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We analyze directed, unweighted graphs obtained from $x_i\in \mathbb{R}^d$ byconnecting vertex $i$ to $j$ iff $|x_i - x_j| < \epsilon(x_i)$. Examples ofsuch graphs include $k$-nearest neighbor graphs, where $\epsilon(x_i)$ variesfrom point to point, and, arguably, many real world graphs such asco-purchasing graphs. We ask whether we can recover the underlying Euclideanmetric $\epsilon(x_i)$ and the associated density $p(x_i)$ given only thedirected graph and $d$.  We show that consistent recovery is possible up to isometric scaling when thevertex degree is at least $\omega(n^{2/(2+d)}\log(n)^{d/(d+2)})$. Our estimatoris based on a careful characterization of a random walk over the directed graphand the associated continuum limit. As an algorithm, it resembles the PageRankcentrality metric. We demonstrate empirically that the estimator performs wellon simulated examples as well as on real-world co-purchasing graphs even with asmall number of points and degree scaling as low as $\log(n)$.

Metric recovery from directed unweighted graphs