Spectral clustering for multiclass Erdös-Rényi graphs

In this article, we study the properties of the spectral analysis of multiclass Erdös-Rényi graphs. With a view towards using the embedding afforded by the decomposition of the graph Laplacian for subsequent processing, we analyze two basic geometric properties, namely interclass intersection and interclass distance. We will first study the dyadic two-class case in details and observe the existence of a phase transition for the interclass intersection. We then focus on the general multiclass case, where we introduce an appropriate notion of diagonal concentration and derive a statistical model that allows sampling graphs whose expected diagonal concentration is fixed. The simulations provided yield useful guidelines for practitioners to choose appropriately parameters in the context of spectral clustering.