  Random graphs with latent geometric structure are popular models of social
and biological networks, with applications ranging from network user profiling
to circuit design. These graphs are also of purely theoretical interest within
computer science, probability and statistics. A fundamental initial question
regarding these models is: when are these random graphs affected by their
latent geometry and when are they indistinguishable from simpler models without
latent structure, such as the Erd\H{o}s-R\'{e}nyi graph $\mathcal{G}(n, p)$? We
address this question for two of the most well-studied models of random graphs
with latent geometry -- the random intersection and random geometric graph.
  Our results are as follows: (1) we prove that the random intersection graph
converges in total variation to $\mathcal{G}(n, p)$ when $d =
\tilde{\omega}(n^3)$, and does not if $d = o(n^3)$, resolving an open problem
in Fill et al. (2000), Rybarczyk (2011) and Kim et al. (2018); (2) we provide
conditions under which the matrix of intersection sizes of random family of
sets converges in total variation to a symmetric matrix with independent
Poisson entries, yielding the first total variation convergence result for
$\tau$-random intersection graphs to $\mathcal{G}(n, p)$; and (3) we show that
the random geometric graph on $\mathbb{S}^{d - 1}$ with edge density $p$
converges in total variation to $\mathcal{G}(n, p)$ when $d =
\tilde{\omega}\left(\min\{ pn^3, p^2 n^{7/2} \} \right)$, yielding the first
progress towards a conjecture of Bubeck et al. (2016). The first of these three
results was obtained simultaneously and independently by Bubeck, Racz and
Richey.
