Dynamical systems enjoy a rich variety of mathematical representations, from interconnections of convolution operators or rational functions of a complex variable to systems of (possibly stochastic) differential or differential-algebraic equations. Although many of these representations can describe the same behavior, i.e. represent the same constraints on manifest variables, each one may characterize a different notion of system structure. This paper introduces a method for interpreting the semantics of different representations of a network system by exploring the set of realizations consistent with each. We then focus on signal structure, extending its definition, and demonstrate that its semantics differ from other network representations in important and useful ways. In particular, the information cost for identifying a system's signal structure from data can be considerably less than that needed for identifying a system's subsystem structure.