$n \rightarrow \infty$) in the way the density $\rho$ affects the differential operator $\Delta_\rho$ (which in general will differ from ). We expand our discussion on the choice of different normalizations in Section \[sec:differentweights\]; for an overview of a more general family of normalizations see [@FHBHAS]. In the context of this paper, it is enough to say that $\Delta_n$ and $\Delta_\rho$ (as defined in ) are closely related to each other according to Theorem \[TheoremTranps\].

In order to state and prove our second main result we make some additional assumptions on the ground-truth measure $\nu$, on the number of data points $n$ and the connectivity parameter ${\varepsilon}$.

\[assumption-on-rho-contrast\] We assume that ${\mathcal{M}}$ is a smooth, compact, orientable, connected $m$-dimensional manifold without a boundary. Furthermore, we assume that $\rho\in C^1({\mathcal{M}})$, and that there exist positive numbers $0<\rho^-< \rho^+$ such that $$\rho^- \le \rho(x) \le \rho^+, \qquad \forall x \in \mathcal{M}.$$ We denote by $C_{Lip}>0$ the Lipschitz constant of $\rho$.

We also assume that $n$ is large enough and ${\varepsilon}$ is small enough, so that $$\label{ineq:assump}
{\varepsilon}\left(1+ \sqrt{\lambda_N}\right) + \frac{\log(n)^{p_m}}{n^{1/m}{\varepsilon}} \leq \min \{ C, \frac{1}{2}(\lambda_{N+1}- \lambda_N) \} \,,$$ where $p_m= 3/4$ if $m=2$ and $p_m= 1/m$ when $m\geq 3$, $C>0$ is a 