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We study the exciton localization and resulting optical response for disordered tubular aggregates of optically active molecules. It has previously been shown that such tubular structures allow for excitons delocalized over more than a thousand molecules, owing to the combined effects of long-range dipole–dipole interactions and the higher-dimensional (not truly one-dimensional) nature of the aggregate. Such large delocalization sizes prompt the question to what extent in experimental systems the delocalization may still be determined by the aggregate size (diameter and length) and how this affects the aggregate's optical response and dynamics. We perform a systematic study of the size effects on the localization properties using numerical simulations of the exciton states in a cylindrical model structure inspired by the previously derived geometry of a cylindrical aggregate of cyanine dye molecules (C8S3). To characterize the exciton localization, we calculate the participation ratio and the autocorrelation function of the exciton wave function. We also calculate the density of states and absorption spectrum. We find strong effects of the tube's radius on the localization and optical properties in the range of parameters relevant to the experiment. In addition, surprisingly, we find that even for tubes as long as 750 nm, the localization size is limited by the tube's length for disorder values that are relevant to experimental circumstances, while observable effects of the tube's length in the absorption spectrum still occur for tube lengths up to about 150 nm. The latter may explain the changes in the optical spectra observed during the aging process of bromine-substituted C8S3 aggregates. For weak disorder, the exciton wave functions exhibit a scattered, fractal-like nature, similar to the quasi-particles in two-dimensional disordered systems. [ABSTRACT FROM AUTHOR]