Showing papers on "Spectral graph theory published in 2006"

Finding community structure in networks using the eigenvectors of matrices

Abstract: We consider the problem of detecting communities or modules in networks, groups of vertices with a higher-than-average density of edges connecting them. Previous work indicates that a robust approach to this problem is the maximization of the benefit function known as ``modularity'' over possible divisions of a network. Here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix, which plays a role in community detection similar to that played by the graph Laplacian in graph partitioning calculations. This result leads us to a number of possible algorithms for detecting community structure, as well as several other results, including a spectral measure of bipartite structure in networks and a centrality measure that identifies vertices that occupy central positions within the communities to which they belong. The algorithms and measures proposed are illustrated with applications to a variety of real-world complex networks.

Graph Effective Resistance and Distributed Control: Spectral Properties and Applications

Abstract: We introduce the concept of matrix-valued effective resistance for undirected matrix-weighted graphs. Effective resistances are defined to be the square blocks that appear in the diagonal of the inverse of the matrix-weighted Dirichlet graph Laplacian matrix. However, they can also be obtained from a "generalized" electrical network that is constructed from the graph, and for which currents, voltages and resistances take matrix values. Effective resistances play an important role in several problems related to distributed control and estimation. They appear in least-squares estimation problems in which one attempts to reconstruct global information from relative noisy measurements; as well as in motion control problems in which agents attempt to control their positions towards a desired formation, based on noisy local measurements. We show that in either of these problems, the effective resistances have a direct physical interpretation. We also show that effective resistances provide bounds on the spectrum of the graph Laplacian matrix and the Dirichlet graph Laplacian. These bounds can be used to characterize the stability and convergence rate of several distributed algorithms that appeared in the literature.

Optimal network topologies: Expanders, Cages, Ramanujan graphs, Entangled networks and all that.

Abstract: We report on some recent developments in the search for optimal network topologies First we review some basic concepts on spectral graph theory, including adjacency and Laplacian matrices, paying special attention to the topological implications of having large spectral gaps We also introduce related concepts such as 'expanders', Ramanujan, and Cage graphs Afterwards, we discuss two different dynamical features of Networks, synchronizability and flow of random walkers, so that they are optimized if the corresponding Laplacian matrix has a large spectral gap From this, we show, by developing a numerical optimization algorithm, that maximum synchronizability and fast random walk spreading are obtained for a particular type of extremely homogeneous regular networks, with long loops and poor modular structure, that we call entangled networks These turn out to be related to Ramanujan and Cage graphs We argue also that these graphs are very good finite-size approximations to Bethe lattices, and provide optimal or almost optimal solutions to many other problems, for instance searchability in the presence of congestion or performance of neural networks Finally, we study how these results are modified when studying dynamical processes controlled by a normalized (weighted and directed) dynamics; much more heterogeneous graphs are optimal in this case Finally, a critical discussion of the limitations and possible extensions of this work is presented

Learning on Graph with Laplacian Regularization

Abstract: We consider a general form of transductive learning on graphs with Laplacian regularization, and derive margin-based generalization bounds using appropriate geometric properties of the graph. We use this analysis to obtain a better understanding of the role of normalization of the graph Laplacian matrix as well as the effect of dimension reduction. The results suggest a limitation of the standard degree-based normalization. We propose a remedy from our analysis and demonstrate empirically that the remedy leads to improved classification performance.

Atomic branching in molecules

Abstract: A graph theoretic measure of extended atomic branching is defined that accounts for the effects of all atoms in the molecule, giving higher weight to the nearest neighbors. It is based on the counting of all substructures in which an atom takes part in a molecule. We prove a theorem that permits the exact calculation of this measure based on the eigenvalues and eigenvectors of the adjacency matrix of the graph representing a molecule. The definition of this measure within the context of the Huckel molecular orbital (HMO) and its calculation for benzenoid hydrocarbons are also studied. We show that the extended atomic branching can be defined using any real symmetric matrix, as well as any Hermitian (self-adjoint) matrix, which permits its calculation in topological, geometrical, and quantum chemical contexts. © 2005 Wiley

Exponential Approximation by Stein's Method and Spectral Graph Theory

Abstract: General Berry-Esseen bounds are developed for the exponential distri- bution using Stein's method and a new concentration inequality approach. As an application, a sharp error term is obtained for Hora's result that the spectrum of the Johnson graph has an exponential limit.

