Characterization of minimal network structures modeling stochastic processes

Identifying the underlying structure of a network from observed data is an important problem across various disciplines. Given the general ill-posed nature of the problem, since in many cases, multiple plausible network models can explain the data, this article concentrates on characterizing classes of models providing possible explanations. Specifically, we explore linear models that can account for observed data in the form of wide-sense stationary processes accommodating the potential presence of feedback loops and direct feedthroughs. To achieve this, we leverage key insights from the theory of graphical models. In particular, we extensively employ Pearl-Verma Theorem in causal discovery which allows one to recover all minimal network structures compatible with the observed data. We adapt such a result to deal with stochastic processes and reinterpret it as a Gram-Schmidt orthogonalization procedure in a suitable Hilbert space. This reinterpretation allows us to characterize all minimal networks explaining a set of data, which have the property of not having any algebraic loops.