A note on estimating the dimension from a random geometric graph

Abstract

Let $G_n$ be a random geometric graph with vertex set $[n]$ based on n i.i.d. random vectors $X_1,\dots ,X_n$ drawn from an unknown density f on ${\mathbb{R}}^d$. An edge $(i,j)$ is present when $\Vert X_i-X_j\Vert \le r_n$, for a given threshold $r_n$ possibly depending upon n, where $\Vert \cdot \Vert$ denotes Euclidean distance. We study the problem of estimating the dimension d of the underlying space when we have access to the adjacency matrix of the graph but do not know $r_n$ or the vectors $X_i$. The main result of the paper is that there exists an estimator of d that converges to d in probability as $n\to \infty$ for all densities with $\int f^5<\infty$ whenever $n^{3∕2}{r}_n^d\to \infty$ and $r_n=o(1)$. The conditions allow very sparse graphs since when $n^{3∕2}{r}_n^d\to 0$, the graph contains isolated edges only, with high probability. We also show that, without any condition on the density, a consistent estimator of d exists when $n{r}_n^d\to \infty$ and $r_n=o(1)$.