A generative model—the preferential attachment scheme

The preferential attachment scheme is often attributed to Herbert Simon. In his paper [111] of 1955, he gave a model for word distribution using the preferential attachment scheme and derived Zipf 's law. Namely, the probability of a word having occurred exactly i times is proportional to 1/i. The basic setup for the preferential attachment scheme is a simple local growth rule which leads to a global consequence — a power law distribution. Since this local growth rule gives preferences to vertices with large degrees, the scheme is often described by " the rich get richer ". In this chapter, we shall give a clean and rigorous treatment of the preferential attachment scheme. Of interest is to determine the exponent of the power law from the parameters of the local growth rule. 3.1. Basic steps of the preferential attachment scheme There are two parameters for the preferential attachment model: • A probability p, where 0 ≤ p ≤ 1. • An initial graph G 0 , that we have at time 0. Usually, G 0 is taken to be the graph formed by one vertex having one loop. (We consider the degree of this vertex to be 1, and in general a loop adds 1 to the degree of a vertex.) Note, in this model multiple edges and loops are allowed. We also have two operations we can do on a graph: • Vertex-step — Add a new vertex v, and add an edge {u, v} from v by randomly and independently choosing u in proportion to the degree of u in the current graph. • Edge-step — Add a new edge {r, s} by independently choosing vertices r and s with probability proportional to their degrees. Note that for the edge-step, r and s could be the same vertex. Thus loops could be created. However, as the graph gets large, the probability of adding a loop can be well bounded and is quite small.