Abstract
Graphs and network data are ubiquitous across a wide spectrum of scientific and application domains. Often in practice, an input graph can be considered as an observed snapshot of a (potentially
continuous) hidden domain or process. Subsequent analysis, processing, and inferences are then performed on this observed graph. In this paper we advocate the perspective that an observed graph is often a noisy version of some discretized 1skeleton of a hidden domain, and specifically we will consider the following natural network model: We assume that there is a true graph G^* which is a certain proximity graph for points sampled from a hidden domain X; while the observed graph G is an ErdosRenyi type perturbed version of G^*.
Our network model is related to, and slightly generalizes, the muchcelebrated smallworld network model originally proposed by Watts and Strogatz. However, the main question we aim to answer is orthogonal to the usual studies of network models (which often focuses on characterizing / predicting behaviors and properties of realworld networks). Specifically, we aim to recover the metric structure of G^* (which reflects that of the hidden space X as we will show) from the observed graph G. Our main result is that a simple filtering process based on the Jaccard index can recover this metric within a multiplicative factor of 2 under our network model. Our work makes one step towards the general question of inferring structure of a hidden space from its observed noisy graph representation. In addition, our results also provide a theoretical understanding for JaccardIndexbased denoising approaches.
BibTeX  Entry
@InProceedings{parthasarathy_et_al:LIPIcs:2017:7211,
author = {Srinivasan Parthasarathy and David Sivakoff and Minghao Tian and Yusu Wang},
title = {{A Quest to Unravel the Metric Structure Behind Perturbed Networks}},
booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)},
pages = {53:153:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770385},
ISSN = {18688969},
year = {2017},
volume = {77},
editor = {Boris Aronov and Matthew J. Katz},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/7211},
URN = {urn:nbn:de:0030drops72112},
doi = {10.4230/LIPIcs.SoCG.2017.53},
annote = {Keywords: metric structure, Erd{\"o}sR{\'e}nyi perturbation, graphs, doubling measure}
}
Keywords: 

metric structure, ErdösRényi perturbation, graphs, doubling measure 
Seminar: 

33rd International Symposium on Computational Geometry (SoCG 2017) 
Issue Date: 

2017 
Date of publication: 

08.06.2017 