On Ensemble Theory Of Complex Networks And Its Applications

The research on complicated networks is a quite important interdiscipline consisting of mathematics, physics, economics and even sociology etc., is specially regarded both at home and abroad. This paper studies on the complicated networks mainly by means of the thought of the ensemble theory. We construct some relative models of networks by the method of the ensemble theory, and classify the models. The paper includes that: (1) writing out a unified formula for the ensemble models of networks and classifying them; (2) giving the definitions of a canonical ensemble of networks, a grand canon-ical ensemble of networks, a micro-canonical ensemble of networks and an ensemble of networks in product weight function, and presenting the mathematical derivation of computing their partition function in detail, and dealing with the ensemble of pseudo-graphs; (3) discussing about the impact from product weight function formations upon the topological characteristics, such as from the number of edges and the node's degree distribution of a network etc., and putting a method of specifying the functions in order to approaching any degree distribution given; (4) deducing the formation of degree dis-tributions by a partition function, and proving that the ensemble of networks with the degree distribution given can be generated by specifying a suitable formation of product weight functions; (5) constructing algorithms of Monte Carlo simulation for the ensemble models of networks, and giving the transition probabilities between graphs, and proving that the Markov chain (sequence) generated converges into a equilibrium distribution; (6) programming the computer program of the MCMC algorithm, and simulating some en-semble models of networks digitally and comparing the results with the theoretical ones; (7) putting up a method of linked-state-matrix-based spectral analysis for the networks in order to analyzing the characteristics of the networks in the ensemble models, this method can reduce computation effectively, and giving some related inequalities for spectral es-timates and the application to an economical network; (8) trying to apply the method of the ensemble of networks into building models of economical networks, and giving some models and their analysis.In conclusion, the research in the paper is done on the strict mathematical base and demonstrated by simulation on computer. Our research indicates that the method of the ensemble of networks is a suitable measure which constructs a kind of wide network models and computes the characteristics of networks and specially a kind of potential models to control the characteristics of networks.