Inference of directed biological networks is an important but notoriously challenging problem. The recent proliferation of large-scale CRISPR perturbation data provides a new opportunity to tackle this problem by leveraging the transcriptional response to the presence of a gene-targeting guide. Here, we introduce inverse sparse regression (inspre), an approach to learning causal networks that leverages large-scale intervention-response data. Applied to 788 genes from the genome-wide perturb-seq dataset, inspre discovers a network with small-world and scale-free properties. We integrate our network estimate with external data, finding relationships between gene eigencentrality and both measures of gene essentiality and gene expression heritability. Our analysis helps to elucidate the structure of networks that may underlie complex traits.