Showing papers on "Spectral graph theory published in 2010"

Spectral clustering and the high-dimensional stochastic blockmodel

Abstract: Networks or graphs can easily represent a diverse set of data sources that are characterized by interacting units or actors. Social networks, representing people who communicate with each other, are one example. Communities or clusters of highly connected actors form an essential feature in the structure of several empirical networks. Spectral clustering is a popular and computationally feasible method to discover these communities. The stochastic blockmodel [Social Networks 5 (1983) 109--137] is a social network model with well-defined communities; each node is a member of one community. For a network generated from the Stochastic Blockmodel, we bound the number of nodes "misclustered" by spectral clustering. The asymptotic results in this paper are the first clustering results that allow the number of clusters in the model to grow with the number of nodes, hence the name high-dimensional. In order to study spectral clustering under the stochastic blockmodel, we first show that under the more general latent space model, the eigenvectors of the normalized graph Laplacian asymptotically converge to the eigenvectors of a "population" normalized graph Laplacian. Aside from the implication for spectral clustering, this provides insight into a graph visualization technique. Our method of studying the eigenvectors of random matrices is original.

Concentration of the adjacency matrix and of the Laplacian in random graphs with independent edges

Abstract: Consider any random graph model where potential edges appear independently, with possibly different probabilities, and assume that the minimum expected degree is !(lnn). We prove that the adjacency matrix and the Laplacian of that random graph are concentrated around the corresponding matrices of the weighted graph whose edge weights are the probabilities in the random model. We apply this result to two different settings. In bond percolation, we show that, whenever the minimum expected degree in the random model is not too small, the Laplacian of the percolated graph is typically close to that of the original graph. As a corollary, we improve upon a bound for the spectral gap of the percolated graph due to Chung and Horn.

Know Thy Neighbor: Towards Optimal Mapping of Contacts to Social Graphs for DTN Routing

Abstract: Delay Tolerant Networks (DTN) are networks of self-organizing wireless nodes, where end-to-end connectivity is intermittent. In these networks, forwarding decisions are generally made using locally collected knowledge about node behavior (e.g., past contacts between nodes) to predict future contact opportunities. The use of complex network analysis has been recently suggested to perform this prediction task and improve the performance of DTN routing. Contacts seen in the past are aggregated to a social graph, and a variety of metrics (e.g., centrality and similarity) or algorithms (e.g., community detection) have been proposed to assess the utility of a node to deliver a content or bring it closer to the destination. In this paper, we argue that it is not so much the choice or sophistication of social metrics and algorithms that bears the most weight on performance, but rather the mapping from the mobility process generating contacts to the aggregated social graph. We first study two well-known DTN routing algorithms - SimBet and BubbleRap - that rely on such complex network analysis, and show that their performance heavily depends on how the mapping (contact aggregation) is performed. What is more, for a range of synthetic mobility models and real traces, we show that improved performances (up to a factor of 4 in terms of delivery ratio) are consistently achieved for a relatively narrow range of aggregation levels only, where the aggregated graph most closely reflects the underlying mobility structure. To this end, we propose an online algorithm that uses concepts from unsupervised learning and spectral graph theory to infer this 'correct' graph structure; this algorithm allows each node to locally identify and adjust to the optimal operating point, and achieves good performance in all scenarios considered.

Optimization Strategies for the Vulnerability Analysis of the Electric Power Grid

Abstract: Identifying small groups of lines, whose removal would cause a severe blackout, is critical for the secure operation of the electric power grid. We show how power grid vulnerability analysis can be studied as a bilevel mixed integer nonlinear programming problem. Our analysis reveals a special structure in the formulation that can be exploited to avoid nonlinearity and approximate the original problem as a pure combinatorial problem. The key new observation behind our analysis is the correspondence between the Jacobian matrix (a representation of the feasibility boundary of the equations that describe the flow of power in the network) and the Laplacian matrix in spectral graph theory (a representation of the graph of the power grid). The reduced combinatorial problem is known as the network inhibition problem, for which we present a mixed integer linear programming formulation. Our experiments on benchmark power grids show that the reduced combinatorial model provides an accurate approximation, to enable vulnerability analyses of real-sized problems with more than 16,520 power lines.

Towards a spectral theory of graphs based on the signless Laplacian, III

Abstract: This part of our work further extends our project of building a new spectral theory of graphs (based on the signless Laplacian) by some results on graph angles, by several comments and by a short survey of recent results.

