A note on estimating the dimension from a random geometric graph

Let $Gn$ be a random geometric graph with vertex set $[n]$ based on $n$ i.i.d.\ random vectors $X1,\ldots,Xn$ drawn from an unknown density $f$ on $\R^d$. An edge $(i,j)$ is present when $|Xi -Xj| \le rn$, for a given threshold $rn$ possibly depending upon $n$, where $| \cdot |$ 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 $rn$ or the vectors $Xi$. 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 $\int f⁵ < \infty$ whenever $n^{3/2} rn^d \to \infty$ and $rn = o(1)$. The conditions allow very sparse graphs since when $n^{3/2} rn^d \to 0$, 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 $n rn^d \to \infty$ and $rn = o(1)$.