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Volume Doubling Condition and a Local Poincaré Inequality on Unweighted Random Geometric Graphs

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.

Dates and versions

hal-03082496 , version 1 (18-12-2020)

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Franziska Göbel, Gilles Blanchard. Volume Doubling Condition and a Local Poincaré Inequality on Unweighted Random Geometric Graphs. 2020. ⟨hal-03082496⟩
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