The degree sequences of an asymmetrical growing network

In this paper, we use utility to describe the attractive effect and then study simple asymmetrical evolving model, considering both preferential attachment and the randomness of the utility. The model is defined so that, at each integer time t, a new vertex, with m edges attached to it, is added to the graph. The new edges added at time t are then preferentially connected to older vertices, i.e., conditionally on G(t−1), the probability that a given edge is connected to vertex i is proportional to its utility at time t−1. The main result is that the asymptotical degree sequence for this process is a power law with exponent 2+1/p.