On minimal realisations of dynamical structure functions

Motivated by the fact that transfer functions do not contain structural information about networks (dependency of state variables), dynamical structure functions were introduced to capture causal relationships between measured nodes in networks. From the dynamical structure functions, (a) we show that the actual number of hidden states can be larger than the number of hidden states estimated from the corresponding transfer function; (b) we can obtain partial information about the true state-space equation, which cannot in general be obtained from the transfer function. Based on these properties, this paper proposes algorithms to find minimal realisations for a given dynamical structure function. This helps to estimate the minimal number of hidden states, to better understand the complexity of the network, and to identify potential targets for new measurements.