Google matrix of the citation network of Physical Review (1310.5624v2)

Abstract: We study the statistical properties of spectrum and eigenstates of the Google matrix of the citation network of Physical Review for the period 1893 - 2009. The main fraction of complex eigenvalues with largest modulus is determined numerically by different methods based on high precision computations with up to $p=16384$ binary digits that allows to resolve hard numerical problems for small eigenvalues. The nearly nilpotent matrix structure allows to obtain a semi-analytical computation of eigenvalues. We find that the spectrum is characterized by the fractal Weyl law with a fractal dimension $d_f \approx 1$. It is found that the majority of eigenvectors are located in a localized phase. The statistical distribution of articles in the PageRank-CheiRank plane is established providing a better understanding of information flows on the network. The concept of ImpactRank is proposed to determine an influence domain of a given article. We also discuss the properties of random matrix models of Perron-Frobenius operators.