Research On The Important Measure Of Complex Hyper - Network

In the real world, many systems, such as the Internet, information networks, traffic networks, electric power grids, social networks and protein-protein interaction networks, etc., can be described by models of complex networks. So far, complex networks have been intensively investigated across many fields of science and en-gineering. With the rapid expansion of the network size and the more complex of connection, there are many large complex networks whose the numbers of vertices and edges between vertices are numerous, complex and dynamic. In some cases, the use of simple or directed graphs to represent complex networks does not provide a complete description of the real-world systems under investigated. A natural way of representing these systems is to use a generalization of graphs known as hyper-graph. It provides a new perspective for the research of these systems. Among many related aspects, topological characterization is very basic and significant for the s-tudy in hypernetwork field. The Estrada index and spectral radius, are regarded as important measures in characterizing and understanding topology and dynamic properties. Here, the limit properties for several classes of graphs are explored. In addition, we construct a class of typical deterministic small-world hypernetwork, and study some related topology properties. Specifically, the main research work and contribution of this dissertation are summarized as follows.1. The distributions of subgraph centralities of three classes of complex net-works are investigated. Based on theory-guided extensive numerical simulations, we analyze similarities and differences of three classes of complex networks, namely the ER random graph networks, WS small-world networks and BA scale-free networks.2. The bounds of the Estrada index of k-uniform linear hypertrees are studied. Let H(n, k) be the set of k-uniform linear hypertrees on n vertices. For k=3,4 and any graph H ∈ H(n,k), by considering its the adjacency matrix, and using induction and the technique of graft transformation of graphs, we determine the unique k-uniform linear hypertrees with the maximum and minimum Estrada indices among all hypertrees in H(n,k), respectively. Furthermore,as a corollary, the first two hypertress with the greatest Estrada index are given. Let H△(n, k) be the set of k-uniform linear hypertrees on n vertices with fixed maximum △, using similar method, the bound is obtained among all hypertrees in H△(n, k).3. The spectral radius of k-uniform linear hypertrees is investigated. Let H(n, k) be the set of k-uniform linear hypertrees on n vertices. For any graph H ∈ H(n, k), by considering its the adjacency matrix, and using tools of spectral graph theory, we obtain the unique 3-uniform linear hypertrees with the maximum spectral radius and the minimum maximum spectral radius in H(n,3). Meanwhile, we determine the unique 4-uniform linear hypertree with the maximum spectral radius among all hypertrees in H(n,4).4. The main characteristics of a family of typical deterministic small-world hypernetwork is studied. By algorithm of vertex iterations, a class of deterministic small-world hypernetwork growing model is proposed. Analytical expressions for their hyperdegree distribution, diameter and the average path are derived. Using method in matrix theory, the recursion relations of the eigenvalues of the adjacency and Laplacian matrix are presented, respectively. Also, spectral distributions of the hypernetwork are analyzed by numerical simulation.