We make the first steps towards generalizing the theory of stochastic block models, in the sparse regime, towards a model where the discrete community structure is replaced by an underlying geometry. We consider a geometric random graph over a homogeneous metric space where the probability of two vertices to be connected is an arbitrary function of the distance. We give sufficient conditions under which the locations can be recovered (up to an isomorphism of the space) in the sparse regime. Moreover, we define a geometric counterpart of the model of flow of information on trees, due to Mossel and Peres, in which one considers a branching random walk on a sphere and the goal is to recover the location of the root based on the locations of leaves. We give some sufficient conditions for percolation and for non-percolation of information in this model.

翻译：我们采取初步步骤,在稀少的政权中,将零星区块模型的理论推广到以基本几何测量法取代离散社区结构的模型。我们考虑在单一度空间上绘制几何随机图,在这种空间中,连接两个脊椎的概率是距离的任意功能。我们为在稀少的政权中恢复位置(直至空间的无形态化)提供了充分的条件。此外,我们界定了由于莫塞尔和佩雷斯的树木信息流动模型的几何对应方,在这种模型中,我们考虑在球体上随机行走,目标是根据叶子的位置恢复根的位置。我们为这一模型中的信息的渗透和不相容提供了一些充分的条件。