Spectral Expanding Expanders

Gil Cohen*, Itay Cohen*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

Dinitz, Schapira, and Valadarsky [5] introduced the intriguing notion of expanding expanders -a family of expander graphs with the property that every two consecutive graphs in the family differ only on a small number of edges. Such a family allows one to add and remove vertices with only few edge updates, making them useful in dynamic settings such as for datacenter network topologies and for the design of distributed algorithms for self-healing expanders. [5] constructed explicit expanding-expanders based on the Bilu-Linial construction of spectral expanders [3]. The construction of expanding expanders, however, ends up being of edge expanders, thus, an open problem raised by [5] is to construct spectral expanding expanders (SEE). In this work, we resolve this question by constructing SEE with spectral expansion which, like [3], is optimal up to a poly-logarithmic factor, and the number of edge updates is optimal up to a constant. We further give a simple proof for the existence of SEE that are close to Ramanujan up to a small additive term. As in [5], our construction is based on interpolating between a graph and its lift. However, to establish spectral expansion, we carefully weigh the interpolated graphs, dubbed partial lifts, in a way that enables us to conduct a delicate analysis of their spectrum. In particular, at a crucial point in the analysis, we consider the eigenvectors structure of the partial lifts.

Original languageEnglish
Title of host publication38th Computational Complexity Conference, CCC 2023
EditorsAmnon Ta-Shma
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772822
DOIs
StatePublished - Jul 2023
Event38th Computational Complexity Conference, CCC 2023 - Warwick, United Kingdom
Duration: 17 Jul 202320 Jul 2023

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume264
ISSN (Print)1868-8969

Conference

Conference38th Computational Complexity Conference, CCC 2023
Country/TerritoryUnited Kingdom
CityWarwick
Period17/07/2320/07/23

Funding

FundersFunder number
European Research Council949499
Israel Science Foundation1569/18

    Keywords

    • Expanders
    • Normalized Random Walk
    • Spectral Analysis

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