The expected adjacency and modularity matrices in the degree corrected stochastic block model
Special Matrices, vol. 6, pp. 110–121, 2018
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 …
Localization of dominant eigenpairs and planted communities by means of Frobenius inner products
Czechoslovak Mathematical Journal, vol. 66, pp. 881–893, 2016
We propose a new localization result for the leading eigenvalue and eigenvector of a symmetric matrix $A$. The result exploits the Frobenius inner product between $A$ and a given rank-one landmark matrix $X$. Different choices for $X$ may be used, depending on the problem under investigation. In particular, we show …
Generalized modularity matrices
Linear Algebra and its Applications, vol. 502, pp. 327–345, 2016
Various modularity matrices appeared in the recent literature on network analysis and algebraic graph theory. Their purpose is to allow writing as quadratic forms certain combinatorial functions appearing in the framework of graph clustering problems. In this paper we put in evidence certain common traits of various …