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.
Proceedings of the Royal Society A, 475(2222), 2019, pdf.
This is joint work with Barmpalias, Huang, Li, Li, Pan and Roughgarden.