This paper proposes a new definition of cross entropy for uncertain random variables, and derives a formula. Moreover, this paper introduces generalized cross entropy for uncertain random variables, and discusses its properties. Based on the definition of cross entropy, a cross entropy chance distribution model of degree-constrained minimum spanning tree (DCMST) problem is proposed. An algorithm is designed here to solve the model. Finally, a numerical example is provided to illustrate the effectiveness of the proposed model and algorithm.

In many practical problems, randomness and uncertainty simultaneously appear in one complex system or network. When graph theory is applied to these problems, these complex systems or networks are usually represented by uncertain random graphs, in which some edges exist with degrees in probability measure, and some other edges exist with degrees in uncertain measure. In this paper, we focus on the connectivity of uncertain random graphs with respect to vertices. We propose the concepts of k- vertex-connectivity index and the connectivity of an uncertain random graph. The former is the chance measure that an uncertain random graph is k-vertex-connected, and the latter is an uncertain random variable, which characterizes the connectivity of the uncertain random graph with respect to vertices. We discuss some properties of these concepts. Methods and formulas are also presented for calculating the k-vertex-connectivity index, and the distribution and expected value of the connectivity of an uncertain random graph.

Uncertain partial differential equation is a type of partial differential equation driven by Liu processes. This paper first provides two methods for solving the first order linear uncertain partial differential equation and the inverse uncertainty distribution of solution is discussed. Moreover, on the basis of the study of uncertainty theory and the propagation process of internet public opinion, three types of internet public opinion (IPO) problem are proposed based on uncertain partial differential equation. Finally, some numerical examples are given.