Modeling the topology of a dynamical network via Wiener filtering approach

The paper considers the problem of determining a suitable link structure for a set of networked interdependent processes, then providing a simplified description for their unknown underlying topology, also giving useful insights about their mutual influences. There are many scenarios where this problem has a prominent relevance. Indeed, it is often possible to measure the outputs of a large number of systems which are not independent, with no a priori knowledge of what the interconnections are. Examples can be found in fields as diverse as Economics, Biology, Ecology and Neural Sciences. The main idea of this work is to provide both a qualitative and quantitative description of the links among the processes in terms of modeling errors, assuming no a priori knowledge about the network features. To this aim, Wiener filtering and graph theory are exploited in a linear framework, in order to reconstruct a suitable connected and acyclic scheme for the internal connections of the whole system. Moreover, we show the consistency of the proposed technique, when the underlying network is actually connected and acyclic, proving that the structure obtained through the identification method coincides with the actual one and that this goal cannot be achieved via simple Wiener filtering. An application to real data illustrates the effectiveness of the suggested approach.