Showing papers on "Spectral graph theory published in 2011"

Spectral Sparsification of Graphs

Abstract: We introduce a new notion of graph sparsification based on spectral similarity of graph Laplacians: spectral sparsification requires that the Laplacian quadratic form of the sparsifier approximate that of the original. This is equivalent to saying that the Laplacian of the sparsifier is a good preconditioner for the Laplacian of the original. We prove that every graph has a spectral sparsifier of nearly linear size. Moreover, we present an algorithm that produces spectral sparsifiers in time $O(m\log^{c}m)$, where $m$ is the number of edges in the original graph and $c$ is some absolute constant. This construction is a key component of a nearly linear time algorithm for solving linear equations in diagonally dominant matrices. Our sparsification algorithm makes use of a nearly linear time algorithm for graph partitioning that satisfies a strong guarantee: if the partition it outputs is very unbalanced, then the larger part is contained in a subgraph of high conductance.

Algorithms, Graph Theory, and Linear Equations in Laplacian Matrices

Abstract: The Laplacian matrices of graphs are fundamental. In addition to facilitating the application of linear algebra to graph theory, they arise in many practical problems. In this talk we survey recent progress on the design of provably fast algorithms for solving linear equations in the Laplacian matrices of graphs. These algorithms motivate and rely upon fascinating primitives in graph theory, including low-stretch spanning trees, graph sparsifiers, ultra-sparsifiers, and local graph clustering. These are all connected by a definition of what it means for one graph to approximate another. While this definition is dictated by Numerical Linear Algebra, it proves useful and natural from a graph theoretic perspective. Mathematics Subject Classification (2010). Primary 68Q25; Secondary 65F08.

Chebyshev polynomial approximation for distributed signal processing

Abstract: Unions of graph Fourier multipliers are an important class of linear operators for processing signals defined on graphs. We present a novel method to efficiently distribute the application of these operators to the high-dimensional signals collected by sensor networks. The proposed method features approximations of the graph Fourier multipliers by shifted Chebyshev polynomials, whose recurrence relations make them readily amenable to distributed computation. We demonstrate how the proposed method can be used in a distributed denoising task, and show that the communication requirements of the method scale gracefully with the size of the network.

Two Laplacians for the Distance Matrix of a Graph

Abstract: We introduce a Laplacian and a signless Laplacian for the distance matrix of a connected graph, called the distance Laplacian and distance signless Laplacian , respectively. We show the equivalence between the distance signless Laplacian, distance Laplacian and the distance spectra for the class of transmission regular graphs. There is also an equivalence between the Laplacian spectrum and the distance Laplacian spectrum of any connected graph of diameter 2. Similarities between n , as a distance Laplacian eigenvalue, and the algebraic connectivity are established.

Multi-way spectral partitioning and higher-order Cheeger inequalities

Abstract: A basic fact in spectral graph theory is that the number of connected components in an undirected graph is equal to the multiplicity of the eigenvalue zero in the Laplacian matrix of the graph. In particular, the graph is disconnected if and only if there are at least two eigenvalues equal to zero. Cheeger's inequality and its variants provide an approximate version of the latter fact; they state that a graph has a sparse cut if and only if there are at least two eigenvalues that are close to zero. It has been conjectured that an analogous characterization holds for higher multiplicities, i.e., there are $k$ eigenvalues close to zero if and only if the vertex set can be partitioned into $k$ subsets, each defining a sparse cut. We resolve this conjecture. Our result provides a theoretical justification for clustering algorithms that use the bottom $k$ eigenvectors to embed the vertices into $\mathbb R^k$, and then apply geometric considerations to the embedding. We also show that these techniques yield a nearly optimal tradeoff between the expansion of sets of size $\approx n/k$, and the $k$th smallest eigenvalue of the normalized Laplacian matrix, denoted $\lambda_k$. In particular, we show that in every graph there is a set of size at most $2n/k$ which has expansion at most $O(\sqrt{\lambda_k \log k})$. This bound is tight, up to constant factors, for the "noisy hypercube" graphs.

Spectra of Uniform Hypergraphs

Abstract: We present a spectral theory of hypergraphs that closely parallels Spectral Graph Theory. A number of recent developments building upon classical work has led to a rich understanding of "hyperdeterminants" of hypermatrices, a.k.a. multidimensional arrays. Hyperdeterminants share many properties with determinants, but the context of multilinear algebra is substantially more complicated than the linear algebra required to address Spectral Graph Theory (i.e., ordinary matrices). Nonetheless, it is possible to define eigenvalues of a hypermatrix via its characteristic polynomial as well as variationally. We apply this notion to the "adjacency hypermatrix" of a uniform hypergraph, and prove a number of natural analogues of basic results in Spectral Graph Theory. Open problems abound, and we present a number of directions for further study.

