The degree sequence of a scale-free random graph process
Recently, Barabási and Albert  suggested modeling complex real-world networks such as the worldwide web as follows: consider a random graph process in which vertices are added to the graph one at a time and joined to a fixed number of earlier vertices, selected with probabilities proportional to their degrees. In  and, with Jeong, in , Barabási and Albert suggested that after many steps the proportion P(d) of vertices with degree d should obey a power law P(d) α d-γ. They obtained γ - 2.9 ± 0.1 by experiment and gave a simple heuristic argument suggesting that γ = 3. Here we obtain P(d) asymptotically for all d ≤ n1/15, where n is the number of vertices, proving as a consequence that γ = 3.
The Structure and Dynamics of Networks
Bollobás, B., Riordan, O., Spencer, J., & Tusnády, G. (2011). The degree sequence of a scale-free random graph process. The Structure and Dynamics of Networks, 9781400841356, 385-395. Retrieved from https://digitalcommons.memphis.edu/facpubs/5852