  We study the spectrum of an asymmetric random matrix with block structured
variances. The rows and columns of the random square matrix are divided into
$D$ partitions with arbitrary size (linear in $N$). The parameters of the model
are the variances of elements in each block, summarized in
$g\in\mathbb{R}^{D\times D}_+$. Using the Hermitization approach and by
studying the matrix-valued Stieltjes transform we show that these matrices have
a circularly symmetric spectrum, we give an explicit formula for their spectral
radius and a set of implicit equations for the full density function. We
discuss applications of this model to neural networks.
