Volume Doubling Condition and a Local Poincar\'e Inequality on Unweighted Random Geometric Graphs

TL;DR

This paper proves that certain random geometric graphs built from points on a submanifold satisfy key geometric and functional inequalities, specifically volume doubling and local Poincaré inequalities, with high probability.

Contribution

It establishes the conditions under which volume doubling and local Poincaré inequalities hold for unweighted random geometric graphs on submanifolds.

Findings

Volume doubling condition holds with high probability.

Local Poincaré inequality is valid under regularity conditions.

Results apply uniformly over all shortest path balls within a certain radius.

Abstract

The aim of this paper is to establish two fundamental measure-metric properties of particular random geometric graphs. We consider $\varepsilon$-neighborhood graphs whose vertices are drawn independently and identically distributed from a common distribution defined on a regular submanifold of $\mathbb{R}^K$. We show that a volume doubling condition (VD) and local Poincar\'e inequality (LPI) hold for the random geometric graph (with high probability, and uniformly over all shortest path distance balls in a certain radius range) under suitable regularity conditions of the underlying submanifold and the sampling distribution.

Full text

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