Spectra of adjacency matrices of random geometric graphs (original) (raw)
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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.
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.
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Inspired by the prospect of having discretized spaces emerge from random graphs, we construct a collection of simple and explicit exponential random graph models that enjoy, in an appropriate parameter regime, a roughly constant vertex degree and form very large numbers of simple polygons (triangles or squares). The models avoid the collapse phenomena that plague naive graph Hamiltonians based on triangle or square counts. More than that, statistically significant numbers of other geometric primitives (small pieces of regular lattices, cubes) emerge in our ensemble, even though they are not in any way explicitly pre-programmed into the formulation of the graph Hamiltonian, which only depends on properties of paths of length 2. While much of our motivation comes from hopes to construct a graph-based theory of random geometry (Euclidean quantum gravity), our presentation is completely self-contained within the context of exponential random graph theory, and the range of potential applications is considerably more broad.
The degree sequences and spectra of scale-free random graphs
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Statistical mechanics of random graphs
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We discuss various aspects of the statistical formulation of the theory of random graphs, with emphasis on results obtained in a series of our recent publications.
Random Graphs from Random Matrices
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We use the order complex corresponding to a symmetric matrix (defined by Giusti et al in 2015). In this note, we use it to define a class of models of random graphs, and show some surprising experimental results, showing sharp phase transitions.
On the evolution of random geometric graphs on spaces of negative curvature
2013
We study a family of random geometric graphs on hyperbolic spaces. In this setting, N points are chosen randomly on the hyperbolic plane of curvature −ζ 2 and any two of them are joined by an edge with probability that depends on their hyperbolic distance, independently of every other pair. In particular, when the positions of the points have been fixed, the distribution over the set of graphs on these points is the Boltzmann distribution, where the Hamiltonian is given by the sum of weighted indicator functions for each pair of points. The weight is proportional to a real parameter β > 0 (interpreted as the inverse temperature) as well as to the hyperbolic distance between the corresponding points. The N points are distributed according to a quasi-uniform distribution which is a distorted version of the uniform distribution and depends on a parameter α > 0-when α = ζ it coincides with the uniform distribution. This class of random graphs was proposed by Krioukov et al. [17] as a model for complex networks. We provide a rigorous analysis of aspects of this model for all values of the ratio ζ/α, focusing on its dependence on the parameter β. We show that a phase transition occurs around β = 1. More specifically, let us assume that 0 < ζ/α < 2. In this case, we show that when β > 1 the degree of a typical vertex is bounded in probability (in fact it follows a distribution which for large values exhibits a power-law tail whose exponent depends only on ζ and α), whereas for β < 1 the degree is a random variable whose expected value grows polynomially in N. When β = 1, we establish logarithmic growth. For the case β > 1, we establish a connection with a class of inhomogeneous random graphs known as the Chung-Lu model. We show that the probability that two given points are adjacent is expressed through the kernel of this inhomogeneous random graph. We also consider the case ζ/α ≥ 2.
Physical Review E, 2009
Parameter-dependent statistical properties of spectra of totally connected irregular quantum graphs with Neumann boundary conditions are studied. The autocorrelation functions of level velocities c(x) andc(ω, x) as well as the distributions of level curvatures and avoided crossing gaps are calculated. The numerical results are compared with the predictions of Random Matrix Theory (RMT) for Gaussian Orthogonal Ensemble (GOE) and for coupled GOE matrices. The application of coupled GOE matrices was justified by studying localization phenomena in graphs' wave functions Ψ(x) using the Inverse Participation Ratio (IPR) and the amplitude distribution P (Ψ(x)).
Physical Review E, 2002
We analyse graphs in which each vertex is assigned random coordinates in a geometric space of arbitrary dimensionality and only edges between adjacent points are present. The critical connectivity is found numerically by examining the size of the largest cluster. We derive an analytical expression for the cluster coefficient which shows that the graphs are distinctly different from standard random graphs, even for infinite dimensionality. Insights relevant for graph bi-partitioning are included.
Statistical mechanics of random geometric graphs: Geometry-induced first-order phase transition
Physical review. E, Statistical, nonlinear, and soft matter physics, 2015
Random geometric graphs (RGGs) can be formalized as hidden-variables models where the hidden variables are the coordinates of the nodes. Here we develop a general approach to extract the typical configurations of a generic hidden-variables model and apply the resulting equations to RGGs. For any RGG, defined through a rigid or a soft geometric rule, the method reduces to a nontrivial satisfaction problem: Given N nodes, a domain D, and a desired average connectivity 〈k〉, find, if any, the distribution of nodes having support in D and average connectivity 〈k〉. We find out that, in the thermodynamic limit, nodes are either uniformly distributed or highly condensed in a small region, the two regimes being separated by a first-order phase transition characterized by a O(N) jump of 〈k〉. Other intermediate values of 〈k〉 correspond to very rare graph realizations. The phase transition is observed as a function of a parameter a∈[0,1] that tunes the underlying geometry. In particular, a=1 in...