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Graphs and network data are ubiquitous across a wide spectrum of scientificand application domains. Often in practice, an input graph can be considered asan observed snapshot of a (potentially continuous) hidden domain or process.Subsequent analysis, processing, and inferences are then performed on thisobserved graph. In this paper we advocate the perspective that an observedgraph is often a noisy version of some discretized 1-skeleton of a hiddendomain, and specifically we will consider the following natural network model:We assume that there is a true graph ${G^*}$ which is a certain proximity graphfor points sampled from a hidden domain $\mathcal{X}$; while the observed graph$G$ is an Erd$\"{o}$s-R$\'{e}$nyi type perturbed version of ${G^*}$.  Our network model is related to, and slightly generalizes, themuch-celebrated small-world network model originally proposed by Watts andStrogatz. However, the main question we aim to answer is orthogonal to theusual studies of network models (which often focuses on characterizing /predicting behaviors and properties of real-world networks). Specifically, weaim to recover the metric structure of ${G^*}$ (which reflects that of thehidden space $\mathcal{X}$ as we will show) from the observed graph $G$. Ourmain result is that a simple filtering process based on the \emph{Jaccardindex} can recover this metric within a multiplicative factor of $2$ under ournetwork model. Our work makes one step towards the general question ofinferring structure of a hidden space from its observed noisy graphrepresentation. In addition, our results also provide a theoreticalunderstanding for Jaccard-Index-based denoising approaches.

A quest to unravel the metric structure behind perturbed networks