Construction Of Complex Network Models And Analysis Of Their Properties

As a high level of abstraction in describing complex system, complex network has always received great attention. The study of complex network is originated from Graph Theory. Nowadays it is based on ER random network model, WS and NW small-world network models, BA scale-free network model and other classical network models. Small-world and scale-free are regarded as two main properties of complex networks. Their present opens up a new era for the study of complex network. In recent years, self-similarity considered as the third property of complex network has received more and more attention.The improvement of classical network models is one focus in the study of complex network. Nowadays, there is a need to adopt new methods and new perspective on the study of complex networks. Firstly, this dissertation studies BA scale-free network model and presents an improved model for its shortcomings. Then four network models named Complex Network, Hypernetwork, Supernetwork, and Super-Hyper Network are proposed based on adjacency matrix and correlation matrix with different matrix operations. Degree distribution polynomials are used to describe degree distributions of different networks, especially self-similariy networks and random networks. The network models based on matrix operation are compared with classical network models and existing network theory. Their characteristics are described. The law of number, the evolution mechanism, and disturbance and stability of four network models are also studied. The main achievements in this dissertation are described as follows.(1) A BE network model with a bimodal degree distribution is obtained by introducing the maximum number of connections, having a sub-linear growth in the number of connections of new nodes and using Logistic function on BA scale-free network model. The shifting and zooming of the peak of BE network model may be achieved by adjusting its parameters. It can be applied to explain the socio-economic polarization in the real world well. And it is degenerated to BA network model in the limiting case.(2) A Complex Network model is constructed by application of Kronecker Product and Kronecker Sum on the adjacency matrix of graph. By the degree distribution polynomial, the degree of the proposed Complex Network can be calculated. The fractal dimension of self-similarity Complex Network does not exceed 2 and the degree distribution of random Complex Network is a Gaussion Distribution.(3) A Hypernetwork is constructed by application of Tracy-Singh Product and Tracy-Singh Sum on the correlation matrix of hyper graph. By polynomials of node degree, node hyper-degree, and hyper-edge degree distributions, the node degree, node hyper-degree and hyper-edge degree of the proposed Hypernetwork can be calculated. The fractal dimension of self-similarity Hypernetwork does not exceed 2 and the node degree, node hyper-degree, and hyper-edge degree distributions of random Hypernetwork are all Gaussion Distribution.(4) A Supernetwork is constructed by application of Khatri-Rao Product and Khatri-Rao Sum on the adjacency matrix of hierarchical graph. By polynomials of marginal degree and joint degree distributions, the marginal degree and joint degree of the proposed Supernetwork can be calculated. The fractal dimension of self-similarity Supernetwork does not exceed 3 and the marginal degree distribution of random Supernetwork is a Gaussion Distribution while the joint degree distribution is a multi-dimension Gaussion Distribution.(5) A Super-Hyper Network is constructed by application of Khatri-Rao Product and Khatri-Rao Sum on the correlation matrix of hierarchical hyper graph. By polynomials of marginal node degree, marginal node hyper-degree, marginal hyper-edge degree distributions and joint node degree, joint node hyper-degree, joint hyper-edge degree distributions, the marginal node degree, marginal node hyper-degree, marginal hyper-edge degree and joint node degree, joint node hyper-degree, joint hyper-edge degree of the proposed Super-Hyper Network can be calculated. The fractal dimension of self-similarity Super-Hyper Network does not exceed 3, and the marginal distributions of node degree, node hyper-degree, and hyper-edge degree of random Super-Hyper Network are all Gaussion Distribution while the joint distributions of node degree, node hyper-degree, and hyper-edge degree are all multi-dimension Gaussion Distribution.(6) The techniques of cloud computing are introduced to matrix-based network models. The framework of MapReduce is applied to develop parallelization methods for matrix operation and calculation of degree distribution polynomials.The proposed four network models in this dissertation are improvement and supplement of existing classical network models. Their feasibility and rationality are validated by comparisions of them with classical network models. The achievements in this dissertation extend the existing network theory and provide new methods and solutions for the future study of network.