Connectedness Index of uncertain Graph

The connectivity of graph is a basic concept of graph theory. Since originated in seven bridges problem by Euler in 1736, the main study of graph theory unfolds on the assumption that the graph is connected. In classic graph theory, the edges and the vertices are all deterministic, and connectivity of the graph can be easily verified. However, in practical application, non-deterministic factors appear in graphs, which leads to new situations. Such as in shortest path problem, if the length of each arc is non-deterministic, the optimization objective can not simply expressed and conventional algorithms can not be used. Sometimes, whether two vertices are joined by an edge can not be completely determined. As a result, whether the graph is connected can not be completely determined. Then, in how much belief degree we can regard a non-deterministic graph is connected? This problem was first investigated by Erdös and Rényi. They thought that whether two vertices are joined can be described as a random variable. Under the assumption, Erdös did further research on random graph, and more details can be obtained in his works.