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Network reconstruction, i.e., obtaining network structure from data, is acentral theme in systems biology, economics and engineering. In some previouswork, we introduced dynamical structure functions as a tool for posing andsolving the problem of network reconstruction between measured states. Whilerecovering the network structure between hidden states is not possible sincethey are not measured, in many situations it is important to estimate theminimal number of hidden states in order to understand the complexity of thenetwork under investigation and help identify potential targets formeasurements. Estimating the minimal number of hidden states is also crucial toobtain the simplest state-space model that captures the network structure andis coherent with the measured data. This paper characterizes minimal orderstate-space realizations that are consistent with a given dynamical structurefunction by exploring properties of dynamical structure functions anddeveloping an algorithm to explicitly obtain such a minimal realization.

Minimal realization of the dynamical structure function and its application to network reconstruction