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Large Low-Diameter Graphs are Good Expanders

Authors Michael Dinitz, Michael Schapira, Gal Shahaf

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Author Details

Michael Dinitz
  • Dept. of Computer Science, Johns Hopkins University, Baltimore, US
Michael Schapira
  • School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel
Gal Shahaf
  • Dept. of Mathematics, The Hebrew University, Jerusalem, Israel.

Funding

  • Dinitz, Michael: Supported in part by NSF awards 1464239 and 1535887

Cite AsGet BibTex

Michael Dinitz, Michael Schapira, and Gal Shahaf. Large Low-Diameter Graphs are Good Expanders. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 71:1-71:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.ESA.2018.71

Abstract

We revisit the classical question of the relationship between the diameter of a graph and its expansion properties. One direction is well understood: expander graphs exhibit essentially the lowest possible diameter. We focus on the reverse direction, showing that "sufficiently large" graphs of fixed diameter and degree must be "good" expanders. We prove this statement for various definitions of "sufficiently large" (multiplicative/additive factor from the largest possible size), for different forms of expansion (edge, vertex, and spectral expansion), and for both directed and undirected graphs. A recurring theme is that the lower the diameter of the graph and (more importantly) the larger its size, the better the expansion guarantees. Aside from inherent theoretical interest, our motivation stems from the domain of network design. Both low-diameter networks and expanders are prominent approaches to designing high-performance networks in parallel computing, HPC, datacenter networking, and beyond. Our results establish that these two approaches are, in fact, inextricably intertwined. We leave the reader with many intriguing questions for future research.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Spectra of graphs
  • Mathematics of computing → Extremal graph theory
  • Networks → Network design principles
  • Networks → Network structure
Keywords
  • Network design
  • Expander graphs
  • Spectral graph theory

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