In this paper my coauthors and I introduce the 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 also consider how satisfaction of the idemetric property is relevant to algorithm design.

To appear in the *Proceedings of the Royal Society A*: pdf.

This is joint work with Barmpalias, Huang, Li, Li, Pan and Roughgarden.