Signed Laplacians for Constrained Graph Clustering

摘要

Given two weighted graphs $G = (V, E, w_G)$ and $H = (V, F, w_H)$ defined on the same vertex set, the constrained clustering problem seeks to find a subset $S \subset V$ that minimises the cut ratio between $w_G(S, V \setminus S)$ and $w_H(S, V \setminus S)$. In this work, we establish a Cheeger-type inequality that relates the solution of the constrained clustering problem to the spectral properties of $ G$ and $H$. To reduce computational complexity, we utilise the signed Laplacian of $H$, streamlining calculations while maintaining accuracy. By solving a generalised eigenvalue problem, our proposed algorithm achieves notable performance improvements, particularly in challenging scenarios where traditional spectral clustering methods struggle. We demonstrate its practical effectiveness through experiments on both synthetic and real-world datasets.