Vibrational modes and spectrum of oscillators on a scale-free network

We study vibrational modes and spectrum of a model system of atoms and springs on a scale-free network where we assume that the atoms and springs are distributed as nodes and links of a scale-free network. To understand the nature of excitations with many degrees of freedom on the scale-free network, we adopt a particular model that we assign the mass M(i) and the specific oscillation frequency omega(i) of the ith atom and the spring constant K(ij) between the ith and jth atoms. We show that the density of states of the spectrum follows a scaling law P (omega(2)) proportional, variant (omega(2))(-gamma), where gamma = 3 and that as the number of nodes N is increasing, the maximum eigenvalue grows as fast as sqrt[N].