Spectral statistics of real asymmetric matrices in sparse Gaussian ensembles

Theoretical analysis of artificial and biological neural networks, such as modeling synaptic or weight matrices, requires taking into account generic real-asymmetric matrix ensembles, which are those with different orders of matrix elements, such as banded or sparse structures. Using a complexity parameter approach, the ensemble averaged spectral densities for both real and complex eigenvalues are derived by analyzing the spectral statistics of the multiparametric Gaussian ensembles of real asymmetric matrices. By taking into account the matrix members with variable mean and variance choices, we can freely simulate the desired ensemble sparsity. Our formulation exposes a profound commonality across sparse real-asymmetric ensembles and offers a common mathematical explanation of the spectral statistics for a broad variety of them.