Matching Correlated Inhomogeneous Random Graphs using the $k$-core Estimator

TL;DR

This paper introduces a $k$-core based method for estimating vertex correspondence in correlated inhomogeneous random graphs, providing theoretical recovery guarantees and applying to various graph models.

Contribution

It develops a general framework for the $k$-core estimator and derives new recovery conditions for multiple inhomogeneous graph models.

Findings

Conditions for exact and partial recovery established

Application to stochastic block models and Chung-Lu graphs

Theoretical guarantees for the $k$-core estimator

Abstract

We consider the task of estimating the latent vertex correspondence between two edge-correlated random graphs with generic, inhomogeneous structure. We study the so-called \emph{$k$-core estimator}, which outputs a vertex correspondence that induces a large, common subgraph of both graphs which has minimum degree at least $k$. We derive sufficient conditions under which the $k$-core estimator exactly or partially recovers the latent vertex correspondence. Finally, we specialize our general framework to derive new results on exact and partial recovery in correlated stochastic block models, correlated Chung-Lu graphs, and correlated random geometric graphs.