Dynamical scaling behavior of percolation clusters in scale-free networks.

In this work we investigate the spectra of Laplacian matrices that determine many dynamic properties of scale-free networks below and at the percolation threshold. We use a replica formalism to develop analytically, based on an integral equation, a systematic way to determine the ensemble averaged eigenvalue spectrum for a general type of treelike networks. Close to the percolation threshold we find characteristic scaling functions for the density of states rho(lambda) of scale-free networks. rho(lambda) shows characteristic power laws rho (lambda) approximately lambda (alpha(1) ) or rho (lambda) approximately lambda (d(2) ) for small lambda, where alpha(1) holds below and alpha(2) at the percolation threshold. In the range where the spectra are accessible from a numerical diagonalization procedure the two methods lead to very similar results.

Dynamics of randomly branched polymers: configuration averages and solvable models.

Treating the relaxation dynamics of an ensemble of random hyperbranched macromolecules in dilute solution represents a challenge even in the framework of Rouse-type approaches, which focus on generalized Gaussian structures (GGSs). The problem is that one has to average over a large class of realizations of molecular structures, and that each molecule undergoes its own dynamics. We show that a replica formalism allows to develop analytically, based on an integral equation, a systematic way to determine the ensemble averaged eigenvalue spectrum. Interestingly, for a specific probability distribution of the spring strengths of the GGSs, the integral equation takes a particularly simple form. Given that several dynamical observables, such as the mechanical moduli G'(omega) and G"(omega), as well as the averaged monomer displacement are relatively simple functions of the eigenvalues, we can use the obtained spectra to compute the corresponding averaged dynamical forms. Comparing the results obtained from this approach and from extensive diagonalizations of hyperbranched GGSs we find a very good agreement.

Dynamics of Vicsek fractals, models for hyperbranched polymers.

We consider the dynamics of Vicsek fractals of arbitrary connectivity, models for hyperbranched polymers. Their basic dynamical properties depend on their eigenvalue spectra, which can be determined iteratively. This paves the way for theoretical studies to very high precision for regular, finite, arbitrarily large hyperbranched structures.