  Dynamic gene-regulatory networks are complex since the number of potential
components involved in the system is very large. Estimating dynamic networks is
an important task because they compromise valuable information about
interactions among genes. Graphical models are a powerful class of models to
estimate conditional independence among random variables, e.g. interactions in
dynamic systems. Indeed, these interactions tend to vary over time. However,
the literature has been focused on static networks, which can only reveal
overall structures. Time-course experiments are performed in order to tease out
significant changes in networks. It is typically reasonable to assume that
changes in genomic networks are few because systems in biology tend to be
stable. We introduce a new model for estimating slowly changes in dynamic
gene-regulatory networks which is suitable for a high-dimensional dataset, e.g.
time-course genomic data. Our method is based on i) the penalized likelihood
with $\ell_1$-norm, ii) the penalized differences between conditional
independence elements across time points and iii) the heuristic search strategy
to find optimal smoothing parameters. We implement a set of linear constraints
necessary to estimate sparse graphs and penalized changing in dynamic networks.
These constraints are not in the linear form. For this reason, we introduce
slack variables to re-write our problem into a standard convex optimization
problem subject to equality linear constraints. We show that GL$_\Delta$
performs well in a simulation study. Finally, we apply the proposed model to a
time-course genetic dataset T-cell.
