# Metric recovery from directed unweighted graphs

arXiv: https://arxiv.org/abs/1411.5720 · HTML: https://mlsanity.com/paper/1411.5720/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Nov 2014.

## Built on by (most cited first, 4 of 4)

- [Comparing Foundation Models using Data Kernels](https://mlsanity.com/paper/2305.05126/) (2023)
- [Generalization Error Bound for Hyperbolic Ordinal Embedding](https://mlsanity.com/paper/2105.10475/) (2021)
- [Graph Neural Networks can Recover the Hidden Features Solely from the Graph Structure](https://mlsanity.com/paper/2301.10956/) (2023)
- [Particle Filter Bridge Interpolation](https://mlsanity.com/paper/2103.14963/) (2021)

## Abstract (arXiv)

We analyze directed, unweighted graphs obtained from xi∈ ℝd by connecting vertex i to j iff |xi - xj| < ε(xi). Examples of such graphs include k-nearest neighbor graphs, where ε(xi) varies from point to point, and, arguably, many real world graphs such as co-purchasing graphs. We ask whether we can recover the underlying Euclidean metric ε(xi) and the associated density p(xi) given only the directed graph and d. We show that consistent recovery is possible up to isometric scaling when the vertex degree is at least ω(n2/(2+d)log(n)d/(d+2)). Our estimator is based on a careful characterization of a random walk over the directed graph and the associated continuum limit. As an algorithm, it resembles the PageRank centrality metric. We demonstrate empirically that the estimator performs well on simulated examples as well as on real-world co-purchasing graphs even with a small number of points and degree scaling as low as log(n).