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.

Problems in Signal Processing and Inference on Graphs

Abstract: Modern datasets are often massive due to the sharp decrease in the cost of collecting and storing data. Many are endowed with relational structure modeled by a graph, an object comprising a set of points and a set of pairwise connections between them. A “signal on a graph” has elements related to each other through a graph—it could model, for example, measurements from a sensor network. In this dissertation we study several problems in signal processing and inference on graphs. We begin by introducing an analogue to Heisenberg’s time-frequency uncertainty principle for signals on graphs. We use spectral graph theory and the standard extension of Fourier analysis to graphs. Our spectral graph uncertainty principle makes precise the notion that a highly localized signal on a graph must have a broad spectrum, and vice versa. Next, we consider the problem of detecting a randomwalk on a graph from noisy observations. We characterize the performance of the optimal detector through the (type-II) error exponent, borrowing techniques from statistical physics to develop a lower bound exhibiting a phase transition. Strong performance is only guaranteed when the signal to noise ratio exceeds twice the random walk’s entropy rate. Monte Carlo simulations show that the lower bound is quite close to the true exponent. iii

Bounds on the Expansion Properties of Tanner Graphs

Abstract: This work focuses on the expansion properties of a Tanner Graph because they are known to be related to the performance of associated iterative message-passing algorithms over various channels. By analyzing the eigenvalues and corresponding eigenvectors of the normalized incidence matrix representing a Tanner Graph, lower bounds on these expansion properties are derived. Specifically, for the binary erasure channel, these results lead to two lower bounds on stopping distance for any given binary linear code and an upper bound on stopping redundancy for the family of difference-set codes (type-I 2-D projective geometry low-density parity-check (LDPC) codes).

Graph adjacency matrix learning for irregularly sampled Markovian natural images

Abstract: The boost of signal processing on graph has recently solicited research on the problem of identifying (learning) the graph underlying the observed signal values according to given criteria, such as graph smoothness or graph sparsity. This paper proposes a procedure for learning the adjacency matrix of a graph providing support to a set of irregularly sampled image values. Our approach to the graph adjacency matrix learning takes into account both the image luminance and the spatial samples' distances, and leads to a flexible and computationally light parametric procedure. We show that, under mild conditions, the proposed procedure identifies a near optimal graph for Markovian fields; specifically, the links identified by the learning procedure minimize the potential energy of the Markov random field for the signal samples under concern. We also show, by numerical simulations, that the learned adjacency matrix leads to a higly compact spectral wavelet graph transform of the so obtained signal on graph and favourably compares to state-of-the-art graph learning procedures, definetly matching the intrinsic signal structure.

Spectral Graph Analysis: A Unified Explanation and Modern Perspectives

Abstract: Complex networks or graphs are ubiquitous in sciences and engineering: biological networks, brain networks, transportation networks, social networks, and the World Wide Web, to name a few. Spectral graph theory provides a set of useful techniques and models for understanding `patterns of interconnectedness' in a graph. Our prime focus in this paper is on the following question: Is there a unified explanation and description of the fundamental spectral graph methods? There are at least two reasons to be interested in this question. Firstly, to gain a much deeper and refined understanding of the basic foundational principles, and secondly, to derive rich consequences with practical significance for algorithm design. However, despite half a century of research, this question remains one of the most formidable open issues, if not the core problem in modern network science. The achievement of this paper is to take a step towards answering this question by discovering a simple, yet universal statistical logic of spectral graph analysis. The prescribed viewpoint appears to be good enough to accommodate almost all existing spectral graph techniques as a consequence of just one single formalism and algorithm.