Related papers: Spectra of Modular and Small-World Matrices

The spectral properties of the Laplacian on a class of quantum graphs with random metric structure are studied. Namely, we consider quantum graphs spanned by the simple $\ZZ^d$-lattice with $\delta$-type boundary conditions at the vertices,…

We study the spectrum of adjacency matrices of random graphs. We develop two techniques to lower bound the mass of the continuous part of the spectral measure or the density of states. As an application, we prove that the spectral measure…

Spectral embedding is a procedure which can be used to obtain vector representations of the nodes of a graph. This paper proposes a generalisation of the latent position network model known as the random dot product graph, to allow…

We consider Laplacians on periodic equilateral metric graphs. The spectrum of the Laplacian consists of an absolutely continuous part (which is a union of an infinite number of non-degenerated spectral bands) plus an infinite number of flat…

We study randomly coloured graphs embedded into Euclidean space, whose vertex sets are infinite, uniformly discrete subsets of finite local complexity. We construct the appropriate ergodic dynamical systems, explicitly characterise ergodic…

Modularity is designed to measure the strength of division of a network into clusters (known also as communities). Networks with high modularity have dense connections between the vertices within clusters but sparse connections between…

We propose a framework for the connection between local symmetries of discrete Hamiltonians and the design of compact localized states. Such compact localized states are used for the creation of tunable, local symmetry-induced bound states…

We derive a message passing method for computing the spectra of locally tree-like networks and an approximation to it that allows us to compute closed-form expressions or fast numerical approximates for the spectral density of random graphs…

Determining the effect of structural perturbations on the eigenvalue spectra of networks is an important problem because the spectra characterize not only their topological structures, but also their dynamical behavior, such as…

We describe the complete spectra of Laplacian, signless Laplacian, and adjacency matrices associated with the commuting graphs of a finite group using group theoretic information. We provide a method to find the center of a group by only…

For a simple graph $G$, the generalized adjacency matrix $A_{\alpha}(G)$ is defined as $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G), \alpha\in [0,1]$, where $A(G)$ is the adjacency matrix and $D(G)$ is the diagonal matrix of vertex degrees of…

Spectral methods that are based on eigenvectors and eigenvalues of discrete graph Laplacians, such as Diffusion Maps and Laplacian Eigenmaps are often used for manifold learning and non-linear dimensionality reduction. It was previously…

We derive the spectral properties of adjacency matrix of complex networks and of their Laplacian by the replica method combined with a dynamical population algorithm. By assuming the order parameter to be a product of Gaussian…

Generative models for graphs have been typically committed to strong prior assumptions concerning the form of the modeled distributions. Moreover, the vast majority of currently available models are either only suitable for characterizing…

We present examples of rooted tree graphs for which the Laplacian has singular continuous spectral measures. For some of these examples we further establish fractional Hausdorff dimensions. The singular continuous components, in these…

We derive exact equations that determine the spectra of undirected and directed sparsely connected regular graphs containing loops of arbitrary length. The implications of our results to the structural and dynamical properties of networks…

We take on a Random Matrix theory viewpoint to study the spectrum of certain reversible Markov chains in random environment. As the number of states tends to infinity, we consider the global behavior of the spectrum, and the local behavior…

We study spectra of directed networks with inhibitory and excitatory couplings. We investigate in particular eigenvector localization properties of various model networks for different value of correlation among their entries. Spectra of…

We introduce an unsupervised graph embedding that trades off local node similarity and connectivity, and global structure. The embedding is based on a generalized graph Laplacian, whose eigenvectors compactly capture both network structure…

We analyze eigenvalues fluctuations of the Laplacian of various networks under the random matrix theory framework. Analyses of random networks, scale-free networks and small-world networks show that nearest neighbor spacing distribution of…