Constrained laplacian score for semi-supervised feature selection

Abstract: In this paper, we address the problem of semi-supervised feature selection from high-dimensional data. It aims to select the most discriminative and informative features for data analysis. This is a recent addressed challenge in feature selection research when dealing with small labeled data sampled with large unlabeled data in the same set. We present a filter based approach by constraining the known Laplacian score. We evaluate the relevance of a feature according to its locality preserving and constraints preserving ability. The problem is then presented in the spectral graph theory framework with a study of the complexity of the proposed algorithm. Finally, experimental results will be provided for validating our proposal in comparison with other known feature selection methods.

The Grone-Merris Conjecture

Abstract: In spectral graph theory, the Grone-Merris Conjecture asserts that the spectrum of the Laplacian matrix of a finite graph is majorized by the conjugate degree sequence of this graph. We give a complete proof for this conjecture.

An iterated graph laplacian approach for ranking on manifolds

Abstract: Ranking is one of the key problems in information retrieval. Recently, there has been significant interest in a class of ranking algorithms based on the assumption that data is sampled from a low dimensional manifold embedded in a higher dimensional Euclidean space.In this paper, we study a popular graph Laplacian based ranking algorithm [23] using an analytical method, which provides theoretical insights into the ranking algorithm going beyond the intuitive idea of "diffusion." Our analysis shows that the algorithm is sensitive to a commonly used parameter due to the use of symmetric normalized graph Laplacian. We also show that the ranking function may diverge to infinity at the query point in the limit of infinite samples. To address these issues, we propose an improved ranking algorithm on manifolds using Green's function of an iterated unnormalized graph Laplacian, which is more robust and density adaptive, as well as pointwise continuous in the limit of infinite samples.We also for the first time in the ranking literature empirically explore two variants from a family of twice normalized graph Laplacians. Experimental results on text and image data support our analysis, which also suggest the potential value of twice normalized graph Laplacians in practice.

Perturbation results for the Estrada index in weighted networks

Abstract: The logarithm of the Estrada index has been proposed recently as a spectral measure to character efficiently the robustness of complex networks. In this paper, we explore the Estrada index in weighted networks and develop various perturbation results based on spectral graph theory. It is shown that the robustness of a network may be enhanced even when some edge weights are reduced. This is of particular theoretical and practical significance to network design and optimization.

Robustness and vulnerability analysis of water distribution networks using graph theoretic and complex network principles

Abstract: Water distribution systems are regarded as large sparse graphs with complex network characteristics. Topological aspects of system resilience, viewed as the antonym to structural vulnerability, are assessed in connection with the network architecture, robustness and loop redundancy. Deterministic techniques from complex networks and spectral graph theory are utilized to quantify well-connectedness and estimate loop redundancy in the studied benchmark networks. By using graph connectivity and expansion properties, system robustness against node/link failures and isolation of the demand nodes from the source(s) are assessed and network tolerance against random failures and targeted attacks on their bridges and cut sets are analyzed. Among other measurements, two metrics of meshed-ness and algebraic connectivity are proposed as candidates for quantification of redundancy and robustness, respectively, in optimization design models. A brief discussion on the scope and relevance of the provided measurements in the analysis of resilience of water distribution networks is presented.

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.

Cheeger constants, growth and spectrum of locally tessellating planar graphs

Abstract: In this article, we study relations between the local geometry of planar graphs (combinatorial curvature) and global geometric invariants, namely the Cheeger constants and the exponential growth. We also discuss spectral applications.

Wavelet frames on graphs defined by fMRI functional connectivity

Abstract: Multiscale representations such as the wavelet transform are useful for many signal processing tasks. Graphs are flexible models to represent complex networks and a spectral graph wavelet transform (SGWT) has recently been developed as a generalization of conventional wavelet designs.

