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 $\rho \left(\lambda \right)$ 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 follows a very different pattern, which can be used as a fingerprint of modularity. Of particular importance is the value $\rho \left(0\right)$, 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.

• Received 18 June 2004

DOI:https://doi.org/10.1103/PhysRevE.71.016106