Network regression links a network's structures to covariates of interest, modeling pairwise conditional dependencies of interacting units as a function of covariates. For instance, in gene network analysis of a certain lung cancer, the network structures may vary over clinical attributes differentiating four different subtypes of the cancer. Within the framework of Gaussian structure equation models, we infer a network's structures, de?ned by an undirected graph, in relation to covariates, through testing regression coefficients. To increase the power of hypothesis testing, we decorrelate the structure equation models, develop a combined constrained likelihood ratio test, combining independent marginal likelihoods and unregularizing hypothesized parameters whereas regularizing nuisance parameters through L0-constraints controlling the individual degree of sparseness. This is joint work with Xiaotong Shen.