Block diagonalization of adjacency and Laplacian matrices for graph product; applications in structural mechanics

Abstract: Eigenvalues and eigenvectors of graphs have many applications in structural mechanics and combinatorial optimization. For a regular space structure, the visualization of its graph model as the product of two simple graphs results in a substantial simplification in the solution of the corresponding eigenproblems. In this paper, the adjacency and Laplacian matrices of four graph products, namely, Cartesian, strong Cartesian, direct and lexicographic products are diagonalized and efficient methods are obtained for calculating their eigenvalues and eigenvectors. An exceptionally efficient method is developed for the eigensolution of the Laplacian matrices of strong Cartesian and direct products. Special attention is paid to the lexicographic product, which is not studied in the past as extensively as the other three graph products. Examples are provided to illustrate some applications of the methods in structural mechanics. Copyright © 2006 John Wiley & Sons, Ltd.

De novo analysis of peptide tandem mass spectra by spectral graph partitioning.

Abstract: We report on a new de novo peptide sequencing algorithm that uses spectral graph partitioning. In this approach, relationships between m/z peaks are represented by attractive and repulsive springs, and the vibrational modes of the spring system are used to infer information about the peaks (such as "likely b-ion" or "likely y-ion"). We demonstrate the effectiveness of this approach by comparison with other de novo sequencers on test sets of ion-trap and QTOF spectra, including spectra of mixtures of peptides. On all datasets, we outperform the other sequencers. Along with spectral graph theory techniques, the new de novo sequencer EigenMS incorporates another improvement of independent interest: robust statistical methods for recalibration of time-of-flight mass measurements. Robust recalibration greatly outperforms simple least-squares recalibration, achieving about three times the accuracy for one QTOF dataset.

Graph kernels by spectral transforms

Abstract: Many graph-based semi-supervised learning methods can be viewed as imposing smoothness conditions on the target function with respect to a graph representing the data points to be labeled. The smoothness properties of the functions are encoded in terms of Mercer kernels over the graph. The central quantity in such regularization is the spectral decomposition of the graph Laplacian, a matrix derived from the graph's edge weights. The eigenvectors with small eigenvalues are smooth, and ideally represent large cluster structures within the data. The eigenvectors having large eigenvalues are rugged, and considered noise. Different weightings of the eigenvectors of the graph Laplacian lead to different measures of smoothness. Such weightings can be viewed as spectral transforms, that is, as transformations of the standard eigenspectrum that lead to different regularizers over the graph. Familiar kernels, such as the diffusion kernel resulting by solving a discrete heat equation on the graph, can be seen as simple parametric spectral transforms. The question naturally arises whether one can obtain effective spectral transforms automatically. In this paper we develop an approach to searching over a nonparametric family of spectral transforms by using convex optimization to maximize kernel alignment to the labeled data. Order constraints are imposed to encode a preference for smoothness with respect to the graph structure. This results in a flexible family of kernels that is more data-driven than the standard parametric spectral transforms. Our approach relies on a quadratically constrained quadratic program (QCQP), and is computationally practical for large datasets.

Consistent observation grouping for generating metric-topological maps that improves robot localization

Abstract: Recently, hybrid maps that combine metric and topological world information have been proposed as a powerful representation of mobile robot environments. Among others, these maps are of special interest for efficiently managing large-scale environments, and for accurate localization. For achieving that, local geometric maps are stored in the nodes of a graph-based global map. In this paper we present a novel approach for automatically obtaining those local maps from observations. The method considers the space sensed in each observation as a node of a graph with arcs representing the space overlap between observations. The recursive partition (cut) of this graph produces groups of strongly connected nodes from which consistent local maps for accurate localization are derived. The proposed partition technique is well-grounded in the spectral graph theory of, and it is formulated for any type of sensor observation. We depict an implementation for grouping 2D laser scans, and show experimental results with real data that demonstrate the performance of the method

A novel way of computing similarities between nodes of a graph, with application to collaborative filtering and subspace projection of the graph nodes

Abstract: This work presents a new perspective on characterizing the similarity between elements of a database or, more generally, nodes of a weighted and undirected graph. It is based on a Markov-chain model of random walk through the database. More precisely, we compute quantities (the average commute time, the pseudoinverse of the Laplacian matrix of the graph, etc) that provide similarities between any pair of nodes, having the nice property of increasing when the number of paths connect- ing those elements increases and when the “length” of paths decreases. It turns out that the square root of the average commute time is a Eu- clidean distance and that the pseudoinverse of the Laplacian matrix is a kernel (its elements are inner products closely related to commute times). A procedure for computing the subspace projection of the node vectors of the graph that preserves as much variance as possible in terms of the commute-time distance – a principal component analysis (PCA) of the graph – is also introduced. This graph PCA provides a nice interpreta- tion to the “Fiedler vector”, widely used for graph partitioning. The model is evaluated on a collaborative-recommendation task where sug- gestions are made about which movies people should watch based upon what they watched in the past. Experimental results on the MovieLens database show that the Laplacian-based similarities (the pseudoinverse of the Laplacian matrix and the “random-forest matrix”) perform well in comparison with other methods. The model, which nicely fits into the so-called “statistical relational learning” framework, could also be usedto compute document or word similarities, and, more generally, it could be applied to machine-learning and pattern-recognition tasks involving a database.