Semi-Supervised Learning with Spectral Graph Wavelets

Abstract: We consider the transductive learning problem when the labels belong to a continuous space. Through the use of spectral graph wavelets, we explore the benefits of multiresolution analysis on a graph constructed from the labeled and unlabeled data. The spectral graph wavelets behave like discrete multiscale differential operators on graphs, and thus can sparsely approximate piecewise smooth signals. Therefore, rather than enforce a prior belief that the labels are globally smooth with respect to the intrinsic structure of the graph, we enforce sparse priors on the spectral graph wavelet coefficients. One issue that arises when the proportion of data with labels is low is that the fine scale wavelets that are useful in sparsely representing discontinuities are largely masked, making it difficult to recover the high frequency components of the label sequence. We discuss this challenge, and propose one method to use the structured sparsity of the wavelet coefficients to aid label reconstruction.

Structural Analysis of Laplacian Spectral Properties of Large-Scale Networks

Abstract: Using methods from algebraic graph theory and convex optimization, we study the relationship between local structural features of a network and spectral properties of its Laplacian matrix. In particular, we derive expressions for the so-called spectral moments of the Laplacian matrix of a network in terms of a collection of local structural measurements. Furthermore, we propose a series of semidefinite programs to compute bounds on the spectral radius and the spectral gap of the Laplacian matrix from a truncated sequence of Laplacian spectral moments. Our analysis shows that the Laplacian spectral moments and spectral radius are strongly constrained by local structural features of the network. On the other hand, we illustrate how local structural features are usually not enough to estimate the Laplacian spectral gap.

Toward the Optimization of Normalized Graph Laplacian

Abstract: Normalized graph Laplacian has been widely used in many practical machine learning algorithms, e.g., spectral clustering and semisupervised learning. However, all of them use the Euclidean distance to construct the graph Laplacian, which does not necessarily reflect the inherent distribution of the data. In this brief, we propose a method to directly optimize the normalized graph Laplacian by using pairwise constraints. The learned graph is consistent with equivalence and nonequivalence pairwise relationships, and thus it can better represent similarity between samples. Meanwhile, our approach, unlike metric learning, automatically determines the scale factor during the optimization. The learned normalized Laplacian matrix can be directly applied in spectral clustering and semisupervised learning algorithms. Comprehensive experiments demonstrate the effectiveness of the proposed approach.

Spectral Analysis of Certain Spherically Homogeneous Graphs

Abstract: We study operators on rooted graphs with a certain spherical homogeneity. These graphs are called path commuting and allow for a decomposition of the adjacency matrix and the Laplacian into a direct sum of Jacobi matrices which reflect the structure of the graph. Thus, the spectral properties of the adjacency matrix and the Laplacian can be analyzed by means of the elaborated theory of Jacobi matrices. For some examples which include antitrees, we derive the decomposition explicitly and present a zoo of spectral behavior induced by the geometry of the graph. In particular, these examples show that spectral types are not at all stable under rough isometries.

New Nordhaus-Gaddum-type results for the Kirchhoff index

Abstract: Let G be a connected graph. The resistance distance between any two vertices of G is defined as the net effective resistance between them if each edge of G is replaced by a unit resistor. The Kirchhoff index is the sum of resistance distances between all pairs of vertices in G. Zhou and Trinajstic (Chem Phys Lett 455(1–3):120–123, 2008) obtained a Nordhaus-Gaddum-type result for the Kirchhoff index by obtaining lower and upper bounds for the sum of the Kirchhoff index of a graph and its complement. In this paper, by making use of the Cauchy-Schwarz inequality, spectral graph theory and Foster’s formula, we give better lower and upper bounds. In particular, the lower bound turns out to be tight. Furthermore, we establish lower and upper bounds on the product of the Kirchhoff index of a graph and its complement.

On the Spectral Evolution of Large Networks

Abstract: In this thesis, I study the spectral characteristics of large dynamic networks and formulate the spectral evolution model. The spectral evolution model applies to networks that evolve over time, and describes their spectral decompositions such as the eigenvalue and singular value decomposition. The spectral evolution model states that over time, the eigenvalues of a network change while its eigenvectors stay approximately constant. I validate the spectral evolution model empirically on over a hundred network datasets, and theoretically by showing that it generalizes arncertain number of known link prediction functions, including graph kernels, path counting methods, rank reduction and triangle closing. The collection of datasets I use contains 118 distinct network datasets. One dataset, the signed social network of the Slashdot Zoo, was specifically extracted during work on this thesis. I also show that the spectral evolution model can be understood as a generalization of the preferential attachment model, if we consider growth in latent dimensions of a network individually. As applications of the spectral evolution model, I introduce two new link prediction algorithms that can be used for recommender systems, search engines, collaborative filtering, rating prediction, link sign prediction and more. The first link prediction algorithm reduces to a one-dimensional curve fitting problem from which a spectral transformation is learned. The second method uses extrapolation of eigenvalues to predict future eigenvalues. As special cases, I show that the spectral evolution model applies to directed, undirected, weighted, unweighted, signed and bipartite networks. For signed graphs, I introduce new applications of the Laplacian matrix for graph drawing, spectral clustering, and describe new Laplacian graph kernels. I also define the algebraic conflict, a measure of the conflict present in a signed graph based on the signed graph Laplacian. I describe the problem of link sign prediction spectrally, and introduce the signed resistance distance. For bipartite and directed graphs, I introduce the hyperbolic sine and odd Neumann kernels, which generalize the exponential and Neumann kernels for undirected unipartite graphs. I show that the problem of directed and bipartite link prediction are related by the fact that both can be solved by considering spectral evolution in the singular value decomposition.

