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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.


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Journal ArticleDOI
TL;DR: A unified theoretical framework for estimating various transmission costs in wireless networks is developed by generalizing the random walk theory that has been primarily developed for undirected graphs to digraphs and demonstrating that the proposed digraph-based analytical model can achieve more accurate transmission cost estimation over existing methods.
Abstract: Various applications in wireless networks, such as routing and query processing, can be formulated as random walks on graphs. Many results have been obtained for such applications by utilizing the theory of random walks (or spectral graph theory), which is mostly developed for undirected graphs. However, this formalism neglects the fact that the underlying (wireless) networks in practice contain asymmetric links, which are best characterized by directed graphs (digraphs). Therefore, random walk on digraphs is a more appropriate model to consider for such networks. In this paper, by generalizing the random walk theory (or spectral graph theory) that has been primarily developed for undirected graphs to digraphs, we show how various transmission costs in wireless networks can be formulated in terms of hitting times and cover times of random walks on digraphs. Using these results, we develop a unified theoretical framework for estimating various transmission costs in wireless networks. Our framework can be applied to random walk query processing strategy and the three routing paradigms--best path routing, opportunistic routing, and stateless routing--to which nearly all existing routing protocols belong. Extensive simulations demonstrate that the proposed digraph-based analytical model can achieve more accurate transmission cost estimation over existing methods.

39 citations

Book
Bogdan Nica1
27 Jul 2018
TL;DR: In this paper, the authors introduce spectral graph theory as a powerful tool for graph-theoretic applications, including the proof of the Friendship Theorem, which states that if any two persons have exactly one common friend, then there is a person who is everybody's friend.
Abstract: Spectral graph theory starts by associating matrices to graphs, notably, the adjacency matrix and the laplacian matrix. The general theme is then, firstly, to compute or estimate the eigenvalues of such matrices, and secondly, to relate the eigenvalues to structural properties of graphs. As it turns out, the spectral perspective is a powerful tool. Some of its loveliest applications concern facts that are, in principle, purely graph-theoretic or combinatorial. To give just one example, spectral ideas are a key ingredient in the proof of the so-called Friendship Theorem: if, in a group of people, any two persons have exactly one common friend, then there is a person who is everybody’s friend. This text is an introduction to spectral graph theory, but it could also be seen as an invitation to algebraic graph theory. On the one hand, there is, of course, the linear algebra that underlies the spectral ideas in graph theory. On the other hand, most of our examples are graphs of algebraic origin. The two recurring sources are Cayley graphs of groups, and graphs built out of finite fields. In the study of such graphs, some further algebraic ingredients (e.g., characters) naturally come up. The table of contents gives, as it should, a good glimpse of where is this text going. Very broadly, the first half is devoted to graphs, finite fields, and how they come together. This part is meant as an appealing and meaningful motivation. It provides a context that frames and fuels much of the second, spectral, half. Most sections have one or two exercises. Their position within the text is a hint. The exercises are optional, in the sense that virtually nothing in the main body depends on them. But the exercises are often of the non-trivial variety, and they should enhance the text in an interesting way. The hope is that the reader will enjoy them. We assume a basic familiarity with linear algebra, finite fields, and groups, but not necessarily with graph theory. This, again, betrays our algebraic perspective. This text is based on a course I taught in Göttingen, in the Fall of 2015. I would like to thank Jerome Baum for his help with some of the drawings. The present version is preliminary, and comments are welcome (email: bogdan.nica@gmail.com).

39 citations

Posted Content
TL;DR: This paper introduces a unified graph learning framework lying at the integration of Gaussian graphical models and spectral graph theory, and develops an optimization framework that leverages graph learning with specific structures via spectral constraints on graph matrices.
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.

39 citations

Journal ArticleDOI
TL;DR: This paper uses algebraic graph theory and convex optimization to study how structural properties influence the spectrum of eigenvalues of the network, and can compute, with low computational overhead, global spectral properties of a network from its local structural properties.
Abstract: The eigenvalues of matrices representing the structure of large-scale complex networks present a wide range of applications, from the analysis of dynamical processes taking place in the network to spectral techniques aiming to rank the importance of nodes in the network. A common approach to study the relationship between the structure of a network and its eigenvalues is to use synthetic random networks in which structural properties of interest, such as degree distributions, are prescribed. Although very common, synthetic models present two major flaws: 1) These models are only suitable to study a very limited range of structural properties; and 2) they implicitly induce structural properties that are not directly controlled and can deceivingly influence the network eigenvalue spectrum. In this paper, we propose an alternative approach to overcome these limitations. Our approach is not based on synthetic models. Instead, we use algebraic graph theory and convex optimization to study how structural properties influence the spectrum of eigenvalues of the network. Using our approach, we can compute, with low computational overhead, global spectral properties of a network from its local structural properties. We illustrate our approach by studying how structural properties of online social networks influence their eigenvalue spectra.

38 citations

Posted Content
TL;DR: This paper interprets neighborhood graphs of pixel patches as discrete counterparts of Riemannian manifolds and performs analysis in the continuous domain, providing insights into several fundamental aspects of graph Laplacian regularization.
Abstract: Inverse imaging problems are inherently under-determined, and hence it is important to employ appropriate image priors for regularization. One recent popular prior---the graph Laplacian regularizer---assumes that the target pixel patch is smooth with respect to an appropriately chosen graph. However, the mechanisms and implications of imposing the graph Laplacian regularizer on the original inverse problem are not well understood. To address this problem, in this paper we interpret neighborhood graphs of pixel patches as discrete counterparts of Riemannian manifolds and perform analysis in the continuous domain, providing insights into several fundamental aspects of graph Laplacian regularization. Specifically, we first show the convergence of the graph Laplacian regularizer to a continuous-domain functional, integrating a norm measured in a locally adaptive metric space. Focusing on image denoising, we derive an optimal metric space assuming nonlocal self-similarity of pixel patches, leading to an optimal graph Laplacian regularizer for denoising in the discrete domain. We then interpret graph Laplacian regularization as an anisotropic diffusion scheme to explain its behavior during iterations, e.g., its tendency to promote piecewise smooth signals under certain settings. To verify our analysis, an iterative image denoising algorithm is developed. Experimental results show that our algorithm performs competitively with state-of-the-art denoising methods such as BM3D for natural images, and outperforms them significantly for piecewise smooth images.

38 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20241
202316
202236
202153
202086
201981