Nonlinear Barab\'asi-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 Barab\'asi-Albert model. Here we investigate the nonlinear Barab\'asi-Albert growing network. In this model, a new node connects to a vertex of degree $k$ with a probability proportional to $k^{\alpha}$ ($\alpha$ 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 $\alpha \le 1$. In the limit $\alpha \to -\infty$ the network is homogeneous. If $\alpha > 1$ 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 Barab\'asi-Albert network is assortative (disassortative) if $\alpha < 1$ ($\alpha > 1$) and no assortative only when $\alpha = 1$. In the limit $\alpha \to -\infty$ the assortativity coefficient can be exactly calculated. We find $r=7/13$ when $m=2$. Finally, the minimum average shortest path length $l_{min}$ is numerically evaluated. Increasing the network size, $l_{min}$ diverges for $\alpha \le 1$ and it is equal to 1 when $\alpha > 1$.