Granger-faithfulness and link orientation in network reconstruction

TL;DR

This paper introduces Granger-faithfulness, a condition ensuring accurate network reconstruction from observational data in dynamic systems, and provides algorithms with guarantees for correct edge inference under this assumption.

Contribution

It formalizes Granger-faithfulness for dynamic networks and proves that most systems are unfaithful, while offering a reconstruction algorithm with no false positives or negatives under faithfulness.

Findings

Most dynamic systems are unfaithful to their networks.

The proposed algorithm guarantees accurate topology reconstruction under Granger-faithfulness.

Orientation rules are consistent for some inferred edges under the same assumption.

Abstract

Networked dynamic systems are often abstracted as directed graphs, where the observed system processes form the vertex set and directed edges are used to represent non-zero transfer functions. Recovering the exact underlying graph structure of such a networked dynamic system, given only observational data, is a challenging task. Under relatively mild well-posedness assumptions on the network dynamics, there are state-of-the-art methods which can guarantee the absence of false positives. However, in this article we prove that under the same well-posedness assumptions, there are instances of networks for which any method is susceptible to inferring false negative edges or false positive edges. Borrowing a terminology from the theory of graphical models, we say those systems are unfaithful to their networks. We formalize a variant of faithfulness for dynamic systems, called Granger-faithfulness, and for a large class of dynamic networks, we show that Granger-unfaithful systems constitute a Lebesgue zero-measure set. For the same class of networks, under the Granger-faithfulness assumption, we provide an algorithm that reconstructs the network topology with guarantees for no false positive and no false negative edges in its output. We augment the topology reconstruction algorithm with orientation rules for some of the inferred edges, and we prove the rules are consistent under the Granger-faithfulness assumption.