The degree sequence of a scale-free random graph process

Recently, Barabasi and Albert [2] 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 [2] and, with Jeong, in [3], Barabasi and Albert suggested that after many steps the proportion P(d) of vertices with degree d should obey a power law P(d) alpha d(-gamma). They obtained gamma = 2.9 +/- 0.1 by experiment and gave a simple heuristic argument suggesting that gamma = 3. Here we obtain P(d) asymptotically for all d less than or equal to n(1/15), where n is the number of vertices, proving as a consequence that gamma = 3. (C) 2001 John Wiley & Sons, Inc.