Universality in Complex Networks: Random Matrix Analysis

We apply random matrix theory to complex networks. We show that nearest neighbor spacing distribution of the eigenvalues of the adjacency matrices of various model networks, namely scale-free, small-world and random networks follow universal Gaussian orthogonal ensemble statistics of random matrix theory. Secondly we show an analogy between the onset of small-world behavior, quantified by the structural properties of networks, and the transition from Poisson to Gaussian orthogonal ensemble statistics, quantified by Brody parameter characterizing a spectral property. We also present our analysis for a protein-protein interaction network in budding yeast.Comment: 4+ pages, 4 figures, to appear in PRE, major change in the paper including titl

Random matrix analysis of complex networks

We study complex networks under random matrix theory (RMT) framework. Using nearest-neighbor and next-nearest-neighbor spacing distributions we analyze the eigenvalues of adjacency matrix of various model networks, namely, random, scale-free and small-world networks. These distributions follow Gaussian orthogonal ensemble statistic of RMT. To probe long-range correlations in the eigenvalues we study spectral rigidity via $\Delta_3$ statistic of RMT as well. It follows RMT prediction of linear behavior in semi-logarithmic scale with slope being $\sim 1/\pi^2$. Random and scale-free networks follow RMT prediction for very large scale. Small-world network follows it for sufficiently large scale, but much less than the random and scale-free networks.Comment: accepted in Phys. Rev. E (replaced with the final version

Spectral analysis of deformed random networks

We study spectral behavior of sparsely connected random networks under the random matrix framework. Sub-networks without any connection among them form a network having perfect community structure. As connections among the sub-networks are introduced, the spacing distribution shows a transition from the Poisson statistics to the Gaussian orthogonal ensemble statistics of random matrix theory. The eigenvalue density distribution shows a transition to the Wigner's semicircular behavior for a completely deformed network. The range for which spectral rigidity, measured by the Dyson-Mehta $\Delta_3$ statistics, follows the Gaussian orthogonal ensemble statistics depends upon the deformation of the network from the perfect community structure. The spacing distribution is particularly useful to track very slight deformations of the network from a perfect community structure, whereas the density distribution and the $\Delta_3$ statistics remain identical to the undeformed network. On the other hand the $\Delta_3$ statistics is useful for the larger deformation strengths. Finally, we analyze the spectrum of a protein-protein interaction network for Helicobacter, and compare the spectral behavior with those of the model networks.Comment: accepted for publication in Phys. Rev. E (replaced with the final version

Spectral statistics of random geometric graphs

We use random matrix theory to study the spectrum of random geometric graphs, a fundamental model of spatial networks. Considering ensembles of random geometric graphs we look at short range correlations in the level spacings of the spectrum via the nearest neighbour and next nearest neighbour spacing distribution and long range correlations via the spectral rigidity Delta_3 statistic. These correlations in the level spacings give information about localisation of eigenvectors, level of community structure and the level of randomness within the networks. We find a parameter dependent transition between Poisson and Gaussian orthogonal ensemble statistics. That is the spectral statistics of spatial random geometric graphs fits the universality of random matrix theory found in other models such as Erdos-Renyi, Barabasi-Albert and Watts-Strogatz random graph.Comment: 19 pages, 6 figures. Substantially updated from previous versio

Phase separation in coupled chaotic maps on fractal networks

The phase ordering dynamics of coupled chaotic maps on fractal networks are investigated. The statistical properties of the systems are characterized by means of the persistence probability of equivalent spin variables that define the phases. The persistence saturates and phase domains freeze for all values of the coupling parameter as a consequence of the fractal structure of the networks, in contrast to the phase transition behavior previously observed in regular Euclidean lattices. Several discontinuities and other features found in the saturation persistence curve as a function of the coupling are explained in terms of changes of stability of local phase configurations on the fractals.Comment: (4 pages, 4 Figs, Submitted to PRE

Modification in CSF specific gravity in acutely decompensated cirrhosis and acute on chronic liver failure independent of encephalopathy, evidences for an early blood-CSF barrier dysfunction in cirrhosis

