Consider an unweighted k-nearest neighbor graph on n points that have been sampled i.i.d.

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

Consider a weighted or unweighted k-nearest neighbor graph that has been built on n data points drawn randomly according to some density p on R^d. We study the convergence of the shortest path distance in such graphs as the sample size tends to infinity. We prove that for unweighted kNN graphs, this distance converges to an unpleasant distance function on the underlying space whose properties are detrimental to machine learning. We also study the behavior of the shortest path distance in weighted kNN graphs.