begin{array}{cccc}0& 1& 0& 1\\ 1& 0& 0& 0\\ 0& 0& 0& 0\\ 1& 0& 0& 0\end{array}\right),$$

we obtain that

$$\prod _{(v_i,v_j)\in \mathcal{R}}{\nu }_{ij}\left(Y\right)=0.028,R^{*}\left(Y\right)=0.2.$$

When

$$Y=\left(\begin{array}{cccc}0& 1& 1& 0\\ 1& 0& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right),$$

we obtain that

$$\prod _{(v_i,v_j)\in \mathcal{R}}{\nu }_{ij}\left(Y\right)=0.378,R^{*}\left(Y\right)=0.2.$$

It follows from Theorem 2 that the 2-regularity index is

$${\mu }_2\left(\mathbb{G}\right)=0.018\times 0.3+0.028\times 0.2+0.378\times 0.2=0.0866.$$

If a model has the following adjacency matrix,

$$A_8=\left(\begin{array}{cccc}0& 1& 1& 1\\ 1& 0& 1& 1\\ 1& 1& 0& 1\\ 1& 1& 1& 0\end{array}\right),$$

then the model is 3-regular. Thus the matrix Y is

$$Y=\left(\begin{array}{cccc}0& 1& 1& 1\\ 1& 0& 0& 0\\ 1& 0& 0& 0\\ 1& 0& 0& 0\end{array}\right).$$

We obtain that

$$\prod _{(v_i,v_j)\in \mathcal{R}}{\nu }_{ij}\left(Y\right)=0.042,R^{*}\left(Y\right)=0.2.$$

It follows from Theorem 2 that the 3-regularity index is

$${\mu }_2\left(\mathbb{G}\right)=0.042\times 0.2=0.0084.$$

A model is regarded as irregular if it has one of the additional 56 adjacency matrices. Consequent to Corollary 1, the index of regularity is

$$\mu \left(\mathbb{G}\right)={\mu }_0\left(\mathbb{G}\right)+{\mu }_1\left(\mathbb{G}\right)+{\mu }_2\left(\mathbb{G}\right)+{\mu }_3\left(\mathbb{G}\right)=0.0324+0.1026+0.0866+0.0084=0.23.$$

5. Conclusions

This paper contributed to the study of the regularity index of URG. We first put forward the concept of the k-regularity index and the regularity index of URG. After that, the methods for calculating the k-regularity index and the regularity index of URG were given. In addition, we investigated and discussed some properties of the k-regularity index and the regularity index. In addition, we designed an adjacency matrix-based algorithm to compute the k-regularity index. Finally, the abovementioned approach and algorithm were then explained with the help of a few numerical examples.

This paper has the following three main contributions. First, different from previous studies, this paper investigates the problem of regularization for an uncertain random graph within the framework of chance theory and symmetry. We propose the concepts of the k-regularity index and the regularity index of uncertain random graphs, and use the chance measure to describe the likelihood of an uncertain random graph being a regular one. Secondly, we investigate the relationship between the k-regularity index and the regularity index of an uncertain random graph. We find that the regularity index of an uncertain random graph is the sum of all k-regularity indices of the graph. Finally, we put forward an adjacency matrix-based algorithm to compute the regularity index of an uncertain random graph efficiently. We perform three experiments with different orders and sizes for uncertain random graphs. The experimental results show that the proposed adjacency matrix-based algorithm can obtain the regularity index of uncertain random graphs efficiently.

This paper can be extended in the following directions. First, future research could study the degree sequence of URG, and investigate the properties of the degree sequence. Secondly, it would be interesting to investigate the diameter distribution function of URG. Thirdly, designing an efficient algorithm to obtain the regularity index of larger-scale URG should be investigated. Finally, we can investigate the applications of uncertain random graphs in social networks and transportation networks, as well as the vision of its combination with symmetry in future research.