Although hepatic encephalopathy (HE) on the background of acute on chronic liver failure (ACLF) is associated with high mortality rates, it is unknown whether this is due to increased blood-brain barrier permeability. Specific gravity of cerebrospinal fluid measured by CT is able to estimate blood-cerebrospinal fluid-barrier permeability. This study aimed to assess cerebrospinal fluid specific gravity in acutely decompensated cirrhosis and to compare it in patients with or without ACLF and with or without hepatic encephalopathy. We identified all the patients admitted for acute decompensation of cirrhosis who underwent a brain CT-scan. Those patients could present acute decompensation with or without ACLF. The presence of hepatic encephalopathy was noted. They were compared to a group of stable cirrhotic patients and healthy controls. Quantitative brain CT analysis used the Brainview software that gives the weight, the volume and the specific gravity of each determined brain regions. Results are given as median and interquartile ranges and as relative variation compared to the control/baseline group. 36 patients presented an acute decompensation of cirrhosis. Among them, 25 presented with ACLF and 11 without ACLF; 20 presented with hepatic encephalopathy grade ≥ 2. They were compared to 31 stable cirrhosis patients and 61 healthy controls. Cirrhotic patients had increased cerebrospinal fluid specific gravity (CSF-SG) compared to healthy controls (+0.4 %, p < 0.0001). Cirrhotic patients with ACLF have decreased CSF-SG as compared to cirrhotic patients without ACLF (−0.2 %, p = 0.0030) that remained higher than in healthy controls. The presence of hepatic encephalopathy did not modify CSF-SG (−0.09 %, p = 0.1757). Specific gravity did not differ between different brain regions according to the presence or absence of either ACLF or HE. In patients with acute decompensation of cirrhosis, and those with ACLF, CSF specific gravity is modified compared to both stable cirrhotic patients and healthy controls. This pattern is observed even in the absence of hepatic encephalopathy suggesting that blood-CSF barrier impairment is manifest even in absence of overt hepatic encephalopathy

Statistical properties of power-law random banded unitary matrices in the delocalization-localization transition regime

Power-law random banded unitary matrices (PRBUM), whose matrix elements decay in a power-law fashion, were recently proposed to model the critical statistics of the Floquet eigenstates of periodically driven quantum systems. In this work, we numerically study in detail the statistical properties of PRBUM ensembles in the delocalization-localization transition regime. In particular, implications of the delocalization-localization transition for the fractal dimension of the eigenvectors, for the distribution function of the eigenvector components, and for the nearest neighbor spacing statistics of the eigenphases are examined. On the one hand, our results further indicate that a PRBUM ensemble can serve as a unitary analog of the power-law random Hermitian matrix model for Anderson transition. On the other hand, some statistical features unseen before are found from PRBUM. For example, the dependence of the fractal dimension of the eigenvectors of PRBUM upon one ensemble parameter displays features that are quite different from that for the power-law random Hermitian matrix model. Furthermore, in the time-reversal symmetric case the nearest neighbor spacing distribution of PRBUM eigenphases is found to obey a semi-Poisson distribution for a broad range, but display an anomalous level repulsion in the absence of time-reversal symmetry.Comment: 10 pages + 13 fig

Correspondence:

We questioned 180 patients with end-stage renal disease on maintenance haemodialysis, chronic ambulatory peritoneal dialysis and those who had undergone renal transplantation at the Department of Nephrology, General Hospital, Kuala Lumpur. Twelve patients (6.7%) had consumed excessive quantities ofanalgesics prior to the institution oflong-term dialysis or transplantation. Primary renal disease was considered to be analgesic nephropathy in seven patients (3.9%); in five patients (2.8%), analgesic abuse could have been a contributory factor to end-stage renal failure. Analgesic nephropathy is hence an uncommon cause of end-stage renal disease in Malaysia. However, it is important to be aware of the problem and to institute preventive measures as the cost of treatment for end-stage renal disease is prohibitive

Interconversion of intrinsic defects in SrTiO3(001)SrTiO_3(001)

Photoemission features associated with states deep in the band gap of n−SrTiO₃ (001) are found to be ubiquitous in bulk crystals and epitaxial films. These features are present even when there is little signal near the Fermi level. Analysis reveals that these states are deep-level traps associated with defects. The commonly investigated defects—O vacancies, Sr vacancies, and aliovalent impurity cations on the Ti sites—cannot account for these features. Rather, ab initio modeling points to these states resulting from interstitial oxygen and its interaction with donor electrons