Study Of The Size Effect And The Degree Distribution Of Growing Networks With Mixed Attachment Mechanisms

Recently, the subject of complex networks has received a great wave of interest since the publication, in 1999, Barabás and Albert proposed the BA model of scale-free networks. This is the main mechanism that explains the power-law distribution of degrees appearing in the description of many complex networks. As growing and preferential attachment mechanisms are incorporated in the evolution of BA network model. But in some real networks, which are not made up of growing and preferential attachment mechanisms, but their evolution.In this paper, we mainly analysis the growing networks with mixed attachment mechanisms .Their papers only give the degree distribution of the model at infinite time, but they don't give the degree distribution at arbitrary time in the evolution of this model. In this paper, using nonequilibrium statistic method, we give the master equation satisfied by degree distribution of the two models in differential format, and obtain its rigorous analytical solution. We obtain that the degree distribution has an extended power-law form for growing networks with mixed attachment mechanisms. When the number of edges of a node is much larger than a certain value, the degree distribution reduces to the power-law form; and when the degree value is much smaller than a certain value, the degree distribution degenerates into the exponential form. It has been found that degree distribution possesses this extended power-law form for many real networks, in some a condition, the influence of size effect is caused, we find the essential reason of the size effect is growing and preferential attachment mechanisms. We study farther the size effect on the basis of the BA model, the concept of critical time is proposed.