  Spectral decomposition of dynamical systems is a popular methodology to
investigate the fundamental qualitative and quantitative properties of these
systems and their solutions. In this chapter, we consider a class of nonlinear
cooperative protocols, which consist of multiple agents that are coupled
together via an undirected state-dependent graph. We develop a representation
of the system solution by decomposing the nonlinear system utilizing ideas from
the Koopman operator theory and its spectral analysis. We use recent results on
the extensions of the well-known Hartman theorem for hyperbolic systems to
establish a connection between the original nonlinear dynamics and the
linearized dynamics in terms of Koopman spectral properties. The expected value
of the output energy of the nonlinear protocol, which is related to the 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 the performance of this class of nonlinear
dynamical networks as a function of their graph topology. Finally, we propose a
scalable computational method for approximation of the components of the
Koopman mode decomposition, which is necessary to evaluate the systemic
performance measure of the nonlinear dynamic network.
