Nonlinear Barabási–Albert network

In recent years there has been considerable interest in the structure and dynamics of complex networks. One of the most studied networks is the linear Barabási–Albert model. Here we investigate the nonlinear Barabási–Albert growing network. In this model, a new node connects to a vertex of degree k with a probability proportional to kα (α real). Each vertex adds m new edges to the network. We derive an analytic expression for the degree distribution P(k) which is valid for all values of m and α⩽1. In the limit α→−∞ the network is homogeneous. If α>1 there is a gel phase with m super-connected nodes. It is proposed a formula for the clustering coefficient which is in good agreement with numerical simulations. The assortativity coefficient r is determined and it is shown that the nonlinear Barabási–Albert network is assortative (disassortative) if α<1(α>1) and no assortative only when α=1. In the limit α→−∞ the assortativity coefficient can be exactly calculated. We find r=713 when m=2. Finally, the minimum average shortest path length lmin is numerically evaluated. Increasing the network size, lmin diverges for α⩽1 and it is equal to 1 when α>1.