Relating the Network Graphs of State-Space Representations to Granger Causality Conditions

In this chapter, we will discuss the problem of estimating theNetworkgraphs network graphsGraphnetwork graphs of state-space representations based on observed data: we observe the output generated by each node of a network of state-space representations, and we would like to reconstruct the communication graph of the network, i.e., we would like to find out which nodes exchange information. The potential exchange of information is not assumed to be observable, i.e., it may take place via hidden internal states. We present an approach based on the notion of Granger causalityGranger causality. The essence of this approach is that there exists a communication link between two nodes, if the outputs generated by the corresponding nodes are related by Granger causalityGranger causality. More precisely, we show an equivalence between the existence of state-space representation in which subsystems corresponding to certain nodes exchange information, and the presence of Granger causalityGranger causality relation between the outputs generated by those subsystems. Since Granger causalityGranger causality can be checked based on observed data, these results open up the possibility of data-driven reverse engineering of the communication graph. We will discuss the case of stochastic linear time-invariant systems, and then the case of stochastic bilinear/LPV/switched systems.