Hyperbranched polymers are typically treelike macromolecules with a very disordered structure. Here we construct hyperbranched polymers based on the degree distribution of the small-world networks. This algorithm allows us to study a transition from monodisperse linear chains to structurally-disordered dendritic polymers by varying the parameter (), which measures the randomness and the degree of branching of the network. Employing the framework of generalized Gaussian structures, we determine for the obtained structures the relaxation spectra, which are exemplified on the mechanical relaxation moduli (storage and loss moduli). We monitor these physical quantities for networks of different sizes and for various values of the parameter . In the intermediate frequency domain, we encounter macroscopically distinguishable behaviours