The real-time-propagation formulation of time-dependent density-functional theory (RT-TDDFT) is an efficient method for modeling the optical response of molecules and nanoparticles. Compared to the widely adopted linear-response TDDFT approaches based on, e.g., the Casida equations, RT-TDDFT appears, however, lacking efficient analysis methods. This applies in particular to a decomposition of the response in the basis of the underlying single-electron states. In this work, we overcome this limitation by developing an analysis method for obtaining the Kohn-Sham electron-hole decomposition in RT-TDDFT. We demonstrate the equivalence between the developed method and the Casida approach by a benchmark on small benzene derivatives. Then, we use the method for analyzing the plasmonic response of icosahedral silver nanoparticles up to Ag$_{561}$. Based on the analysis, we conclude that in small nanoparticles individual single-electron transitions can split the plasmon into multiple resonances due to strong single-electron-plasmon coupling whereas in larger nanoparticles a distinct plasmon resonance is formed.

11 pages, 3 figures