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 $\alpha \le 1$\mathrm{.}Inthelimit$\alpha \to -\infty$thenetworkishomogeneous\mathrm{.}If$\alpha > 1$thereisagelphasewith$m$ 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 $\alpha \to -\mathrm{\infty }$ the assortativity coefficient can be exactly calculated. We find $r=7\mathrm{/}13$ 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 $\alpha >1$.