Density estimation from unweighted k-nearest neighbor graphs: a roadmap

Consider an unweighted k-nearest neighbor graph on n points that have been sampled i.i.d. from some unknown density p on R. We prove how one can estimate the density p just from the unweighted adjacency matrix of the graph, without knowing the points themselves or any distance or similarity scores. The key insights are that local differences in link numbers can be used to estimate a local function of the gradient of p, and that integrating this function along shortest paths leads to an estimate of the underlying density.