  We consider bipartite tight-binding graphs composed by $N$ nodes split into
two sets of equal size: one set containing nodes with on-site loss, the other
set having nodes with on-site gain. The nodes are connected randomly with
probability $p$. We give a rationale for the relevance of such "throttle/brake"
coupled systems (physically open systems) to grasp the stability issues of
complex networks in areas such as biochemistry, neurons or economy, for which
their modelling in terms of non-hermitian Hamiltonians is still in infancy.
Specifically, we measure the connectivity between the two sets with the
parameter $\alpha$, which is the ratio of current adjacent pairs over the total
number of possible adjacent pairs between the sets. For general
undirected-graph setups, the non-hermitian Hamiltonian $H(\gamma,\alpha,N)$ of
this model presents pseudo-Hermiticity, where $\gamma$ is the loss/gain
strength. However, we show that for a given graph setup $H(\gamma,\alpha,N)$
becomes ${\cal PT}-$symmetric. In both scenarios (pseudo-Hermiticity and ${\cal
PT}-$symmetric), depending on the parameter combination, the spectra of
$H(\gamma,\alpha,N)$ can be real even when it is non-hermitian. Thus, we
numerically characterize the average fractions of real and imaginary
eigenvalues of $H(\gamma,\alpha,N)$ as a function of the parameter set
$\{\gamma,\alpha,N\}$. We demonstrate, for both setups, that there is a well
defined sector of the $\gamma\alpha-$plane (which grows with $N$) where the
spectrum of $H(\gamma,\alpha,N)$ is predominantly real.
