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We study the scenario of graph-based clustering algorithms such as spectralclustering. Given a set of data points, one first has to construct a graph onthe data points and then apply a graph clustering algorithm to find a suitablepartition of the graph. Our main question is if and how the construction of thegraph (choice of the graph, choice of parameters, choice of weights) influencesthe outcome of the final clustering result. To this end we study theconvergence of cluster quality measures such as the normalized cut or theCheeger cut on various kinds of random geometric graphs as the sample sizetends to infinity. It turns out that the limit values of the same objectivefunction are systematically different on different types of graphs. Thisimplies that clustering results systematically depend on the graph and can bevery different for different types of graph. We provide examples to illustratethe implications on spectral clustering.

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