Dynamics of "comb-of-comb" networks

TL;DR

This paper analytically investigates the Laplacian spectra of deterministic 'comb-of-comb' networks, revealing their spectral dimension approaches 2 as size increases, and explores implications for various dynamical processes.

Contribution

It provides analytical equations for the Laplacian spectra of 'comb-of-comb' networks using Chebyshev polynomials and analyzes their dynamical implications.

Findings

Spectral dimension approaches 2 in the infinite size limit

Derived analytical equations for Laplacian spectra using Chebyshev polynomials

Demonstrated effects on polymer network dynamics, including relaxation and depolarization

Abstract

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.