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Motivated by the increasing need for developing automated decision-support tools for cyber-physical networks subject to uncertainties, we have been pursuing development of a new control-theoretic framework for network security and vulnerability. In this paper, we build on the proposed framework to put forth concrete definitions for security and (dually) discoverability, for a class of models that can represent dynamics of numerous cyber-physical networks of interest: namely, dynamical network spread models. These security and discoverability definitions capture whether or not, and to what extent, a stakeholder can infer the temporal dynamics of the spread from localized and noisy measurements. We then equivalence these security and security-level definitions to the control-theoretic notions of observability and optimal estimation, and so obtain explicit algebraic and spectral conditions for security and analyses of the security level. Further drawing on graph-theory constructs, a series of graphical conditions for security, as well as characterizations of security levels, are derived. A case study on zoonotic disease spread is also included, to illustrate concrete application of the analyses in management of cyber-physical infrastructure networks.