  High-dimensional networks play a key role in understanding complex
relationships. These relationships are often dynamic in nature and can change
with multiple external factors (e.g., time and groups). Methods for estimating
graphical models are often restricted to static graphs or graphs that can
change with a single covariate (e.g., time). We propose a novel class of
graphical models, the covariate-varying network (CVN), that can change with
multiple external covariates.
  In order to introduce sparsity, we apply a $L_1$-penalty to the precision
matrices of $m \geq 2$ graphs we want to estimate. These graphs often show a
level of similarity. In order to model this 'smoothness', we introduce the
concept of a 'meta-graph' where each node in the meta-graph corresponds to an
individual graph in the CVN. The (weighted) adjacency matrix of the meta-graph
represents the strength with which similarity is enforced between the $m$
graphs.
  The resulting optimization problem is solved by employing an alternating
direction method of multipliers. We test our method using a simulation study
and we show its applicability by applying it to a real-world data set, the gene
expression networks from the study 'German Cancer in childhood and
molecular-epidemiology' (KiKme). An implementation of the algorithm in R is
publicly available under https://github.com/bips-hb/cvn
