The Maximum Degree of the Barabasi-Albert Random Tree

In the classical Erdős–Rényi model of random graphs, when the number of edges is proportional to the number of vertices, the degree distribution is approximately Poisson with a tail decreasing even faster than exponentially. However, in many real life networks power law degree distributions were observed with different exponents. To introduce a more realistic model for the evolution of random networks, Barabási and Albert [1] proposed the following one, which they called scale free. In the beginning, at the first step, we only have a single edge. At every further step we start a new (undirected) edge from one of the vertices created so far. The other endpoint of the edge is a new vertex, while the starting point is chosen from the existing vertices at random, in such a way that each vertex is selected with probability proportional to its degree (in other words, to choose an existing vertex we first choose one of the edges with equal probability, then one of the endpoints of that edge). In this model the asymptotic proportion of vertices with degree k decreases as k−3, which is the same power law that was observed in the World Wide Web. A couple of papers has recently been devoted to the study of this random graph as well as to other similar models, all different from the classical Erdős–Rényi construction. Here we only mention [2]. A generalization of this model was investigated in [9]. There, at the n-th step, a vertex of degree k was chosen with probability proportional to k + β, where β was a fixed parameter of the model, β > −1. Thus, a vertex of degree k was selected with probability k + β Sn , where Sn denoted the sum of weights over all vertices of the random tree with n edges and n+1 vertices; that is, Sn = 2n+(n+1)β = (2+β)n+β. In [9] the proportion of vertices of degree k was shown to converge almost surely