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.

Spectral determinant of Schrödinger operators on graphs

Abstract: We study the spectral properties of the operator (- +V (x )) on a graph. ( is the Laplacian and V (x ) is some potential defined on the graph). In particular, we derive an expression for the spectral determinant that generalizes one previously obtained for the Laplacian operator.

Techniques for Constructing Biorthogonal Bipartite Graph Filter Banks

Abstract: The processing of data defined on irregular discrete domains, i.e., graph signals, is becoming an emerging area with great application potential. Using spectral graph theory, Narang and Ortega (2013) laid the framework for two channel filter banks with critical sampling for bipartite graph signals. The bipartite graph filter bank can be extended to any arbitrary graph using the notion of separable filtering. The design of the biorthogonal filter banks by Narang and Ortega (2013) is based on the factorization of a maximally flat polynomial. The factorization technique does not allow much control of the spectral response of the graph filters, resulting in response asymmetry. In this paper, we present a generic framework for constructing biorthogonal graph filter banks that does not require factorization. We introduce the notion of polyphase representation and ladder structures for graph filter banks. We show that filters having virtual spectral symmetry and almost energy preservation can be constructed without any sophisticated optimization. Fine control of the spectral response can also be achieved with ease.

A Unified Framework for Structured Graph Learning via Spectral Constraints

Abstract: Graph learning from data represents a canonical problem that has received substantial attention in the literature. However, insufficient work has been done in incorporating prior structural knowledge onto the learning of underlying graphical models from data. Learning a graph with a specific structure is essential for interpretability and identification of the relationships among data. Useful structured graphs include the multi-component graph, bipartite graph, connected graph, sparse graph, and regular graph. In general, structured graph learning is an NP-hard combinatorial problem, therefore, designing a general tractable optimization method is extremely challenging. In this paper, we introduce a unified graph learning framework lying at the integration of Gaussian graphical models and spectral graph theory. To impose a particular structure on a graph, we first show how to formulate the combinatorial constraints as an analytical property of the graph matrix. Then we develop an optimization framework that leverages graph learning with specific structures via spectral constraints on graph matrices. The proposed algorithms are provably convergent, computationally efficient, and practically amenable for numerous graph-based tasks. Extensive numerical experiments with both synthetic and real data sets illustrate the effectiveness of the proposed algorithms. The code for all the simulations is made available as an open source repository.

Network growth and the spectral evolution model

Abstract: We introduce and study the spectral evolution model, which characterizes the growth of large networks in terms of the eigenvalue decomposition of their adjacency matrices: In large networks, changes over time result in a change of a graph's spectrum, leaving the eigenvectors unchanged. We validate this hypothesis for several large social, collaboration, authorship, rating, citation, communication and tagging networks, covering unipartite, bipartite, signed and unsigned graphs. Following these observations, we introduce a link prediction algorithm based on the extrapolation of a network's spectral evolution. This new link prediction method generalizes several common graph kernels that can be expressed as spectral transformations. In contrast to these graph kernels, the spectral extrapolation algorithm does not make assumptions about specific growth patterns beyond the spectral evolution model. We thus show that it performs particularly well for networks with irregular, but spectral, growth patterns.

The problem of deficiency indices for discrete Schrödinger operators on locally finite graphs

Abstract: The number of self-adjoint extensions of a symmetric operator acting on a complex Hilbert space is characterized by its deficiency indices. Given a locally finite unoriented simple tree, we prove that the deficiency indices of any discrete Schrodinger operator are either null or infinite. We also prove that all deterministic discrete Schrodinger operators which act on a random tree are almost surely self-adjoint. Furthermore, we provide several criteria of essential self-adjointness. We also address some importance to the case of the adjacency matrix and conjecture that, given a locally finite unoriented simple graph, its deficiency indices are either null or infinite. Besides that, we consider some generalizations of trees and weighted graphs.