The Study Of Phase Transitions In The Evolution Of Complex Networks

In this thesis, the influence of preferential attachment on evolving networks and the phase transitions of the networks are studied. The summary of the thesis is as follows:1. The historical background and the main properties of complex networks are introduced, and the extended model proposed by Barabasi and Albert (also called EBA model) is discussed.2. The extended model without preferential attachment is discussed. The model is built by local events, including the addition of new links, rewiring, and the addition of new nodes. Using the continuum theory, the connectivity distribution of the network is a generalized exponential. The exponential character of the distribution indicates that the absence of preferential attachment eliminates the scale-free character of the network, and the network belongs to the class defined by the random networks and the small-world ones. The scale-free networks cannot be formed without preferential attachment in the local events.3. The extended model with partial preferential attachment is studied. The model is built by gradually introducing the preferential attachment to the local events. When preferential attachment is only present in the processes of the addition of new nodes or new links, the networks are scale-free. There is only scale-free regime in the phase diagram. When preferential attachment is brought into the rewiring, the exponential regime and the scale-free regime coexist in the phase diagram. There is a phase transition from the scale-free regime to the exponential regime for a certain rewiring probability. For different local events, the boundaries between the two regimes are different. Though growth and preferential attachment are necessary in forming the scale-free networks, the two factors cannot ensure that the network is scale-free.4. The extended model with complete preferential attachment is investigated. Fitness is introduced to the extended model and two kinds of fitness probabilitydistributions are studied. When fitness satisfies the bimodal distribution, the connectivity distribution of the network satisfies a generalized power-law form. When fitness is chosen uniformly, the connectivity distribution follows a generalized power-law form with an inverse logarithmic correction. Though the network is scale-free, the connectivity distribution of the network has been affected by the local events and fitness.