  Reconstructing a network of dynamic systems from observational data is an
active area of research. Many approaches guarantee a consistent reconstruction
under the relatively strong assumption that the network dynamics is governed by
strictly causal transfer functions. However, in many practical scenarios,
strictly causal models are not adequate to describe the system and it is
necessary to consider models with dynamics that include direct feedthrough
terms. In presence of direct feedthroughs, guaranteeing a consistent
reconstruction is a more challenging task. Indeed, under no additional
assumptions on the network, we prove that, even in the limit of infinite data,
any reconstruction method is susceptible to inferring edges that do not exist
in the true network (false positives) or not detecting edges that are present
in the network (false negative). However, for a class of triangle-free networks
introduced in this article, some consistency guarantees can be provided. We
present a method that either exactly recovers the topology of a triangle-free
network certifying its correctness or outputs a graph that is sparser than the
topology of the actual network, specifying that such a graph has no false
positives, but there are false negatives.
