Metric recovery from directed unweighted graphs
This addresses a fundamental challenge in graph analysis for applications like network science and machine learning, though it is incremental as it builds on existing random walk and PageRank methods.
The paper tackles the problem of recovering the underlying Euclidean metric and density from directed unweighted graphs, such as k-nearest neighbor graphs, by showing that consistent recovery is possible up to isometric scaling when vertex degree meets a specified bound, with empirical validation on simulated and real-world co-purchasing graphs.
We analyze directed, unweighted graphs obtained from $x_i\in \mathbb{R}^d$ by connecting vertex $i$ to $j$ iff $|x_i - x_j| < ε(x_i)$. Examples of such graphs include $k$-nearest neighbor graphs, where $ε(x_i)$ 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 $ε(x_i)$ and the associated density $p(x_i)$ 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 $ω(n^{2/(2+d)}\log(n)^{d/(d+2)})$. 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 $\log(n)$.