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Universal geometric non-embedding of random regular graphs

Dylan J. Altschuler, Konstantin Tikhomirov

TL;DR

This work shows that for any fixed degree $\Delta\ge3$, a uniformly random $\Delta$-regular graph on $n$ vertices almost surely cannot be realized as a geometric graph in any $d$-dimensional normed space when $d\le c\log n$, with a universal constant $c$. The authors develop a seeded, multiscale $\varepsilon$-net discretization to reduce continuous embedding problems to finite combinatorial ones and leverage non-linear Poincaré inequalities to relate graph geometry to embedding obstructions. A central technical contribution is a robust non-embedding estimate over the Banach–Mazur compactum, which, together with a net-argument over normed spaces, yields a high-probability non-embedding result without dimension blow-up factors. The findings imply a universal, norm-agnostic obstruction for typical random regular graphs, sharpening our understanding of dimensional limits in metric embeddings and suggesting broader applicability of the discretization framework to other embedding paradigms.

Abstract

Let $Δ\ge 3$ be fixed, $n \ge n_Δ$ be a large integer. It is a classical result that $Δ$--regular expanders on $n$ vertices are not embeddable as geometric (distance) graphs into Euclidean space of dimension less than $c \log n$, for some universal constant $c$. We show that for typical $Δ$-regular graphs, this obstruction is universal with respect to the choice of norm. More precisely, for a uniform random $Δ$-regular graph $G$ on $n$ vertices, it holds with high probability: there is no normed space of dimension less than $c\log n$ which admits a geometric graph isomorphic to $G$. The proof is based on a seeded multiscale $\varepsilon$--net argument.

Universal geometric non-embedding of random regular graphs

TL;DR

This work shows that for any fixed degree , a uniformly random -regular graph on vertices almost surely cannot be realized as a geometric graph in any -dimensional normed space when , with a universal constant . The authors develop a seeded, multiscale -net discretization to reduce continuous embedding problems to finite combinatorial ones and leverage non-linear Poincaré inequalities to relate graph geometry to embedding obstructions. A central technical contribution is a robust non-embedding estimate over the Banach–Mazur compactum, which, together with a net-argument over normed spaces, yields a high-probability non-embedding result without dimension blow-up factors. The findings imply a universal, norm-agnostic obstruction for typical random regular graphs, sharpening our understanding of dimensional limits in metric embeddings and suggesting broader applicability of the discretization framework to other embedding paradigms.

Abstract

Let be fixed, be a large integer. It is a classical result that --regular expanders on vertices are not embeddable as geometric (distance) graphs into Euclidean space of dimension less than , for some universal constant . We show that for typical -regular graphs, this obstruction is universal with respect to the choice of norm. More precisely, for a uniform random -regular graph on vertices, it holds with high probability: there is no normed space of dimension less than which admits a geometric graph isomorphic to . The proof is based on a seeded multiscale --net argument.
Paper Structure (6 sections, 13 theorems, 58 equations)

This paper contains 6 sections, 13 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Non-linear Poincaré inequalities, and geometric embeddings
  3. Construction of the discretization scheme
  4. Proof of Main Result
  5. Preliminaries
  6. Robust non-embedding

Key Result

Proposition 1.1

For every $\beta>0$ there is $c>0$ depending only on $\beta$ with the following property. Let $\Delta\geq 3$, and for a large $n$ with $\Delta n$ even, let $G_n$ be a $\Delta$--regular graph on $n$ vertices with spectral gap at least $\beta$. Then for every $d\leq c \log n$ there is no geometric emb

Theorems & Definitions (33)

  • Proposition 1.1: Corollary of the Poincaré inequality in Euclidean spaces
  • Proposition 1.2: A corollary of Naor2021 and \ref{['eq: suf condition']}
  • Theorem 1.1: Main result
  • Remark 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Proposition 2.1
  • Definition 3.1: $n$--tuples
  • Definition 3.2: Sparse $n$--tuples
  • ...and 23 more