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.

Network Topology Identification from Spectral Templates

Abstract: Network topology inference is a cornerstone problem in statistical analyses of complex systems. In this context, the fresh look advocated here permeates benefits from convex optimization and graph signal processing, to identify the so-termed graph shift operator (encoding the network topology) given only the eigenvectors of the shift. These spectral templates can be obtained, for example, from principal component analysis of a set of graph signals defined on the particular network. The novel idea is to find a graph shift that while being consistent with the provided spectral information, it endows the network structure with certain desired properties such as sparsity. The focus is on developing efficient recovery algorithms along with identifiability conditions for two particular shifts, the adjacency matrix and the normalized graph Laplacian. Application domains include network topology identification from steady-state signals generated by a diffusion process, and design of a graph filter that facilitates the distributed implementation of a prescribed linear network operator. Numerical tests showcase the effectiveness of the proposed algorithms in recovering synthetic and structural brain networks.

Induced Gravity from Theory Space

Abstract: The mass spectrum of a model constructed in a theory space is expressed in terms of eigenvalues of the Laplacian on the graph structure of the theory space. The nature of the one-loop UV divergence in the vacuum energy is then determined by only the degree matrix of the graph. Using these facts, we construct models of induced gravity that do not exhibit quadratic divergences at the one-loop level.

Graph comparison and its application in network synchronization

Abstract: Using the tool of graph comparison from spectral graph theory, we propose new methodologies to guarantee complete synchronization in complex networks. The main idea is to utilize flexibly topological features of a given network so that the eigenvalues of the Laplacian matrix associated with the network can be estimated. The proposed methodologies enable the construction of different coupling-strength combinations in response to different knowledge about sub-networks. The obtained bounds of the network graphs' eigenvalues can be further used to study the robust synchronization problem in face of link failures in networks. Examples are utilized to demonstrate how to apply the methodologies to networks.

The generalized distance spectrum of a graph and applications

Abstract: The generalized distance matrix of a graph is the matrix whose entries depend only on the pairwise distances between vertices, and the generalized distance spectrum is the set of eigenvalues of thi...