  Having in mind the large-scale analysis of gene regulatory networks, we
review a graph decimation algorithm, called "leaf-removal", which can be used
to evaluate the feedback in a random graph ensemble. In doing this, we consider
the possibility of analyzing networks where the diagonal of the adjacency
matrix is structured, that is, has a fixed number of nonzero entries. We test
these ideas on a network model with fixed degree, using both numerical and
analytical calculations. Our results are the following. First, the leaf-removal
behavior for large system size enables to distinguish between different regimes
of feedback. We show their relations and the connection with the onset of
complexity in the graph. Second, the influence of the diagonal structure on
this behavior can be relevant.
