Preferential attachment with power law growth in the number of new edges

The Barabasi-Albert model is used to explain the formation of power laws in the degree distributions of networks. The model assumes that the principle of preferential attachment underlies the growth of networks, that is, new nodes connects to a fixed number of nodes with a probability that is proportional to their degrees. Yet, for empirical networks the number of new edges is often not constant, but varies as more nodes become part of the network. This paper considers an extension to the original Barabasi-Albert model, in which the number of edges established by a new node follows a power law distribution with support in the total number of nodes. While most new nodes connect to a few nodes, some new nodes connect to a larger number. We first characterize the dynamics of growth of the degree of the nodes. Second, we identify sufficient conditions under which the expected value of the average degree of the network is asymptotically stable. Finally, we show how the dynamics of the model resemble the evolution of protein interaction networks, Twitter, and Facebook.