Dynamics of "comb-of-comb" networks

The dynamics of complex networks, being a current hot topic of many scientific fields, is often coded through the corresponding Laplacian matrix. The spectrum of this matrix carries the main features of the networks' dynamics. Here we consider the deterministic networks which can be viewed as "comb-of-comb" iterative structures. For their Laplacian spectra we find analytical equations involving Chebyshev polynomials, whose properties allow one to analyze the spectra in deep. Here, in particular, we find that in the infinite size limit the corresponding spectral dimension goes as $d_s\rightarrow2$. The $d_s$ leaves its fingerprint in many dynamical processes, as we exeplarily show by considering the dynamical properties of the polymer networks, including single monomer displacement under a constant force, mechanical relaxation, and fluorescence depolarization.