Recovering Nonuniform Planted Partitions via Iterated Projection

In the planted partition problem, the n vertices of a random graph are partitioned into k “clusters,” and edges between vertices in the same cluster and different clusters are included with constant probability p and q, respectively (where 0 ≤ q < p ≤ 1). We give an efficient spectral algorithm that recovers the clusters with high probability, provided that the sizes of any two clusters are either very close or separated by ≥ Ω( √ n). We also discuss a generalization of planted partition in which the algorithm’s input is not a random graph, but a random real symmetric matrix with independent above-diagonal entries. Our algorithm is an adaptation of a previous algorithm for the uniform case, i.e., when all clusters are size n/k ≥ Ω( √ n). The original algorithm recovers the clusters one by one via iterated projection: it constructs the orthogonal projection operator onto the dominant kdimensional eigenspace of the random graph’s adjacency matrix, uses it to recover one of the clusters, then deletes it and recurses on the remaining vertices. We show herein that a similar algorithm works in the nonuniform case.