Dynamics of comb-of-comb networks

The dynamics of complex networks, a current hot topic in 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 $d_s\to 2$. The $d_s$ leaves its fingerprint on many dynamical processes, as we exemplarily show by considering the dynamical properties of polymer networks, including single monomer displacement under a constant force, mechanical relaxation, and fluorescence depolarization.

Figure 1

Illustration of network $C_g$ corresponding to $r=4$ and $g=3$ (all beads). Only filled beads represent the network at generation $g=1$; filled and crossed beads together give the network at generation $g=2$; see text for details.

Figure 2

Densities of states $\rho \left(\lambda \right)$ corresponding to the following parameter sets $(r,g)$: (a) $(2,16)$, (b) $(4,8)$, (c) $(16,4)$, and (d) $(65536,1)$. All networks have $N_g=65536$ beads in total [case (d) represents just a linear chain]; see text for details.

Figure 3

Averaged monomer displacement $〈Y\left(t\right)〉$, Eq. (15), for the networks $C_g$ of different $r$ and $g$. The case $r=2^{16}=65536$ and $g=1$ corresponds to a linear chain. All networks have $N_g=65536$ beads in total. The choice of units is given by setting $\sigma =1$ and $\frac{F}{\zeta }=1$; see text for details.

Figure 4

Top: Storage modulus $G^{\prime }\left(\omega \right)$, Eq. (16). Bottom: Loss modulus $G^{\prime \prime }\left(\omega \right)$, Eq. (17). Both panels are for the networks in Fig. 3. Inset: Local slopes of curves with the same symbolic and color code. Here we use $\sigma =1$ and $\frac{\nu k_{B}T}{N_g}=1$; see text for details.

Figure 5

Average probability $\langle P\left(t\right)\rangle$, Eq. (20), for the networks in Fig. 3. Inset: Local slopes of curves with the same symbolic and color code. The time is given in units of $1/\tilde{k}$ by setting $\tilde{k}=1$; see text for details.