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.

Source Localization on Graphs via l1 Recovery and Spectral Graph Theory

Abstract: We cast the problem of source localization on graphs as the simultaneous problem of sparse recovery and diffusion kernel learning. An l1 regularization term enforces the sparsity constraint while we recover the sources of diffusion from a single snapshot of the diffusion process. The diffusion kernel is estimated by assuming the process to be as generic as the standard heat diffusion. We show with synthetic data that we can concomitantly learn the diffusion kernel and the sources, given an estimated initialization. We validate our model with cholera mortality and atmospheric tracer diffusion data, showing also that the accuracy of the solution depends on the construction of the graph from the data points.

Eigenfrequencies of symmetric planar trusses via weighted graph symmetry and new canonical forms

Abstract: Purpose – The purpose of this paper is to describe how the method recently developed for mass‐spring systems and frame structures is modified to include the free vibration of trusses.Design/methodology/approach – Here, two methods are presented for calculating the eigenfrequencies of structures. The first approach is graph theoretical and uses graph symmetry. The graph models are decomposed into submodels and healing processes are employed such that the union of the eigenvalues of the healed submodels contain the eigenvalues of the entire model. The second method has an algebraic nature and uses special canonical forms. The present method is illustrated through three simple examples with odd and even number of bays.Findings – The inter‐relation for the mechanical properties of elements is established using new weighted graphs, enabling easy calculation of the eigenvalues involved. Two methods are presented for calculating the eigenfrequencies of the truss structures.Originality/value – Symmetry is used fo...

Orthogonal Eigenvector Matrix of the Laplacian

Abstract: The orthogonal eigenvector matrix Z of the Laplacian matrix of a graph with N nodes is studied rather than its companion X of the adjacency matrix, because for the Laplacian matrix, the eigenvector matrix Z corresponds to the adjacency companion X of a regular graph, whose properties are easier. In particular, the column sum vector of Z (which we call the fundamental weight vector w) is, for a connected graph, proportional to the basic vector eN = (0,0,, 1), so that more information about the specfics of the graph is contained in the row sum of Z (which we call the dual fundamental weight vector φ). Since little is known about Z (or X), we have tried to understand simple properties of Z such as the number of zeros, the sum of elements, the maximum and minimum element and properties of φ. For the particular class of Erdős-Renyi random graphs, we found that a product of a Gaussian and a super-Gaussian distribution approximates accurately the distribution of φU, a uniformly at random chosen component of the dual fundamental weight vector of Z.

Determining what Questions to Ask, with the Help of Spectral Graph Theory.

Abstract: This paper considers questions and the objects being asked about to be a graph and formulates the knowledge goal of a question-asking agent in terms of connecting this graph. The game of twenty questions can be thought of as a testbed of such a question-asking agent’s knowledge. If the agent’s knowledge of the domain were completely specified, the goal of questionasking would be to find the answer as quickly as possible and could follow a decision tree approach to narrow down the candidate answers. However, if the agent’s knowledge is incomplete, it must have a secondary goal for the questions it plans: to complete its knowledge. We claim that this secondary goal of a question asking agent can be formulated in terms of spectral graph theory. In particular, disconnected portions of the graph must be connected in a principled way. We show how the eigenvalues of a graph Laplacian of the the question-object adjacency graph can identify whether a set of knowledge contains disconnected components and the zero elements of the powers of the question-object adjacency graph provide a way to identify these questions. We illustrate the approach using an emotion description task.

Graph theoretic uncertainty and feasibility

Abstract: We expand upon a graph theoretic set of uncertainty principles with tight bounds for difference estimators acting simultaneously in the graph domain and the frequency domain. We show that the eigenfunctions of a modified graph Laplacian and a modified normalized graph Laplacian operator dictate the upper and lower bounds for the inequalities. Finally, we establish the feasibility region of difference estimator values in $\mathbb{R}^2$.