Spectral properties of the resistance Laplacian with applications to data clustering and anomaly detection

Abstract: The resistance Laplacian is a graph matrix associated with the effective resistance metric and provides a global counterpart of the classical graph Laplacian. Although it inherits several fundamental properties of the ordinary Laplacian, including a connected graph partitioning theorem analogous to that of Fiedler, its intrinsic spectral structure has remained largely unexplored. In this paper, we develop a structural theory of the resistance Laplacian. We derive a canonical decomposition that separates its intrinsic, average, and deviation components, thereby revealing how the global geometry induced by effective resistance differs from the local geometry encoded by the ordinary Laplacian. Building upon this decomposition, we establish several structural and spectral properties of the associated deviation operator, obtain variational characterizations of the largest eigenvalue and its corresponding eigenspace, and express the resistance Laplacian in Laplacian coordinates, thereby elucidating the relationship between the eigenspaces of the two operators. Finally, we formulate resistance-based graph partitioning objectives whose spectral relaxations recover the dominant eigenvector of the resistance Laplacian, providing a variational interpretation of the connected partition theorem. Experimental results on synthetic and real-life datasets demonstrate the effectiveness of the proposed framework for graph partitioning, data clustering, and exploratory anomaly detection.