2015/03/02 by Sam Cole, Shmuel Friedland, Cole, Sam +3
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Data Structures and Algorithms (cs.DS) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #Random Matrices and Applications #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1503.00423
openalex publication_date 2015/03/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we consider the planted partition model, in which n = ks vertices of a random graph are partitioned into k "clusters," each of size s. 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 algorithm that, with high probability, recovers the clusters as long as the cluster sizes are are least Ω(√(n)). Informally, our algorithm constructs the projection operator onto the dominant k-dimensional eigenspace of the graph's adjacency matrix and uses it to recover one cluster at a time. To our knowledge, our algorithm is the first purely spectral algorithm which runs in polynomial time and works even when s = Θ(√ n), though there have been several non-spectral algorithms which accomplish this. Our algorithm is also among the simplest of these spectral algorithms, and its proof of correctness illustrates the usefulness of the Cauchy integral formula in this domain.