  It was recently claimed that on d-dimensional small-world networks with a
density p of shortcuts, the typical separation s(p) ~ p^{-1/d} between
shortcut-ends is a characteristic length for shortest-paths{cond-mat/9904419}.
This contradicts an earlier argument suggesting that no finite characteristic
length can be defined for bilocal observables on these systems
{cont-mat/9903426}. We show analytically, and confirm by numerical simulation,
that shortest-path lengths \ell(r) behave as \ell(r) ~ r for r < r_c, and as
\ell(r) ~ r_c for r > r_c, where r is the Euclidean separation between two
points and r_c(p,L) = p^{-1/d} log(L^dp) is a characteristic length. This shows
that the mean separation s between shortcut-ends is not a relevant length-scale
for shortest-paths. The true characteristic length r_c(p,L) diverges with
system size L no matter the value of p. Therefore no finite characteristic
length can be defined for small-world networks in the thermodynamic limit.
