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Metric Poincaré inequalities for graphs

Dylan J. Altschuler, Pandelis Dodos, Konstantin Tikhomirov, Konstantinos Tyros

Abstract

This article considers embeddings of bounded degree graphs into general metric spaces. Our first main result is a metric analogue of Matoušek's extrapolation that relates the Poincaré constants $γ(G,\varrho^p)$ and $γ(G,\varrho^q)$ for any pair of exponents $0 < p,q < \infty$, any bounded degree expander graph $G$, and any metric space $\mathcal{M}=(M,\varrho)$. Our second main result provides a sharp estimate of the Poincaré constant $γ(G,\varrho)$ in terms of the cardinalities of the vertex set of $G$ and the target metric space $\mathcal{M}=(M,\varrho)$, in the setting of \textit{random} graphs. Both theorems utilize a novel structural dichotomy for metric embeddings of graphs. These results lead to the resolution of several problems within the theory of metric embeddings. Among the applications are estimates on the nonlinear spectral gap of metric snowflakes, optimal estimates on the minimum cardinality of (bi-Lipschitz) universal metric spaces for graphs, and sharp lower bounds on the bi-Lipschitz distortion of random regular graphs into arbitrary metric spaces of large cardinality.

Metric Poincaré inequalities for graphs

Abstract

This article considers embeddings of bounded degree graphs into general metric spaces. Our first main result is a metric analogue of Matoušek's extrapolation that relates the Poincaré constants and for any pair of exponents , any bounded degree expander graph , and any metric space . Our second main result provides a sharp estimate of the Poincaré constant in terms of the cardinalities of the vertex set of and the target metric space , in the setting of \textit{random} graphs. Both theorems utilize a novel structural dichotomy for metric embeddings of graphs. These results lead to the resolution of several problems within the theory of metric embeddings. Among the applications are estimates on the nonlinear spectral gap of metric snowflakes, optimal estimates on the minimum cardinality of (bi-Lipschitz) universal metric spaces for graphs, and sharp lower bounds on the bi-Lipschitz distortion of random regular graphs into arbitrary metric spaces of large cardinality.

Paper Structure

This paper contains 57 sections, 31 theorems, 350 equations.

Table of Contents

  1. Introduction
  2. Matoušek's extrapolation for metric spaces
  3. Metric Poincaré inequalities for random graphs
  4. Applications to metric embeddings
  5. Embeddings of random regular graphs into metric spaces
  6. Bi-Lipschitz universality
  7. Metric snowflakes
  8. Outline of the proofs: key ideas and technical overview
  9. A structural dichotomy
  10. Extrapolation
  11. Left extrapolation for concentrated $f$
  12. Non-concentrated $f$
  13. Metric Poincaré inequalities
  14. Information about $f$
  15. Approximation scheme
  16. ...and 42 more sections

Key Result

Theorem 1.2

Let $d\geqslant 3$ be an integer, let $G$ be a $d$-regular graph with Cheeger constantThe Cheeger constant is defined in Subsection sec3-Cheeger; our asymptotic notation is defined in Subsection sec3-asymptotic.$h(G)>0$, and let $\mathcal{M}=(M,\varrho)$ be an arbitrary metric space. Then, for any $ and where the Poincaré constants $\gamma(G,\varrho^p)$, $\gamma(G,\varrho^q)$ are defined accordin

Theorems & Definitions (89)

  • Definition 1.1: Nonlinear spectral gap
  • Theorem 1.2: Extrapolation for metric spaces
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5: Poincaré inequality for metric spaces
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8: Comparison of normed and metric settings
  • Corollary 1.9
  • Remark 1.10
  • ...and 79 more