Scale-free networks with the power-law exponent between 1 and 3

Basing on the batch arrival concept in the queue theory, this paper proposes a Poisson network model with node batch arrival. The Nodes arrive the system as a Poisson process with rate A. In the first model, the batch is a power function of the batch number with exponent theta(0 <= 0 < + infinity). Using Poisson process theory and continuum approach, we found that the stationary mean degree distribution of this model is a power-law distribution, and its power-law exponent is between 1 and 3. In the second model, the batch is a log function of the batch number and we obtained that the power-law exponent of stationary mean degree distribution is 3 when the batch rises more slowly. So our model is not only the extension of the BA model, but also a theoretical foundation of many real networks of which the power-law exponent is between 1 and 2.