Cookies?
Library Header Image
LSE Research Online LSE Library Services

The idemetric property: when most distances are (almost) the same

Barmpalias, George, Huang, Neng, Lewis-Pye, Andrew, Li, Angsheng, Li, Xuechen, Pan, Yicheng and Roughgarden, Tim (2019) The idemetric property: when most distances are (almost) the same. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. ISSN 1364-5021 (In Press)

[img] Text - Accepted Version
Download (592kB)

Abstract

We 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 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 Ω(n2) 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)).

Item Type: Article
Official URL: https://royalsocietypublishing.org/journal/rspa
Additional Information: © 2019 The Royal Society
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Sets: Departments > Mathematics
Date Deposited: 10 Jan 2019 16:27
Last Modified: 18 Feb 2019 00:05
URI: http://eprints.lse.ac.uk/id/eprint/91674

Actions (login required)

View Item View Item

Downloads

Downloads per month over past year

View more statistics