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 ε-neighborhood graphs whose vertices are drawn independently and identically distributed from a common distribution defined on a regular submanifold of R K . We show that a volume doubling condition (VD) and local Poincaré 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.