Spectral Embeddings of Degree-α Laplacians in Random Dot Product Graphs

Spectral clustering is a standard tool for community detection, yet its performance depends heavily on the choice of the graph matrix used for embedding—typically the adjacency matrix or the symmetric Laplacian. This paper investigates a continuous family of degree-normalized matrices, D^-α AD^-α, to understand how the parameter α influences the geometry and local uncertainty of node embeddings. The authors seek to determine if a systematic preference for α exists and which network features dictate that choice.

The authors employ the Random Dot Product Graph (RDPG) model to derive a row-wise central limit theorem for the degree-α spectral embedding. By analyzing the limiting distribution of the embedded nodes, they characterize how the normalization parameter α affects both the population centers and the local fluctuations (uncertainty) of the nodes. They specifically apply these results to the Degree-Corrected Stochastic Block Model (DCSBM) and use a projected-Gaussian Bayes-error diagnostic to compare the performance of different α values.

The study reveals that while spherical row normalization makes the population community directions identical across different values of α, the local uncertainty—represented by Gaussian probability ellipses—varies significantly. As α increases, the radial dependence on degree-correction parameters is reduced, but the shape and orientation of the local fluctuations change. The authors demonstrate that no single α is universally optimal; rather, the ideal normalization depends on the network's density, community imbalance, and block-probability structure. Specifically, stronger normalization is typically favored in lower-density or more imbalanced settings.

This work provides a unified theoretical framework for understanding the statistical consequences of degree normalization in spectral clustering. By moving beyond simple population-level geometry to analyze second-order distributional deviations, the authors offer a rigorous basis for selecting normalization parameters. This helps practitioners move away from arbitrary choices of graph matrices toward principled, model-informed decisions that can improve clustering accuracy in challenging network environments.