  Graph topology inference of network processes with co-evolving and
interacting time-series is crucial for network studies. Vector autoregressive
models (VAR) are popular approaches for topology inference of directed graphs;
however, in large networks with short time-series, topology estimation becomes
ill-posed. The present paper proposes a novel nonlinearity-preserving topology
inference method for directed networks with co-evolving nodal processes that
solves the ill-posedness problem. The proposed method, large-scale kernelized
Granger causality (lsKGC), uses kernel functions to transform data into a
low-dimensional feature space and solves the autoregressive problem in the
feature space, then finds the pre-images in the input space to infer the
topology. Extensive simulations on synthetic datasets with nonlinear and linear
dependencies and known ground-truth demonstrate significant improvement in the
Area Under the receiver operating characteristic Curve ( AUC ) of the receiver
operating characteristic for network recovery compared to existing methods.
Furthermore, tests on real datasets from a functional magnetic resonance
imaging (fMRI) study demonstrate 96.3 percent accuracy in diagnosis tasks of
schizophrenia patients, which is the highest in the literature with only brain
time-series information.
