Document Open Access Logo

High-Dimensional Expanders from Expanders

Authors Siqi Liu, Sidhanth Mohanty, Elizabeth Yang

PDF

File

Thumbnail PDF
PDF
LIPIcs.ITCS.2020.12.pdf
  • Filesize: 0.98 MB
  • 32 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Expander graphs and randomness extractors
Keywords
  • High-Dimensional Expanders
  • Markov Chains

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

We present an elementary way to transform an expander graph into a simplicial complex where all high order random walks have a constant spectral gap, i.e., they converge rapidly to the stationary distribution. As an upshot, we obtain new constructions, as well as a natural probabilistic model to sample constant degree high-dimensional expanders.
In particular, we show that given an expander graph G, adding self loops to G and taking the tensor product of the modified graph with a high-dimensional expander produces a new high-dimensional expander. Our proof of rapid mixing of high order random walks is based on the decomposable Markov chains framework introduced by [Jerrum et al., 2004].

Cite As Get BibTex

Siqi Liu, Sidhanth Mohanty, and Elizabeth Yang. High-Dimensional Expanders from Expanders. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 12:1-12:32, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ITCS.2020.12

Author Details

Siqi Liu
  • EECS Department, UC Berkeley, CA, United States
Sidhanth Mohanty
  • EECS Department, UC Berkeley, CA, United States
Elizabeth Yang
  • EECS Department, UC Berkeley, CA, United States

Funding

  • Liu, Siqi: This research was supported in part by a Google Faculty Award and donations from the Ethereum Foundation and the Interchain Foundation.
  • Mohanty, Sidhanth: Supported by NSF Grant CCF-1718695.
  • Yang, Elizabeth: Supported by NSF Grant DGE 1752814.

Acknowledgements

We thank Tom Gur for introducing us to this intriguing question and for helpful discussions, and we thank Nikhil Srivastava for insightful conversations. We would also like to thank the Simons Institute for the Theory of Computing where a large portion of this work was done. The third author is supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE 1752814.

References

  1. Nima Anari, Kuikui Liu, Shayan Oveis Gharan, and Cynthia Vinzant. Log-concave polynomials II: high-dimensional walks and an FPRAS for counting bases of a matroid. Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, June 2019. Google Scholar
  2. László Babai. Spectra of Cayley graphs. Journal of Combinatorial Theory, Series B, 27(2):180-189, 1979. Google Scholar
  3. Avraham Ben-Aroya and Amnon Ta-Shma. A combinatorial construction of almost-Ramanujan graphs using the zig-zag product. SIAM Journal on Computing, 40(2):267-290, 2011. Google Scholar
  4. Nathanaël Berestycki. Lectures on mixing times. Cambridge University, 2014. Google Scholar
  5. Michael Chapman, Nati Linial, and Yuval Peled. Expander Graphs-Both Local and Global. arXiv preprint, 2018. URL: http://arxiv.org/abs/1812.11558.
  6. Irit Dinur and Tali Kaufman. High dimensional expanders imply agreement expanders. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pages 974-985. IEEE, 2017. Google Scholar
  7. Shai Evra and Tali Kaufman. Bounded degree cosystolic expanders of every dimension. In Proceedings of the forty-eighth annual ACM symposium on Theory of Computing, pages 36-48. ACM, 2016. Google Scholar
  8. Joel Friedman. A proof of Alon’s second eigenvalue conjecture. In Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, pages 720-724. ACM, 2003. Google Scholar
  9. Mikhail Gromov. Singularities, expanders and topology of maps. Part 2: From combinatorics to topology via algebraic isoperimetry. Geometric and Functional Analysis, 20(2):416-526, 2010. Google Scholar
  10. Shlomo Hoory, Nathan Linial, and Avi Wigderson. Expander graphs and their applications. Bulletin of the American Mathematical Society, 43(4):439-561, 2006. Google Scholar
  11. Mark Jerrum, Jung-Bae Son, Prasad Tetali, Eric Vigoda, et al. Elementary bounds on Poincaré and log-Sobolev constants for decomposable Markov chains. The Annals of Applied Probability, 14(4):1741-1765, 2004. Google Scholar
  12. Tali Kaufman, David Kazhdan, and Alexander Lubotzky. Ramanujan complexes and bounded degree topological expanders. In 2014 IEEE 55th Annual Symposium on Foundations of Computer Science, pages 484-493. IEEE, 2014. Google Scholar
  13. Tali Kaufman and David Mass. High dimensional random walks and colorful expansion. arXiv preprint, 2016. URL: http://arxiv.org/abs/1604.02947.
  14. Tali Kaufman and Izhar Oppenheim. High order random walks: Beyond spectral gap. arXiv preprint, 2017. URL: http://arxiv.org/abs/1707.02799.
  15. Tali Kaufman and Izhar Oppenheim. Construction of new local spectral high dimensional expanders. arXiv preprint, 2019. URL: http://arxiv.org/abs/1710.05304.
  16. David A Levin and Yuval Peres. Markov chains and mixing times, volume 107. American Mathematical Soc., 2017. Google Scholar
  17. Nathan Linial and Roy Meshulam. Homological connectivity of random 2-complexes. Combinatorica, 26(4):475-487, 2006. Google Scholar
  18. Alexander Lubotzky. High dimensional expanders. arXiv preprint, 2017. URL: http://arxiv.org/abs/1712.02526.
  19. Alexander Lubotzky, Ralph Phillips, and Peter Sarnak. Ramanujan graphs. Combinatorica, 8(3):261-277, 1988. Google Scholar
  20. Alexander Lubotzky, Beth Samuels, and Uzi Vishne. Explicit constructions of Ramanujan complexes of type Ad. European Journal of Combinatorics, 26(6):965-993, 2005. Google Scholar
  21. Omer Reingold, Salil Vadhan, and Avi Wigderson. Entropy waves, the zig-zag graph product, and new constant-degree expanders. Annals of mathematics, pages 157-187, 2002. Google Scholar
  22. Horst Sachs. Über teiler, faktoren und charakteristische polynome von graphen. Teil I. Wiss. Z. TH Ilmenau, 12:7-12, 1966. Google Scholar
  23. Michael Sipser and Daniel A Spielman. Expander codes. In Proceedings 35th Annual Symposium on Foundations of Computer Science, pages 566-576. IEEE, 1994. Google Scholar
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact