Degree Distribution Nearby the Origin of a Preferential Attachment Graph

In their paper [1] Barabási and Albert proposed a certain random process of evolving graphs as a model of real-world networks, like the Internet. In their model vertices are added to the graph one by one, and edges connecting the new vertex to the old ones are drawn randomly, with probabilities proportional to the degree of the endpoint. In the particular case where only a single edge is allowed at every step, a recursive tree process, also known as plane oriented recursive trees, is obtained. In fact, that model was introduced more than a decade earlier by Szymański [9], and then a couple of papers have been devoted to it. The interested reader is referred to [2] for a very general model of web graphs. In [5] the asymptotic degree distribution was obtained for a one-parameter generalization of the Barabási–Albert random tree. In [3] the same degree distribution was proved to exist on each of the largest levels of the tree. Surprisingly, in the neighbourhood of the root, on the lower levels a completely different degree distribution was found to emerge [6]. Consider the following modification of the Barabási–Albert random graph. Starting from the very simple graph consisting of two points and the edge between them, at every step we add a new vertex and some (possibly 0) new edges to the graph. For the new edges each old vertex is selected at random, with probability depending linearly on its degree, and independently of the others; then the selected vertices are connected to the new one.