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Local Cliques in ER-Perturbed Random Geometric Graphs

Authors Matthew Kahle, Minghao Tian, Yusu Wang

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Matthew Kahle
  • Department of Mathematics, The Ohio State University, USA
Minghao Tian
  • Computer Science and Engineering Dept., The Ohio State University, USA
Yusu Wang
  • Computer Science and Engineering Dept., The Ohio State University, USA

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Matthew Kahle, Minghao Tian, and Yusu Wang. Local Cliques in ER-Perturbed Random Geometric Graphs. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 29:1-29:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ISAAC.2019.29

Abstract

We study a random graph model introduced in [Srinivasan Parthasarathy et al., 2017] where one adds Erdős - Rényi (ER) type perturbation to a random geometric graph. More precisely, assume G_X^* is a random geometric graph sampled from a nice measure on a metric space X = (X,d). An ER-perturbed random geometric graph G^(p,q) is generated by removing each existing edge from G_X^* with probability p, while inserting each non-existent edge to G_X^* with probability q. We consider a localized version of clique number for G^(p,q): Specifically, we study the edge clique number for each edge in a graph, defined as the size of the largest clique(s) in the graph containing that edge. We show that the edge clique number presents two fundamentally different types of behaviors in G^(p,q), depending on which "type" of randomness it is generated from. As an application of the above results, we show that by a simple filtering process based on the edge clique number, we can recover the shortest-path metric of the random geometric graph G_X^* within a multiplicative factor of 3 from an ER-perturbed observed graph G^(p,q), for a significantly wider range of insertion probability q than what is required in [Srinivasan Parthasarathy et al., 2017].

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Random graphs
  • Theory of computation → Computational geometry
Keywords
  • random graphs
  • random geometric graphs
  • edge clique number
  • the probabilistic method
  • metric recovery

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