Fast approximation algorithms for graph partitioning using spectral and semidefinite-programming techniques

Abstract: Graph partitioning problems are a central topic of research in the study of approximation algorithms. They are of interest to theoretical computer scientists for their far-reaching connections to spectral graph theory, metric embeddings and inapproximability theory. And they are also important for many practitioners, as algorithms for graph partitioning are often fundamental primitives in the solution of other problems, such as image segmentation, clustering and social-network analysis. While many theoretical approximation algorithms exist for graph partitioning, they often rely on multicommodity-flow computations that run quadratic time in the worst case and are too time-consuming for the massive graphs that are prevalent in today's applications. In this thesis, we study the design of approximation algorithms that yield strong approximation ratios, while running in subquadratic time and relying on computational procedures that are often fast in practice. Our algorithms employ spectral and s-t flow operations to explore the cuts of a graph in a very efficient way. A crucial ingredient in their design is the usage of random walks that capture the sparse cuts of a graph better than single eigenvectors. The analysis of our methods is particularly simple, as it relies on a semidefinite programming formulation of the graph partitioning problem of choice. Indeed, we can develop our algorithms as primal-dual methods for solving a semidefinite program and show that certain random walks arise naturally from this approach.

The tau constant and the discrete Laplacian matrix of a metrized graph

Abstract: We express the tau constant of a metrized graph in terms of the discrete Laplacian matrix and its pseudo-inverse.

Analysis of Adjacency Matrix and Neighborhood Associated with Zero Divisor Graph of Finite Commutative Rings

Abstract: The present investigation is concerned with zero divisor graph of direct Product of finite commutative rings and to give some new ideas about its corresponding adjacency matrix. In the first section of the paper, we study about neighborhood set of the zero divisor graph of direct product over finite commutative rings. In the second section we discussed some examples of these ring. Finally, some surprsing results (regarding to the adjacency matrix) and theorems also estabilised. The study on neighborhood may be used to represent graph in computer algorthim, neighborhood are used to determine the clustering cofficient of graph and adjacency martix is useful in computer application.

Spectral Regression with Low-Rank Approximation for Dynamic Graph Link Prediction

Abstract: The paper mentions that the temporal regression model for the dynamic graph link prediction problem rests on spectral graph theory and low-rank approximation for the graph Laplacian matrix.

Eigenvector sensitive feature selection for spectral clustering

Abstract: Spectral clustering is one of the most popular methods for data clustering, and its performance is determined by the quality of the eigenvectors of the related graph Laplacian. Generally, graph Laplacian is constructed using the full features, which will degrade the quality of the related eigenvectors when there are a large number of noisy or irrelevant features in datasets. To solve this problem, we propose a novel unsupervised feature selection method inspired by perturbation analysis theory, which discusses the relationship between the perturbation of the eigenvectors of a matrix and its elements' perturbation. We evaluate the importance of each feature based on the average L1 norm of the perturbation of the first k eigenvectors of graph Laplacian corresponding to the k smallest positive eigenvalues, with respect to the feature's perturbation. Extensive experiments on several high-dimensional multi-class datasets demonstrate the good performance of our method compared with some state-of-the-art unsupervised feature selection methods.

An uncertainty principle for functions defined on graphs

Abstract: The classical uncertainty principle provides a fundamental tradeoff in the localization of a function in the time and frequency domains. In this paper we extend this classical result to functions defined on graphs. We justify the use of the graph Laplacian's eigenbasis as a surrogate for the Fourier basis for graphs, and define the notions of "spread" in the graph and spectral domains. We then establish an analogous uncertainty principle relating the two quantities, showing the degree to which a function can be simultaneously localized in the graph and spectral domains.