  We present two complementary analytical approaches for calculating the
distribution of shortest path lengths in Erdos-R\'enyi networks, based on
recursion equations for the shells around a reference node and for the paths
originating from it. The results are in agreement with numerical simulations
for a broad range of network sizes and connectivities. The average and standard
deviation of the distribution are also obtained. In the case that the mean
degree scales as $N^{\alpha}$ with the network size, the distribution becomes
extremely narrow in the asymptotic limit, namely almost all pairs of nodes are
equidistant, at distance $d=\lfloor 1/\alpha \rfloor$ from each other. The
distribution of shortest path lengths between nodes of degree $m$ and the rest
of the network is calculated. Its average is shown to be a monotonically
decreasing function of $m$, providing an interesting relation between a local
property and a global property of the network. The methodology presented here
can be applied to more general classes of networks.
