We study the level spacing distribution p(s) in the spectrum of random networks. According to our numerical results, the shape of p(s) in the Erdős-Rényi (E-R) random graph is determined by the average degree k and p(s) undergoes a dramatic change when k is varied around the critical point of the percolation transition, k = 1. When k 1, the p(s) is described by the statistics of the Gaussian orthogonal ensemble (GOE), one of the major statistical ensembles in Random Matrix Theory, whereas at k = 1 it follows the Poisson level spacing distribution. Closely above the critical point, p(s) can be described in terms of an intermediate distribution between Poisson and the GOE, the Brodydistribution. Furthermore, below the critical point p(s) can be given with the help of the regularized Gamma-function. Motivated by these results, we analyse the behaviour of p(s) in real networks such as the internet, a word association network and a protein-protein interaction network as well. When the giant component of these networks is destroyed in a node deletion process simulating the networks subjected to intentional attack, their level spacing distribution undergoes a similar transition to that of the E-R graph.
The contrast of electron microscope lattice images of a vanadium monosulphide with superstructures was dynamically calculated on the basis of the multislice method. Interactions of 80 waves were analysed. It was shown that the vacancy-rich vanadium sites were imaged as white spots not only at the very thin part but also at the thicker part inside the first equal-thickness contour, which appeared at the thickness of 60 ,~. A non-stoichiometric vanadium monosulphide, VSl.lss, quenched imperfectly from the preparation temperature at 1217 °C, was examined on the basis of the lattice image observation by a 100 kV high-resolution electron microscope. Orderings in the inter-as well as intralayers could be recognized directly from the arrangement of the white spots. Interpretable lattice images appeared exclusively inside the first equal-thickness contour. A mixture of some different types of images was observed in a crystal fragment; in some band-like regions the 4C-type superstructure was found with almost the same structure as Fe7Ss. Each band extended parallel to the (001) plane with thickness of at most 50 A and the orientation was in a twin relation to that of the adjoining one. The other regions showed less welldefined lattice images, although they gave the broad diffraction peaks ranging from the 4C-to 3C-type reflexions. In order to determine the phase relations at high temperatures the specimen was heated by focusing an electron beam on the portion slightly apart from the area under examination. The technique enabled us to observe only the heating effect. Upon heating, the 4C type first increased in volume, seen by the thickening of some bands. On further heating the 3C type increased in intensity on the diffraction pattern. The structure of the 3C type was estimated to be almost the same as FeTSe8 except that the vacancy concentration fuctuated among the partially occupied vanadium layers.
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