Graph Laplacian Kernels for Object Classification from a Single Example

Abstract: Classification with only one labeled example per class is a challenging problem in machine learning and pattern recognition. While there have been some attempts to address this problem in the context of specific applications, very little work has been done so far on the problem under more general object classification settings. In this paper, we propose a graph-based approach to the problem. Based on a robust path-based similarity measure proposed recently, we construct a weighted graph using the robust path-based similarities as edge weights. A kernel matrix, called graph Laplacian kernel, is then defined based on the graph Laplacian. With the kernel matrix, in principle any kernel-based classifier can be used for classification. In particular, we demonstrate the use of a kernel nearest neighbor classifier on some synthetic data and real-world image sets, showing that our method can successfully solve some difficult classification tasks with only very few labeled examples.

A spectral generative model for graph structure

Abstract: This paper shows how to construct a generative model for graph structure. We commence from a sample of graphs where the correspondences between nodes are unknown ab initio. We also work with graphs where there may be structural differences present, i.e. variations in the number of nodes in each graph and the edge-structure. The idea underpinning the method is to embed the nodes of the graphs into a vector space by performing kernel PCA on the heat kernel. The co-ordinates of the nodes are determined by the eigenvalues and eigenvectors of the Laplacian matrix, together with a time parameter which can be used to scale the embedding. Node correspondences are located by applying Scott and Longuet-Higgins algorithm to the embedded nodes. We capture variations in graph structure using the covariance matrix for corresponding embedded point-positions. We construct a point distribution model for the embedded node positions using the eigenvalues and eigenvectors of the covariance matrix. We show how to use this model to both project individual graphs into the eigenspace of the point-position covariance matrix and how to fit the model to potentially noisy graphs to reconstruct the Laplacian matrix. We illustrate the utility of the resulting method for shape-analysis using data from the COIL database.

Hierarchical Directed Spectral Graph Partitioning MS&E 337 - Information Networks

Abstract: In this report, we examine the generalization of the Laplacian of a graph due to Fan Chung. We show that Fan Chung’s generalization reduces to examining one particular symmetrization of the adjacency matrix for a directed graph. From this result, the directed Cheeger bounds trivially follow. Additionally, we implement and examine the benefits of directed hierarchical spectral clustering empirically on a dataset from Wikipedia. Finally, we examine a set of competing heuristic methods on the same dataset.

Spectral Convergence of the Discrete Laplacian on Models of a Metrized Graph

Abstract: A metrized graph is a compact singular 1-manifold endowed with a metric. A given metrized graph can be modelled by a family of weighted combinatorial graphs. If one chooses a sequence of models from this family such that the vertices become uniformly distributed on the metrized graph, then the ith largest eigenvalue of the Laplacian matrices of these combinatorial graphs converges to the ith largest eigenvalue of the continuous Laplacian operator on the metrized graph upon suitable scaling. The eigenvectors of these matrices can be viewed as functions on the metrized graph by linear interpolation. These interpolated functions form a normal family, any convergent subsequence of which limits to an eigenfunction of the continuous Laplacian operator on the metrized graph.

Riemannian graph diffusion for DT-MRI regularization

Abstract: A new method for diffusion tensor MRI (DT-MRI) regularization is presented that relies on graph diffusion. We represent a DT image using a weighted graph, where the weights of edges are functions of the geodesic distances between tensors. Diffusion across this graph with time is captured by the heat-equation, and the solution, i.e. the heat kernel, is found by exponentiating the Laplacian eigen-system with time. Tensor regularization is accomplished by computing the Riemannian weighted mean using the heat kernel as its weights. The method can efficiently remove noise, while preserving the fine details of images. Experiments on synthetic and real-world datasets illustrate the effectiveness of the method.

Non–bipartite Graphs with Third Largest Laplacian Eigenvalue Less Than Three

Abstract: All bipartite graphs whose third largest Laplacian eigenvalue is less than 3 have been characterized by Zhang. In this paper, all connected non–bipartite graphs with third largest Laplacian eigenvalue less than three are determined.

