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We study numerically the spectrum and eigenstate properties of the Googlematrix of various examples of directed networks such as vocabulary networks ofdictionaries and university World Wide Web networks. The spectra have gaplessstructure in the vicinity of the maximal eigenvalue for Google dampingparameter $\alpha$ equal to unity. The vocabulary networks have relativelyhomogeneous spectral density, while university networks have pronouncedspectral structures which change from one university to another, reflectingspecific properties of the networks. We also determine specific properties ofeigenstates of the Google matrix, including the PageRank. The fidelity of thePageRank is proposed as a new characterization of its stability.

Spectral properties of the Google matrix of the World Wide Web and other directed networks