Fractal Weyl law for Linux Kernel Architecture

LEONARDO ERMANN; ALEXEI D. CHEPELIANSKII; DIMA L. SHEPELYANSKY

EUROPEAN PHYSICAL JOURNAL B - CONDENSED MATTER

SPRINGER

Año: 2011 vol. 79 p. 115 - 120

1434-6028

We study the properties of spectrum and eigenstates of the Google matrix of a directed networkformed by the procedure calls in the Linux Kernel.Our results obtained for various versions of the Linux Kernelshow that the spectrum is characterized by the fractal Weyl lawestablished recently for systems of quantum chaotic scattering andthe Perron-Frobenius operators of dynamical maps.The fractal Weyl exponent is found to be$ u approx 0.65$ that corresponds to the fractal dimensionof the network $d approx 1.3$. An independent computation of the fractal dimension by the cluster growing method, generalized for directed networks, gives a close value $d approx 1.4$.The eigenmodes of the Google matrix of Linux Kernel are localized on certain principal nodes. We argue that the fractal Weyl law should be generic for directed networks with the fractal dimension $d<2$.