Why do two graph analysis methods give different answers?

Adjacency and Laplacian spectral embeddings are both popular for extracting structure from networks, but they frequently produce different results. This work reveals the structural reason: when all nodes have identical degrees (a regular graph), the methods agree perfectly; any deviation introduces disagreement. The amount of disagreement depends on degree heterogeneity pushing them apart and community structure pulling them together, captured in an explicit bound. Validation across thousands of simulated networks confirms the theory and provides practitioners a way to predict when these embeddings can be used interchangeably.