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. To introduce sparsity, we apply a L1-penalty to the precision matrices of m ≥ 2 graphs we estimate. These graphs often show a level of similarity. To model this smoothness, we introduce the concept of a ’meta-graph’ where each node in the meta-graph corresponds to an individual graph within 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 two real-world data sets in childhood cancer and vaccine genomics. An implementation of the algorithm in R is publicly available under https://bips-hb.r-universe.dev/CVN (accessed on 22 February 2026).
- Open Access
- Article
Inferring High-Dimensional Dynamic Networks Changing with Multiple Covariates
Author Information
Received: 23 Feb 2026 | Revised: 26 Mar 2026 | Accepted: 02 Apr 2026 | Published: 21 Apr 2026
Abstract
Keywords
alternating direction method of multipliers | covariate-varying networks | Gaussian graphical model | graphical LASSO | sparsity
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