For inhomogeneous random graphs with a graphon variance profile, the traces of powers of the adjacency matrix, suitably rescaled, converge to Gaussian processes with covariances expressible as graphon homomorphism densities.
A traffic approach for profiled Pennington-Worah matrices
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We study macroscopic observables of large random matrices introduced by Pennington and Worah, defined by applying entry wise a non linear function on a product of matrices with independent entries. We allow the variance of the entries of the matrices to vary from entry to entry. We complement P\'ech\'e perspective from [Electron. Commun. Probab. 24 (2019)] showing a decomposition of these matrices whose and traffic asymptotic traffic-equivalent for their ingredients, when the activation function belongs to the space of odd polynomials. This give a new interpretation of the linear plus chaos phenomenon observed for these matrices.
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Central limit theorems for linear spectral statistics of inhomogeneous random graphs with graphon limits
For inhomogeneous random graphs with a graphon variance profile, the traces of powers of the adjacency matrix, suitably rescaled, converge to Gaussian processes with covariances expressible as graphon homomorphism densities.