Abstract
As the system becomes more and more complex, we are usually in the state of indeterminacy. In the real world, the states of uncertainty and randomness are the two most common types of indeterminacy. An uncertain random graph is applied to describe a graph model with uncertainty and randomness simultaneously. This paper mainly focuses on the connectivity of two vertices in an uncertain random graph. Firstly, a local connectivity index is proposed to unveil the chance measure that two special vertices are connected in an uncertain random graph. Furthermore, a method for calculating the local connectivity index is formulated. In addition, some simplified forms of the method are developed, and an algorithm is designed to obtain the local connectivity index. Finally, the information relevant to the relationship between the local connectivity index and the connectivity index is discussed.
Highlights
In real life, graph theory is widely applied to solve a variety of optimization problems, such as the traveling salesman problem (Li et al [21]), transportation problem (Lv et al [31]), network flow problem (Asadi and Kia [2]), and complex network systems (Cheng et al [5], Li and Daniels [22], Li et al [23])
Inspired by the above discussion, this paper mainly considers the following two issues: First, how likely are the two vertices in an uncertain random graph connected? Second, what is the relationship between the connectivity of two vertices and that of the uncertain random graph? To answer these questions, a local connectivity index is proposed to measure how likely it is for two vertices of an uncertain random graph to be connected
There is currently no related research on the connectivity of two specific vertices in an uncertain random graph. In view of this fact, this paper focuses on investigating the connectivity of two specific vertices of an uncertain random graph
Summary
Graph theory is widely applied to solve a variety of optimization problems, such as the traveling salesman problem (Li et al [21]), transportation problem (Lv et al [31]), network flow problem (Asadi and Kia [2]), and complex network systems (Cheng et al [5], Li and Daniels [22], Li et al [23]). Liu [27] declares that probability theory is not applicable to model belief degree, and presents a counterexample. To deal with complex systems that uncertainty and randomness coexist in graphs, uncertain random graph is defined by Liu [28]. Inspired by the above discussion, this paper mainly considers the following two issues: First, how likely are the two vertices in an uncertain random graph connected? A local connectivity index is proposed to measure how likely it is for two vertices of an uncertain random graph to be connected. This paper analyzes the relationship between the connectivity of two vertices and that of the uncertain random graph.
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