Introducing utility to describe the attractions of nodes in this work, we propose and study a simple asymmetrical evolving network model, considering both preferential attachment and random (controlled by probability p). That is, the utility increment Δui≠Δuj when connecting node i to node j. The simulation results show that the model can reproduce power-law distributions of utility P(u)∼u-a, a=2+1/p, which can be obtained using mean field approximation. Furthermore, the model exhibits exponential networks with respect to small values of p and power-law scale-free networks with respect to big values of p, which is in good agreement with theoretical analysis. In other words, the model represents a transition between exponential and power-law scaling. To better understand the degree correlations of our model, the clustering coefficient C and degree assortativity r depending on probability p are also discussed.
