Differential Analysis of Directed Networks

We developed a novel statistical method to identify structural differences between net- works 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 con- sistent prediction by building the $\ell$2 regular- ized regression of each endogenous variables against pre-screened exogenous variables, cor- recting for potential endogeneity issue. The construction stage consistently selects and es- timates both common and differential effects by undertaking $\ell$1 regularized regression of each endogenous variable against the predicts of other endogenous variables as well as its an- choring exogenous variables. Our method al- lows 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 con- sistency of identified common and differential effects. Our studies on synthetic data demon- strated 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.