  We consider a latent space model for random graphs where a node $i$ is
associated to a random latent point $X_i$ on the Euclidean unit ball. The
probability that an edge exists between two nodes is determined by a ``link''
function, which corresponds to a dot product kernel. For a given class $\F$ of
spherically symmetric distributions for $X_i$, we consider two estimation
problems: latent norm recovery and latent Gram matrix estimation. We construct
an estimator for the latent norms based on the degree of the nodes of an
observed graph in the case of the model where the edge probability is given by
$f(\langle X_i,X_j\rangle)=\mathbbm{1}_{\langle X_i,X_j\rangle\geq \tau}$,
where $0<\tau<1$. We introduce an estimator for the Gram matrix based on the
eigenvectors of observed graph and we establish Frobenius type guarantee for
the error, provided that the link function is sufficiently regular in the
Sobolev sense and that a spectral-gap-type condition holds. We prove that for
certain link functions, the model considered here generates graphs with degree
distribution that have tails with a power-law-type distribution, which can be
seen as an advantage of the model presented here with respect to the classic
Random Geometric Graph model on the Euclidean sphere. We illustrate our results
with numerical experiments.
