Eigenvalue Spectra of Random Geometric Graphs (original) (raw)

Spectral statistics of random geometric graphs

We study the spectrum of random geometric graphs using random matrix theory. We look at short range correlations in the level spacings via the nearest neighbour and next nearest neighbour spacing distribution and long range correlations via the spectral rigidity ∆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 that the spectral statistics of random geometric graphs fits the universality of random matrix theory. In particular, the short range correlations are very close to those found in the Gaussian orthogonal ensemble of random matrix theory. For long range correlations we find deviations from Gaussian orthogonal ensemble statistics towards Poisson. We compare with previous results for Erd˝ os-Rényi, Barabási-Albert and Watts-Strogatz random graphs where similar random matrix theory universality has been found.

Spectral Analysis of Random Networks

Lecture Notes in Physics, 2004

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 ∆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 ∆3 statistics remain identical to the undeformed network. On the other hand the ∆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.

Eigenvalue ratio statistics of complex networks: Disorder versus randomness

Physical Review E

The distribution of the ratios of consecutive eigenvalue spacings of random matrices has emerged as an important tool to study spectral properties of many-body systems. This article numerically investigates the eigenvalue ratios distribution of various model networks, namely, small-world, Erdős-Rényi random, and (dis)assortative random having a diagonal disorder in the corresponding adjacency matrices. Without any diagonal disorder, the eigenvalues ratio distribution of these model networks depict Gaussian orthogonal ensemble (GOE) statistics. Upon adding diagonal disorder, there exists a gradual transition from the GOE to Poisson statistics depending upon the strength of the disorder. The critical disorder (w c) required to procure the Poisson statistics increases with the randomness in the network architecture. We relate w c with the time taken by maximum entropy random walker to reach the steady state. These analyses will be helpful to understand the role of eigenvalues other than the principal one for various network dynamics such as transient behavior.

Spectra of adjacency matrices of random geometric graphs

2006

We investigate the spectral properties of the adjacency matrices of random geomet- ric graphs both theoretically and by simulation, concentrating on the thermodynamic limit. Our results show interesting difierences from those previously found for other models of random graphs. In particular the spectra do not show the symmetry about 0 found for classical and scale-free random graphs, and we flnd

Eigenvalue spectra of complex networks

Journal of Physics A: Mathematical and General, 2005

We examine the eigenvalue spectrum, ρ(µ), of the adjacency matrix of a random scale-free network with an average of p edges per vertex using the replica method. We show how in the dense limit, when p → ∞, one can obtain two relatively simple coupled equations whose solution yields ρ(µ) for an arbitrary complex network. For scale-free graphs, with degree distribution exponent λ, we obtain an exact expression for the eigenvalue spectrum when λ = 3 and show that ρ(µ) ∼ 1/µ 2λ−1 for large µ. In the limit λ → ∞ we recover known results for the Erdös-Rényi random graph.

Spectral transitions in networks

New Journal of Physics, 2006

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 Brody-distribution. Furthermore, below the critical point p(s) can be given with the help of the regularised 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.

Random Matrix Spectra and Relaxation in Complex Networks

Acta Physica Polonica B, 2015

We evaluate spectra of stochastic matrices defined on complex random graphs, with edges (i, j) of a graph being given positive random weights W ij > 0 in such a fashion that column sums are normalized to one. The structure of the graphs and the distribution of the non-zero edge weights W ij are largely arbitrary. We only require that the mean vertex degree remains finite in the thermodynamic limit, and that the W ij satisfy a detailed balance condition. The main motivation for this work derives from the fact that knowing the spectra of stochastic matrices is tantamount to knowing the complete spectrum of relaxation times of stochastic processes described by them. One of the interesting new phenomena uncovered by our study is the appearance of localization transitions and mobility edges in the spectra of stochastic matrices of the type investigated in the present study.

Mesoscopic structures and the Laplacian spectra of random geometric graphs

Journal of Complex Networks, 2015

We investigate the Laplacian spectra of random geometric graphs (RGGs). The spectra are found to consist of both a discrete and a continuous part. The discrete part is a collection of Dirac delta peaks at integer values roughly centered around the mean degree. The peaks are mainly due to the existence of mesoscopic structures that occur far more abundantly in RGGs than in non-spatial networks. The probability of certain mesoscopic structures is analytically calculated for one-dimensional RGGs and they are shown to produce integer-valued eigenvalues that comprise a significant fraction of the spectrum, even in the large network limit. A phenomenon reminiscent of Bose-Einstein condensation in the appearance of zero eigenvalues is also found.

Effect of Unfolding on the Spectral Statistics of Adjacency Matrices of Complex Networks

Procedia Computer Science, 2012

Random matrix theory is finding an increasing number of applications in the context of information theory and communication systems, especially in studying the properties of complex networks. Such properties include short-term and long-term correlation. We study the spectral fluctuations of the adjacency of networks using random-matrix theory. We consider the influence of the spectral unfolding, which is a necessary procedure to remove the secular properties of the spectrum, on different spectral statistics. We find that, while the spacing distribution of the eigenvalues shows little sensitivity to the unfolding method used, the spectral rigidity has greater sensitivity to unfolding.

Random matrix analysis of complex networks

Physical Review E, 2007

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\Delta_3Delta_3 statistic of RMT as well. It follows RMT prediction of linear behavior in semi-logarithmic scale with slope being sim1/pi2\sim 1/\pi^2sim1/pi2. 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.