Differential Analysis of Directed Networks

We developed a novel statistical method to identify structural differences between networks characterized by structural equation models. We propose to reparameterize the model to separate the differential structures from common structures, and then design an algorithm with calibration and construction stages to identify these differential structures. The calibration stage serves to obtain consistent prediction by building the ℓ ₂ regularized regression of each endogenous variables against pre-screened exogenous variables, correcting for potential endogeneity issue. The construction stage consistently selects and estimates both common and differential effects by undertaking ℓ ₁ regularized regression of each endogenous variable against the predicts of other endogenous variables as well as its anchoring exogenous variables. Our method allows easy parallel computation at each stage. Theoretical results are obtained to establish non-asymptotic error bounds of predictions and estimates at both stages, as well as the consistency of identified common and differential effects. Our studies on synthetic data demonstrated that our proposed method performed much better than independently constructing the networks. A real data set is analyzed to illustrate the applicability of our method.