On spectral embedding performance and elucidating network structure in stochastic block model graphs

TL;DR

This paper compares the performance of Laplacian and adjacency spectral embeddings in stochastic block models, revealing their strengths depending on network sparsity and structure, and providing theoretical insights into their relative effectiveness.

Contribution

It offers an asymptotic, information-theoretic analysis of spectral embedding methods, clarifying when each method performs better based on network properties.

Findings

Laplacian spectral embedding favors sparse graphs.

Adjacency spectral embedding favors not-too-sparse graphs.

Adjacency spectral embedding is better for core-periphery structures.

Abstract

Statistical inference on graphs often proceeds via spectral methods involving low-dimensional embeddings of matrix-valued graph representations, such as the graph Laplacian or adjacency matrix. In this paper, we analyze the asymptotic information-theoretic relative performance of Laplacian spectral embedding and adjacency spectral embedding for block assignment recovery in stochastic block model graphs by way of Chernoff information. We investigate the relationship between spectral embedding performance and underlying network structure (e.g.~homogeneity, affinity, core-periphery, (un)balancedness) via a comprehensive treatment of the two-block stochastic block model and the class of $K$-block models exhibiting homogeneous balanced affinity structure. Our findings support the claim that, for a particular notion of sparsity, loosely speaking, "Laplacian spectral embedding favors relatively sparse graphs, whereas adjacency spectral embedding favors not-too-sparse graphs." We also provide evidence in support of the claim that "adjacency spectral embedding favors core-periphery network structure."