A new class of scale free random graphs

Consider the following modification of the Barabási-Albert random graph. At every step a new vertex is added to the graph. It is connected to the old vertices randomly, with probabilities proportional to the degree of the other vertex, and independently of each other. We show that the proportion of vertices of degree k decreases at the rate of k-3. Furthermore, we prove a strong law of large numbers for the maximum degree.