Network reconstruction from intrinsic noise: Minimum-phase systems

This paper considers the problem of inferring the structure and dynamics of an unknown network driven by unknown noise inputs. Equivalently we seek to identify direct causal dependencies among manifest variables only from observations of these variables. We consider linear, time-invariant systems of minimal order and with one noise source per measured state. If the transfer matrix from the inputs to manifest states is known to be minimum phase, this problem is shown to have a unique solution irrespective of the network topology. This is equivalent to there being only one spectral factor (up to a choice of signs of the inputs) of the output spectral density that satisfies these assumptions. Hence for this significant class of systems, the network reconstruction problem is well posed.