Network Reconstruction from Intrinsic Noise

We consider the problem of inferring the structure and dynamics of an unknown network driven by unknown noise inputs. For linear, time-invariant systems of minimal order, with hidden states and one noise source per measured state, we characterise under what conditions this problem is wellposed and present two main results. First, for networks with stable interactions between measured states and where the noise sources act through minimum phase dynamics, we prove that the solution for the network is unique and that the solution for the input dynamics is unique up to sign. If the assumptions of stability and phase-minimality are relaxed, we characterise the set of all solutions and prove that if the noise acts directly on the measured states, then the number of solutions is finite and bounded by a function of the number of hidden states. Lastly we present a method to obtain this set of solutions from data.