The principle of the Internet evolving and the conjecture of the optimal structure of Internet

In this paper we will give the statistical characteristics and general principles of an optimal structure of the Internet, which is a scale-free network. Since the purpose of the Internet is to allow fast and easy communication, the average path length is used to measure the performance of the network, and the number of edges of the network is used as a metric of its cost. Based on this, the goal of this Internet optimization problem is to obtain the highest performance with the lowest cost. A multi goal optimization problem is proposed to model this problem. By using two empirical formulas of \langle k\rangle and \langle l\rangle, we are able to find the statistical characteristics of the optimal structure. There is a critical power law exponent \alpha _\rm c for the Internet with power law degree distribution, at which the Internet can obtain a relatively good performance with a low cost. We find that this \alpha _\rm c is approximately 2.1.