The expected adjacency and modularity matrices in the degree corrected stochastic block model

Abstract

We provide explicit expressions for the eigenvalues and eigenvectors of matrices that can be written as the Hadamard product of a block partitioned matrix with constant blocks and a rank one matrix. Such matrices arise as the expected adjacency or modularity matrices in certain random graph models that are widely used as benchmarks for community detection algorithms.

Keywords

• adjacency matrix
• modularity matrix
• stochastic block model
• inflation product
• 15a18
• 15b99