  We study the mean length $\ell(k)$ of the shortest paths between a vertex of
degree $k$ and other vertices in growing networks, where correlations are
essential. In a number of deterministic scale-free networks we observe a
power-law correction to a logarithmic dependence, $\ell(k) = A\ln
[N/k^{(\gamma-1)/2}] - C k^{\gamma-1}/N + ...$ in a wide range of network
sizes. Here $N$ is the number of vertices in the network, $\gamma$ is the
degree distribution exponent, and the coefficients $A$ and $C$ depend on a
network. We compare this law with a corresponding $\ell(k)$ dependence obtained
for random scale-free networks growing through the preferential attachment
mechanism. In stochastic and deterministic growing trees with an exponential
degree distribution, we observe a linear dependence on degree, $\ell(k) \cong
A\ln N - C k$. We compare our findings for growing networks with those for
uncorrelated graphs.
