Spectral decomposition of dynamical systems is a popular methodology to investigate fundamental qualitative and quantitative properties of dynamical systems and their solutions. In this talk, we consider a class of nonlinear cooperative
protocols that consist of multiple agents that are coupled together via an undirected state-dependent graph. We develop a representation of system solution by decomposing the nonlinear system utilizing ideas from the Koopman theory and its spectral
analysis. We use recent results on the well-known Hartman theorem for hyperbolic systems to establish a connection between the nonlinear and the linearized dynamics in terms of Koopman spectral properties. The expected value of the output energy
of the nonlinear network, which relates to notions of coherence and robustness in dynamical networks, is evaluated and characterized in terms of Koopman eigenvalues, eigenfunctions, and modes. Spectral representation of the performance measure enables us to develop algorithmic methods to assess performance of this class of nonlinear dynamical networks as a function of their graph topology.