Uncertain Hypergraphs: A Conceptual Framework and Some Topological Characteristics Indexes

Regularity Index of Uncertain Random Graph

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.

Hypergraph and Uncertain Hypergraph Representation Learning Theory and Methods

With the advent of big data and the information age, the data magnitude of various complex networks is growing rapidly. Many real-life situations cannot be portrayed by ordinary networks, while hypergraphs have the ability to describe and characterize higher order relationships, which have attracted extensive attention from academia and industry in recent years. Firstly, this paper described the development process, the application areas, and the existing review research of hypergraphs; secondly, introduced the theory of hypergraphs briefly; then, compared the learning methods of ordinary graphs and hypergraphs from three aspects: matrix decomposition, random walk, and deep learning; next, introduced the structural optimization of hypergraphs from three perspectives: dynamic hypergraphs, hyperedge weight optimization, and multimodal hypergraph generation; after that, the applicability of three uncertain hypergraph models were analyzed based on three uncertainty theories: probability theory, fuzzy set, and rough set; finally, the future research directions of hypergraphs and uncertain hypergraphs were prospected.