  We introduce the \emph{idemetric} property, which formalises the idea that
most nodes in a graph have similar distances between them, and which turns out
to be quite standard amongst small-world network models. Modulo reasonable
sparsity assumptions, we are then able to show that a strong form of
idemetricity is actually equivalent to a very weak expander condition (PUMP).
This provides a direct way of providing short proofs that small-world network
models such as the Watts-Strogatz model are strongly idemetric (for a wide
range of parameters), and also provides further evidence that being idemetric
is a common property.
  We then consider how satisfaction of the idemetric property is relevant to
algorithm design. For idemetric graphs we observe, for example, that a single
breadth-first search provides a solution to the all-pairs shortest paths
problem, so long as one is prepared to accept paths which are of stretch close
to 2 with high probability. Since we are able to show that Kleinberg's model is
idemetric, these results contrast nicely with the well known negative results
of Kleinberg concerning efficient decentralised algorithms for finding short
paths: for precisely the same model as Kleinberg's negative results hold, we
are able to show that very efficient (and decentralised) algorithms exist if
one allows for reasonable preprocessing. For deterministic distributed routing
algorithms we are also able to obtain results proving that less routing
information is required for idemetric graphs than the worst case in order to
achieve stretch less than 3 with high probability: while $\Omega(n^2)$ routing
information is required in the worst case for stretch strictly less than 3 on
almost all pairs, for idemetric graphs the total routing information required
is $O(nlog(n))$.
