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 Rd. An edge (i,j) is present when |Xi -Xj| ≤ rn, for a given threshold rn possibly depending upon n, where | · | 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 → ∞ for all densities with int f5 < ∞ whenever n3/2 rnd → ∞ and rn = o(1). The conditions allow very sparse graphs since when n3/2 rnd → 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 rnd → ∞ and rn = o(1).