On the Laplacian spectrum of an infinite graph

Abstract: In this paper, we introduce the notion of Laplacian spectrum of an infinite countable graph in a different way than in the papers by B. Mohar. We prove some basic properties of this type of spectrum. The approach used is in line with our approach to the limiting spectrum of an infinite graph. The technique of the Laplacian spectrum of finite graphs is essential in this approach.

Point pattern matching via spectral geometry

Abstract: In this paper, we describe the use of Riemannian geometry, and in particular the relationship between the Laplace-Beltrami operator and the graph Laplacian, for the purposes of embedding a graph onto a Riemannian manifold. Using the properties of Jacobi fields, we show how to compute an edge-weight matrix in which the elements reflect the sectional curvatures associated with the geodesic paths between nodes on the manifold. We use the resulting edge-weight matrix to embed the nodes of the graph onto a Riemannian manifold of constant sectional curvature. With the set of embedding coordinates at hand, the graph matching problem is cast as that of aligning pairs of manifolds subject to a geometric transformation. We illustrate the utility of the method on image matching using the COIL database.

Increasing statistical power in medical image analysis

Abstract: In this paper, we present a novel method for estimating the effective number of independent variables in imaging applications that require multiple hypothesis testing. The method increases the statistical power of the results by refuting the assumption of independence among variables, while keeping the probability of false positives low. It is based on the spectral graph theory, in which the variables are seen as the vertices of a complete undirected graph and the correlation matrix as the adjacency matrix that weights its edges. By computing the eigenvalues of the correlation matrix, it is possible to obtain valuable information about the dependence levels among the variables of the problem. The method is compared to other available models and its effectiveness illustrated in a case study on the morphology of the human corpus callosum.

Facial Expression Analysis Based on Graph Spectral Decomposition

Abstract: In this paper explore how to use spectral methods of geometry structural graph for analyzing facial expression and clustering in the pattern-space.Use the leading eigenvectors of the weighted graph adjacency matrix to define eigenmodes of the adjacency matrix.For each eigenmode,compute vectors of spectral properties.It includes the leading inter-mode adjacency matrices.Embed these vectors in a pattern-space using multidimensional scaling on the norm for pairs of patten vectors.Illustrate the utility of the embedding methods representing the arrangement of facial expression of dissimilar human face in the pattern-space.

Spectral Radius and maximum degree of connected graphs

Abstract: We give an upper bound on the maximal eigenvalue of the adjacency matrix of a connected graph in terms of its maximum degree, diameter and order This bound is best possible up to a constant factor and improves prevoius results of Stevanovic, Zhang, and Alon and Sudakov

Spectrum of the laplacian of an asymmetric fractal graph

Abstract: We consider a simple self-similar sequence of graphs which does not satisfy the symmetry conditions which imply the existence of a spectral decimation property for the eigenvalues of the graph Laplacians. We show that, for this particular sequence, a very similar property to spectral decimation exists, and obtain a complete description of the spectra of the graphs in the sequence. AMS 2000 subject classiflcation: Primary 28A80

Spectral Multiplicity Tolerant Inexact Graph Matching

Abstract: A graph can be exactly specified by the spectrum and corresponding eigenvectors of its adjacency matrix. This provides a solid foundation for spectrum based graph matching. However, most previous methods ignore the spectral multiplicity, which may significantly affect the matching accuracy. In this paper, we address the problem of minimizing the matching error when graph spectral multiplicity is involved. We first model spectral multiplicity by the sub-eigenspace rotation matrix R, and integrate R into the spectrum based graph matching model. We then focus on the exact graph matching problem, and show how to establish the vertex-to-vertex correspondence by iteratively optimizing the sub-eigenspace rotation matrix R and the permutation matrix P. A reliable matching initialization method is proposed to make this process converge rapidly. Finally, we extend the approach to the inexact graph matching problem by optimally warping two graphs to the same size. The proposed approach is robust and efficient. We support our approach with numerical experiments and demonstrate its effectiveness in the practical application of uncaliberated stereo matching.

Graph Spectral Decomposition and Clustering

Abstract: This paper investigates the spectral description methods for unweighted graph sequences and the clustering of spectral features in feature spaces. First, the corner features in 2D images of 3D polyhedral objects are represented as neighborhood graphs. Adjacency matrices are constructed from Delaunay graphs of the corners. Then the eigenmodes are defined by the leading eigenvectors of the adjacency matrices. For each eigenmode, we compute the vectors of spectral properties, which include the eigenmode perimeter, eigenmode volume, Cheeger number, inter-mode adjacency matrix and inter-mode edge distance. Then these vectors are embedded into a pattern space by multidimensional scaling on the L2 norm for pairs of pattern vectors. Meanwhile, the performances of different embedding methods are compared. Finally, the clustering results by k-means method are shown.