Departure from Regularity: Degree Heterogeneity and Eigengap as the Structural Drivers of ASE-LSE Latent Subspace Disagreement

2026-05-21

AI Summary

This study elucidates the structural origins underlying the divergence between adjacency spectral embedding (ASE) and Laplacian spectral embedding (LSE) in capturing distinct latent subspaces from the same network. Leveraging spectral graph theory, random graph models, and matrix perturbation analysis, the authors establish node degree homogeneity as a sufficient condition for perfect alignment between ASE and LSE. They further demonstrate, for the first time from a structural perspective, that degree heterogeneity amplifies the discrepancy between ASE and LSE, whereas strong community structure mitigates it. To quantify this effect, they propose an interpretable metric based on the ratio of degree heterogeneity to community strength, which effectively predicts the interchangeability of the two embeddings. Extensive simulations validate both the derived error bound and the predictive power of the proposed metric.

Abstract

Two of the most widely used methods for analysing graph data, Adjacency Spectral Embedding and Laplacian Spectral Embedding, often produce different results when applied to the same network. Yet the structural reasons behind this disagreement remain incompletely understood. This paper provides a structural account. We show that regularity is a sufficient condition for perfect agreement: when every node has the same number of connections, the two methods produce identical latent subspaces. Any departure from this regularity introduces disagreement, and we prove an explicit bound whose two terms suggest the structural ingredients controlling it: degree heterogeneity, which pushes the methods apart, and community structure strength, which pulls them back together. We validate both drivers empirically across thousands of simulated networks, confirming that heterogeneity drives disagreement up, community strength suppresses it, and their ratio provides a strong predictor of when the two embeddings can be treated as interchangeable and when they cannot.

Problem

• spectral embedding
• graph data
• degree heterogeneity
• eigengap
• community structure

Innovation

• degree heterogeneity
• eigengap
• spectral embedding
• graph regularity
• community structure