Related papers: Spectral Statistics of Lattice Graph Percolation M…
The relationships between eigenvalues and eigenvectors of a product graph and those of its factor graphs have been known for the standard products, while characterization of Laplacian eigenvalues and eigenvectors of the Kronecker product of…
Random geometric graphs are a popular choice for a latent points generative model for networks. Their definition is based on a sample of $n$ points $X_1,X_2,\cdots,X_n$ on the Euclidean sphere~$\mathbb{S}^{d-1}$ which represents the latent…
Graphs with diverse structural characteristics play a central role in modelling and optimization tasks. The ability to generate different types of graphs that exhibit shared properties is likewise essential for algorithm selection and…
Exact eigendecomposition of large matrices is very expensive, and it is practically impossible to compute exact eigenvalues. Instead, one may set a more modest goal of approaching the empirical distribution of the eigenvalues, recovering…
Representing data residing on a graph as a linear combination of building block signals can enable efficient and insightful visual or statistical analysis of the data, and such representations prove useful as regularizers in signal…
We show that, with very high probability, the random graph Laplacian has simple spectrum. Our method provides a quantitatively effective estimate of the spectral gaps. Along the way, we establish results on affine no-gaps delocalization,…
Quantum graphs have attracted attention from mathematicians for some time. A quantum graph is defined by having a Laplacian on each edge of a metric graph and imposing boundary conditions at the vertices to get an eigenvalue problem. A…
Random graphs defined by an occurrence probability that is invariant under node aggregation have been identified recently in the context of network renormalization. The invariance property requires that edges are drawn with a specific…
Although the spectra of random networks have been studied for a long time, the influence of network topology on the dense limit of network spectra remains poorly understood. By considering the configuration model of networks with four…
Graphs are ubiquitous data structures for representing interactions between entities. With an emphasis on the use of graphs to represent chemical molecules, we explore the task of learning to generate graphs that conform to a distribution…
The operator-theoretic dichotomy underlying diffusion on directed networks is \emph{symmetry versus non-self-adjointness} of the Markov transition operator. In the reversible (detailed-balance) regime, a directed random walk $P$ is…
This paper studies the eigenvalue distribution of the Watts-Strogatz random graph, which is known as the "small-world" random graph. The construction of the small-world random graph starts with a regular ring lattice of n vertices; each has…
Sparse non-Hermitian random matrices arise in the study of disordered physical systems with asymmetric local interactions, and have applications ranging from neural networks to ecosystem dynamics. The spectral characteristics of these…
We consider the adjacency matrices of sparse random graphs from the Chung-Lu model, where edges are added independently between the $N$ vertices with varying probabilities $p_{ij}$. The rank of the matrix $(p_{ij})$ is some fixed positive…
Let $(X,E_X)$ and $(V,E_V)$ be finite connected graphs without loops. We assume that $V$ has two distinguished vertices $a,b$ and an automorphism $\gamma$ which exchanges $a$ and~$b$. The $V$-edge substitution of $X$ is the graph $X[V]$…
The eigenvalue spectrum of the transition matrix of a network encodes important information about its structural and dynamical properties. We study the transition matrix of a family of fractal scale-free networks and analytically determine…
Spectral graph theory gives an algebraical approach to analyze the dynamics of a network by using the matrix that represents the network structure. However, it is not easy for social networks to apply the spectral graph theory because the…
Large graphs commonly appear in social networks, knowledge graphs, recommender systems, life sciences, and decision making problems. Summarizing large graphs by their high level properties is helpful in solving problems in these settings.…
We provide a perturbative expansion for the empirical spectral distribution of a Hermitian matrix with large size perturbed by a random matrix with small operator norm whose entries in the eigenvector basis of the first one are independent…
Singular spectrum analysis is developed as a nonparametric spectral decomposition of a time series. It can be easily extended to the decomposition of multidimensional lattice-like data through the filtering interpretation. In this…