  In this report we show that in a planar exponentially growing network
consisting of $N$ nodes, congestion scales as $O(N^2/\log(N))$ independently of
how flows may be routed. This is in contrast to the $O(N^{3/2})$ scaling of
congestion in a flat polynomially growing network. We also show that without
the planarity condition, congestion in a small world network could scale as low
as $O(N^{1+\epsilon})$, for arbitrarily small $\epsilon$. These extreme results
demonstrate that the small world property by itself cannot provide guidance on
the level of congestion in a network and other characteristics are needed for
better resolution. Finally, we investigate scaling of congestion under the
geodesic flow, that is, when flows are routed on shortest paths based on a link
metric. Here we prove that if the link weights are scaled by arbitrarily small
or large multipliers then considerable changes in congestion may occur.
However, if we constrain the link-weight multipliers to be bounded away from
both zero and infinity, then variations in congestion due to such remetrization
are negligible.
