Differential Analysis of Directed Networks

TL;DR

This paper introduces a new statistical method for comparing directed networks by separating common and differential structures, with theoretical guarantees and demonstrated effectiveness on synthetic and real data.

Contribution

The paper presents a novel reparameterization and a two-stage L1-regularized regression approach for identifying differential network structures with theoretical error bounds.

Findings

Outperforms independent network construction methods on synthetic data

Provides theoretical nonasymptotic error bounds and consistency results

Successfully applied to real data to demonstrate practical utility

Abstract

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 L2 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 L1 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 nonasymptotic 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.