scispace - formally typeset
Search or ask a question
Topic

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.


Papers
More filters
Journal ArticleDOI
TL;DR: In this article, an efficient method for calculating the eigenvalues of space structures with regular topologies is presented, where the topology of a structure is formed as the Cartesian product of its generators.

20 citations

01 Jan 2003
TL;DR: This paper designs support-tree preconditioners for n × n matrices with m nonzeros that are symmetric and diagonally-dominant with a nonnegative diagonal (SDD matrix) and shows that dilexp(G) is always at most n, hence the bound is at most O(m √ n log n) for any Laplacian (or SDD) matrix.
Abstract: In this paper we design support-tree preconditioners for n × n matrices with m nonzeros that are symmetric and diagonally-dominant with a nonnegative diagonal (SDD matrix). This reduces to designing such a preconditioner for a Laplacian matrix, A, which can be interpreted as an undirected nonnegatively-weighted graph, G with n vertices and m edges. Preconditioners accelerate the convergence of iterative methods for solving linear systems, and our preconditioner allows us to analyze the convergence of a particular algorithm, due to Gremban and Miller, called support-tree conjugate gradient (STCG). An advantage of support-tree preconditioners is that STCG parallelizes well. We show that STCG equipped with our preconditioner requires O(m log n · √ dilexp(G)) work and O(m) space to solve the system Ax = b, where dilexp(G) is an edge-expansion-based upper bound on the diameter of G. Existing bounds depend only on the size of the matrix (graph), hence our bound is incomparable. For instance, if G is a bounded-degree expander graph with uniform edge weights, dilexp(G) = O(log 2 n), and the work is O(n log n). This is currently the best known bound for Laplacians of expander graphs. We show that dilexp(G) is always at most n, hence our bound is at most O(m √ n log n) for any Laplacian (or SDD) matrix. For sufficiently dense systems, when m = Ω(n), this bound offers the best known work guarantee of any linear-space method. The main technical contributions of this paper include (i) adapting a recent result of Racke to designing support-tree preconditioners, (ii) extending a power dissipation approach for bounding support numbers of preconditioners, and (iii) applying the methods used in Leighton and Rao’s approximate max-flow min-cut theorem to the “asymmetric” product flows the arise in Racke’s construction.

20 citations

01 Jan 2003
TL;DR: This thesis develops a set of simple yet realistic interactive processing models for perceptual organization, model the processing in the framework of spectral graph theory, with a criterion encoding the overall goodness of perceptual organization.
Abstract: Perceptual organization refers to the process of organizing sensory input into coherent and interpretable perceptual structures. This process is challenging due to the chicken-and-egg nature between the various sub-processes such as image segmentation, figure-ground segregation and object recognition. Low-level processing requires the guidance of high-level knowledge to overcome noise; while high-level processing relies on low-level processes to reduce the computational complexity. Neither process can be sufficient on its own. Consequently, any system that carries out these processes in a sequence is bound to be brittle. An alternative system is one in which all processes interact with each other simultaneously. In this thesis, we develop a set of simple yet realistic interactive processing models for perceptual organization. We model the processing in the framework of spectral graph theory, with a criterion encoding the overall goodness of perceptual organization. We derive fast solutions for near-global optima of the criterion, and demonstrate the efficacy of the models on segmenting a wide range of real images. Through these models, we are able to capture a variety of perceptual phenomena: a unified treatment of various grouping, figure-ground and depth cues to produce popout, region segmentation and depth segregation in one step; and a unified framework for integrating bottom-up and top-down information to produce an object segmentation from spatial and object attention. We achieve these goals by empowering current spectral graph methods with a principled solution for multiclass spectral graph partitioning; expanded repertoire of grouping cues to include similarity, dissimilarity and ordering relationships; a theory for integrating sparse grouping cues; and a model for representing and integrating higher-order relationships. These computational tools are also useful more generally in other domains where data need to be organized effectively.

20 citations

Posted Content
TL;DR: In this article, a unified approach to study convergence and stochastic stability of continuous time consensus protocols (CPs) is presented, which applies to networks with directed information flow; both cooperative and non-cooperative interactions; networks under weak stochastically forcing; and those whose topology and strength of connections may vary in time.
Abstract: A unified approach to studying convergence and stochastic stability of continuous time consensus protocols (CPs) is presented in this work. Our method applies to networks with directed information flow; both cooperative and noncooperative interactions; networks under weak stochastic forcing; and those whose topology and strength of connections may vary in time. The graph theoretic interpretation of the analytical results is emphasized. We show how the spectral properties, such as algebraic connectivity and total effective resistance, as well as the geometric properties, such the dimension and the structure of the cycle subspace of the underlying graph, shape stability of the corresponding CPs. In addition, we explore certain implications of the spectral graph theory to CP design. In particular, we point out that expanders, sparse highly connected graphs, generate CPs whose performance remains uniformly high when the size of the network grows unboundedly. Similarly, we highlight the benefits of using random versus regular network topologies for CP design. We illustrate these observations with numerical examples and refer to the relevant graph-theoretic results. Keywords: consensus protocol, dynamical network, synchronization, robustness to noise, algebraic connectivity, effective resistance, expander, random graph

20 citations

Posted Content
TL;DR: In this paper, the authors present a general procedure that allows for the reduction or expansion of any network (considered as a weighted graph), which maintains the spectrum of the network's adjacency matrix up to a set of eigenvalues known beforehand from its graph structure.
Abstract: In this paper we present a general procedure that allows for the reduction or expansion of any network (considered as a weighted graph). This procedure maintains the spectrum of the network's adjacency matrix up to a set of eigenvalues known beforehand from its graph structure. This procedure can be used to establish new equivalence relations on the class of all weighted graphs (networks) where two graphs are equivalent if they can be reduced to the same graph. Additionally, dynamical networks (or any finite dimensional, discrete time dynamical system) can be analyzed using isospectral transformations. By so doing we obtain stronger results regarding the global stability (strong synchronization) of dynamical networks when compared to other standard methods.

20 citations


Network Information
Related Topics (5)
Bounded function
77.2K papers, 1.3M citations
82% related
Upper and lower bounds
56.9K papers, 1.1M citations
82% related
Iterative method
48.8K papers, 1.2M citations
81% related
Matrix (mathematics)
105.5K papers, 1.9M citations
80% related
Optimization problem
96.4K papers, 2.1M citations
79% related
Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20241
202316
202236
202153
202086
201981