  We study the problem of detecting latent geometric structure in random
graphs. To this end, we consider the soft high-dimensional random geometric
graph $\mathcal{G}(n,p,d,q)$, where each of the $n$ vertices corresponds to an
independent random point distributed uniformly on the sphere
$\mathbb{S}^{d-1}$, and the probability that two vertices are connected by an
edge is a decreasing function of the Euclidean distance between the points. The
probability of connection is parametrized by $q \in [0,1]$, with smaller $q$
corresponding to weaker dependence on the geometry; this can also be
interpreted as the level of noise in the geometric graph. In particular, the
model smoothly interpolates between the spherical hard random geometric graph
$\mathcal{G}(n,p,d)$ (corresponding to $q = 1$) and the Erd\H{o}s-R\'enyi model
$\mathcal{G}(n,p)$ (corresponding to $q = 0$). We focus on the dense regime
(i.e., $p$ is a constant).
  We show that if $nq \to 0$ or $d \gg n^{3} q^{2}$, then geometry is lost:
$\mathcal{G}(n,p,d,q)$ is asymptotically indistinguishable from
$\mathcal{G}(n,p)$. On the other hand, if $d \ll n^{3} q^{6}$, then the signed
triangle statistic provides an asymptotically powerful test for detecting
geometry. These results generalize those of Bubeck, Ding, Eldan, and R\'acz
(2016) for $\mathcal{G}(n,p,d)$, and give quantitative bounds on how the noise
level affects the dimension threshold for losing geometry. We also prove
analogous results under a related but different distributional assumption, and
we further explore generalizations of signed triangles in order to understand
the intermediate regime left open by our results.
