Fiedler-Regularized Causal Discovery for Sparse Connected DAGs

Causal discovery often starts from a set of variables that were measured together because they are believed to describe related parts of the same underlying process. Standard continuous DAG learners encode data fit, sparsity, and acyclicity, but they can still return graphs that fragment into isolated variables or disconnected components. We study a simple structural prior for this setting: the learned causal DAG should remain sparse while its undirected skeleton should be weakly connected. We introduce Fiedler regularization, a differentiable spectral penalty based on the algebraic connectivity of a smooth support-preserving skeleton of the learned weighted adjacency matrix. The resulting penalty can be added modularly to differentiable causal discovery objectives, providing connectivity control without changing the underlying learner. We instantiate the approach for GOLEM, graph autoencoder, and DAG-GNN models. Across sparse connected synthetic DAGs up to 200 variables, including connected-conditioned Erdős-Rényi, sparse connected, scale-free, and small-world graph families, Fiedler-regularized learners improve fragmentation control and structural recovery across learner families. These results position algebraic connectivity as a principled and practical prior for learning sparse causal DAGs in settings where the measured variables are expected to have a connected causal structure.