A Control Theoretic Look at Granger Causality: Extending Topology Reconstruction to Networks with Direct Feedthroughs

Reconstructing the causal structure of a network of dynamic systems from observational data is an important problem in many areas of science. One of the earliest and most prominent approaches to this problem is Granger causality. It has been shown that in a network with linear dynamics and strictly causal transfer functions, Granger causality consistently reconstructs the underlying graph of the network. On the other hand, techniques that allow for the presence of direct feedthroughs usually assume there are no feedback loops in the dynamics of the network. In this article, we develop an extension of Granger causality that provides theoretical guarantees for the reconstruction of a network topology even in the presence of direct feedthroughs and feedback loops. The only required assumption is a relatively mild condition of well-posedness named recursiveness, where at least one strictly causal transfer function needs to be present in every feedback loop. The technique is compared with other state-of-the-art methods on a benchmark example specifically designed to include several dynamic configurations that are challenging to reconstruct.