  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
$\alpha > 1$.
