A brief explanation of the issue of faithfulness and link orientation in network reconstruction

Complex systems can often be understood via a graph abstraction where nodes represent individual components and edges represent input/output relations among them. Recovering the network structure of a complex system from noninvasively observed data plays a central role in many areas of science. A classic approach to this problem is Granger causality. For strictly causal linear dynamic systems, Granger causality guarantees a consistent reconstruction of the network. However, it is a well-established fact that Granger causality, and analogous methods, lead to the inference of spurious links in the presence of direct feedthroughs. On the other hand, graphical model approaches can deal successfully with static operators in acyclic structures. Indeed, in those cases, graphical model tools guarantee a consistent network reconstruction, apart from pathological conditions associated with very specific values of the system parameters. When these pathological conditions do not occur, borrowing terminology from the theory of graphical models, the network is said to be faithful to its graph representation. We discuss the notion of faithfulness and adapt it to the more general case of networks of dynamic systems, in order to combine the main idea behind Granger causality with graphical model techniques. We provide an algorithm which, under faithfulness, has theoretical guarantees for the reconstruction of a large class of linear models containing both direct feedthroughs and feedback loops.