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 inference from consensus dynamics

Abstract: We consider the problem of identifying the topology of a weighted, undirected network G from observing snapshots of multiple independent consensus dynamics. Specifically, we observe the opinion profiles of a group of agents for a set of M independent topics and our goal is to recover the precise relationships between the agents, as specified by the unknown network G. In order to overcome the under-determinacy of the problem at hand, we leverage concepts from spectral graph theory and convex optimization to unveil the underlying network structure. More precisely, we formulate the network inference problem as a convex optimization that seeks to endow the network with certain desired properties — such as sparsity — while being consistent with the spectral information extracted from the observed opinions. This is complemented with theoretical results proving consistency as the number M of topics grows large. We further illustrate our method by numerical experiments, which showcase the effectiveness of the technique in recovering synthetic and real-world networks.

A unified method for eigendecomposition of graph products

Abstract: In this paper, a unified method is developed for calculating the eigenvalues of the weighted adjacency and Laplacian matrices of three different graph products. These products have many applications in computational mechanics, such as ordering, graph partitioning, and subdomaining of finite element models.

First-principles multiway spectral partitioning of graphs

Abstract: We consider the minimum-cut partitioning of a graph into more than two parts using spectral methods. While there exist well-established spectral algorithms for this problem that give good results, they have traditionally not been well motivated. Rather than being derived from first principles by minimizing graph cuts, they are typically presented without direct derivation and then proved after the fact to work. In this paper, we take a contrasting approach in which we start with a matrix formulation of the minimum cut problem and then show, via a relaxed optimization, how it can be mapped onto a spectral embedding defined by the leading eigenvectors of the graph Laplacian. The end result is an algorithm that is similar in spirit to, but different in detail from, previous spectral partitioning approaches. In tests of the algorithm we find that it outperforms previous approaches on certain particularly difficult partitioning problems.