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SYBN: Biological and Social Networks

SYBN 3: Biologische und Soziale Netzwerke, Postersitzung

SYBN 3.11: Poster

Monday, March 7, 2005, 14:00–15:30, Poster TU E

Dynamics of scalefree networks: scaling behavior — •Christian von Ferber, Florian Jasch, and Alexander Blumen — Theoretische Polymerphysik, Physikalisches Institut, Universität Freiburg

We characterize the dynamics of networks for problems ranging from the dynamics of randomly branched polymers and stress relaxation of near citical gels to diffusion and spreading (e.g. of viruses) on general networks. A basic property of a network is its degree distribution p_k, i.e. the probability that an arbitrary vertex is connected to k other vertices. If p_k exhibits a power law p_k∼ k^−γ the network is called scale-free; scale-free networks differ from the classical random graphs, for which the distribution p_k is Poissonian. Recent work clarified that the properties of scale-free networks at the percolation theshold differ markedly from the classical case: for γ<4 nontrivial γ-dependent critical exponents appear [1]. Many time and frequency-dependent observables can be written in terms of the spectrum ρ(λ), the density of eigenvalues of the discrete Laplacian defined on the network. We develop and solve an integral equation for ρ(λ) for random graphs with arbitrary degree distributions [2]. For scale-free networks close to percolation we find scaling forms for ρ(λ). If p_k decays fast ρ(λ) has a Lifshitz tail for λ→ 0 while for p_k∼ k^−γ with γ<4 a power law ρ(λ)∼ λ^d_s/2−1 with a γ-dependend spectral dimension d_s is found [2]. Extensive numerical diagonalizations of simulated ensembles of networks support our analytical findings.

[1] R. Cohen et al. Phys. Rev. E, 66:036113, 2002.

[2] F. Jasch, C. von Ferber, and A. Blumen. Phys. Rev. E, 68:051106, 2003; Phys. Rev. E, 70:016112, 2004