Spectral Analysis and the Dynamic Response of Complex Networks

Abstract

The eigenvalues and eigenvectors of the connectivity matrix of complex networks contain information about its topology and its collective behavior. In particular, the spectral density $\rho (\lambda )$ of this matrix reveals important network characteristics: random networks follow Wigner's semicircular law whereas scale-free networks exhibit a triangular distribution. In this paper we show that the spectral density of hierarchical networks follow a very different pattern, which can be used as a fingerprint of modularity. Of particular importance is the value $\rho (0)$, related to the homeostatic response of the network: it is maximum for random and scale free networks but very small for hierarchical modular networks. It is also large for an actual biological protein-protein interaction network, demonstrating that the current leading model for such networks is not adequate.

4 pages 14 figures

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