Finding the Number of Clusters in a Graph: a Nearly-Linear Time Algorithm

Suranjan De, He Sun

Keywords: spectral clustering, eigen-gap heuristic, number of clusters

TL;DR: We present a nearly-linear time algorithm that compute the number of clusters in a graph.

Abstract:

Given an undirected graph $G$ with the normalised adjacency matrix $N_G$, the well-known eigen-gap heuristic for clustering asserts that $G$ has $k$ clusters if there is a large gap between the $k$th and $(k+1)$th largest eigenvalues of $N_G$. Although this heuristic is well-supported in spectral graph theory and widely applied in practice, determining $k$ often relies on computing the eigenvalues of $N_G$ with high time complexity. This paper addresses this key problem in graph clustering, and shows that the number of clusters
$k$ implied by the eigen-gap heuristic can be computed in nearly-linear time.

Primary Area: learning on graphs and other geometries & topologies

Code Of Ethics: I acknowledge that I and all co-authors of this work have read and commit to adhering to the ICLR Code of Ethics.

Submission Guidelines: I certify that this submission complies with the submission instructions as described on https://iclr.cc/Conferences/2025/AuthorGuide.

Anonymous Url: I certify that there is no URL (e.g., github page) that could be used to find authors’ identity.

No Acknowledgement Section: I certify that there is no acknowledgement section in this submission for double blind review.

Submission Number: 11203