How the result of graph clustering methods depends on the construction of the graph

¹ Max Planck Institute for Intelligent Systems, Spemannstr. 38, 72076 Tübingen, Germany
mmaier@tuebingen.mpg.de
² Department of Computer Science, University of Hamburg, Vogt-Kölln-Str. 30, 22527 Hamburg, Germany
luxburg@informatik.uni-hamburg.de
³ Faculty of Mathematics and Computer Science, Saarland University, Postfach 151150, 66041, Saarbrücken, Germany
hein@cs.uni-sb.de

Received: 23 November 2010
Revised: 21 December 2011

Abstract

We study the scenario of graph-based clustering algorithms such as spectral clustering. Given a set of data points, one first has to construct a graph on the data points and then apply a graph clustering algorithm to find a suitable partition of the graph. Our main question is if and how the construction of the graph (choice of the graph, choice of parameters, choice of weights) influences the outcome of the final clustering result. To this end we study the convergence of cluster quality measures such as the normalized cut or the Cheeger cut on various kinds of random geometric graphs as the sample size tends to infinity. It turns out that the limit values of the same objective function are systematically different on different types of graphs. This implies that clustering results systematically depend on the graph and can be very different for different types of graph. We provide examples to illustrate the implications on spectral clustering.