Spectra of Random Graphs with Planted Partitions

Spectral methods for clustering are now standard, and there are many toy examples in which they can be seen to yield more sensible solutions than classical schemes like vanilla k-means. A more rigorous analysis of these methods has proved elusive, however, and has so far consisted mostly of probabilistic analyses for random inputs with planted clusterings. Such an analysis, typically calls for proving tight asymptotic bounds on the spectrum of the graph in question. In this paper, we study a considerably broad data model first introduced by Feige and Kilian [FK01]: the planted partition graph model. We prove tight bounds on the Laplacian and Adjacency spectrum of those graphs which we think will be crucial to the design and analysis of an exact algorithm for planted partition as well as semi-random graph k-clustering.