Spectral graph theory

About: Spectral graph theory is a research topic. Over the lifetime, 1334 publications have been published within this topic receiving 77373 citations.

Identifying congestion in software-defined networks using spectral graph theory

Abstract: Software-defined networks (SDN) are an emerging technology that offers to simplify networking devices by centralizing the network layer functions and allowing adaptively programmable traffic flows. We propose using spectral graph theory methods to identify and locate congestion in a network. The analysis of the balanced traffic case yields an efficient solution for congestion identification. The unbalanced case demonstrates a distinct drop in connectivity that can be used to determine the onset of congestion. The eigenvectors of the Laplacian matrix are used to locate the congestion and achieve effective graph partitioning.

Network Cluster-Robust Inference

Abstract: Network data commonly consists of observations on a single large network. Accordingly, researchers often partition the network into clusters in order to apply cluster-robust inference methods. All existing such methods require clusters to be asymptotically independent. We show that for this requirement to hold, under certain conditions, it is necessary and sufficient for clusters to have small "conductance," which is the ratio of edge boundary size to volume. This yields a quantitative measure of cluster quality. Unfortunately, there are important classes of networks for which small-conductance clusters appear not to exist. Our simulation results show that for such networks, cluster-robust methods can exhibit substantial size distortion. Based on well-known results in spectral graph theory, we suggest using the eigenvalues of the graph Laplacian to determine the existence and number of small-conductance clusters. We also discuss the use of spectral clustering for constructing clusters in practice.

On The Critical Point Structure of Eigenfunctions Belonging to the First Nonzero Eigenvalue of A Genus Two Closed Hyperbolic Surface

Abstract: We develop a method based on spectral graph theory to approximate the eigenvalues and eigenfunctions of the Laplace-Beltrami operator of a compact riemannian manifold -- The method is applied to a closed hyperbolic surface of genus two -- The results obtained agree with the ones obtained by other authors by different methods, and they serve as experimental evidence supporting the conjectured fact that the generic eigenfunctions belonging to the first nonzero eigenvalue of a closed hyperbolic surface of arbitrary genus are Morse functions having the least possible total number of critical points among all Morse functions admitted by such manifolds

In search for a perfect shape of polyhedra: Buffon transformation

Abstract: For an arbitrary polygon consider a new one by joining the centres of consecutive edges. Iteration of this procedure leads to a shape which is affine equivalent to a regular polygon. This regularisation effect is usually ascribed to Count Buffon (1707-1788). We discuss a natural analogue of this procedure for 3-dimensional polyhedra, which leads to a new notion of affine $B$-regular polyhedra. The main result is the proof of existence of star-shaped affine $B$-regular polyhedra with prescribed combinatorial structure, under partial symmetry and simpliciality assumptions. The proof is based on deep results from spectral graph theory due to Colin de Verdiere and Lovasz.

Graph Adjacency Matrix Associated with a Data Partition

Abstract: A frequently recurring problem in several applications is to compare two or more data sets and evaluate the level of similarity In this paper we describe a technique to compare two data partitions of different data sets The comparison is obtained by means of matrices called Graph Adjacency Matrices which represent the data sets Then, a match coefficient returns an estimation of the level of similarity between the data sets