Strategic Interaction and Networks

Abstract: This paper brings a general network analysis to a wide class of economic games. A network, or interaction matrix, tells who directly interacts with whom. A major challenge is determining how network structure shapes overall outcomes. We have a striking result. Equilibrium conditions depend on a single number: the lowest eigenvalue of a network matrix. Combining tools from potential games, optimization, and spectral graph theory, we study games with linear best replies and characterize the Nash and stable equilibria for any graph and for any impact of players’ actions. When the graph is sufficiently absorptive (as measured by this eigenvalue), there is a unique equilibrium. When it is less absorptive, stable equilibria always involve extreme play where some agents take no actions at all. This paper is the first to show the importance of this measure to social and economic outcomes, and we relate it to different network link patterns.

The laplacian paradigm: emerging algorithms for massive graphs

Abstract: This presentation describes an emerging paradigm for the design of efficient algorithms for massive graphs This paradigm, which we will refer to as the Laplacian Paradigm, is built on a recent suite of nearly-linear time primitives in spectral graph theory developed by Spielman and Teng, especially their solver for linear systems Ax=b, where A is the Laplacian matrix of a weighted, undirected n-vertex graph and b is an n-place vector. In the Laplacian Paradigm for solving a problem (on a massive graph), we reduce the optimization or computational problem to one or multiple linear algebraic problems that can be solved efficiently by applying the nearly-linear time Laplacian solver So far, the Laplacian paradigm already has some successes It has been applied to obtain nearly-linear-time algorithms for applications in semi-supervised learning, image process, web-spam detection, eigenvalue approximation, and for solving elliptic finite element systems It has also been used to design faster algorithms for generalized lossy flow computation and for random sampling of spanning trees. The goal of this presentation is to encourage more researchers to consider the use of the Laplacian Paradigm to develop faster algorithms for solving fundamental problems in combinatorial optimization (e.g., the computation of matchings, flows and cuts), in scientific computing (e.g., spectral approximation), in machine learning and data analysis (such as for web-spam detection and social network analysis), and in other applications that involve massive graphs.

Eigenvalue bounds, spectral partitioning, and metrical deformations via flows

Abstract: We present a new method for upper bounding the second eigenvalue of the Laplacian of graphs. Our approach uses multi-commodity flows to deform the geometry of the graph; we embed the resulting metric into Euclidean space to recover a bound on the Rayleigh quotient. Using this, we show that every n-vertex graph of genus g and maximum degree D satisfies λ2(G)=O((g+1)3D/n). This recovers the O(D/n) bound of Spielman and Teng for planar graphs, and compares to Kelner's bound of O((g+1)poly(D)/n), but our proof does not make use of conformal mappings or circle packings. We are thus able to extend this to resolve positively a conjecture of Spielman and Teng, by proving that λ2(G) = O(Dh6log h/n) whenever G is Kh-minor free. This shows, in particular, that spectral partitioning can be used to recover O(√n)-sized separators in bounded degree graphs that exclude a fixed minor. We extend this further by obtaining nearly optimal bounds on λ2 for graphs that exclude small-depth minors in the sense of Plotkin, Rao, and Smith. Consequently, we show that spectral algorithms find separators of sublinear size in a general class of geometric graphs. Moreover, while the standard “sweep” algorithm applied to the second eigenvector may fail to find good quotient cuts in graphs of unbounded degree, our approach produces a vector that works for arbitrary graphs. This yields an alternate proof of the well-known nonplanar separator theorem of Alon, Seymour, and Thomas that states that every excluded-minor family of graphs has O(√n)-node balanced separators.

The normalized laplacian matrix and general randi c index of graphs

Abstract: To any graph we may associate a matrix which records information about its structure. The goal of spectral graph theory is to see how the eigenvalues of such a matrix representation relate to the structure of a graph. In this thesis, we focus on a particular matrix representation of a graph, called the normalized Laplacian matrix, which is defined as L = D−1/2(D − A)D−1/2, where D is the diagonal matrix of degrees and A is the adjacency matrix of a graph. We first discuss some basic properties about the spectrum and the largest eigenvalue of the normalized Laplacian. We study graphs that are cospectral with respect to the normalized Laplacian eigenvalues. Properties of graphs with few normalized Laplacian eigenvalues are discussed. We then investigate the relationship that the normalized Laplacian eigenvalues have to the general Randic index R−1(G) of a graph, defined as R−1(G) = ∑ x∼y 1 dxdy , where dx is the degree of the vertex x. We next consider the energy of a simple graph with respect to its normalized Laplacian eigenvalues, which we call the L-energy. The L-energy of a graph G is EL(G) = ∑n i=1 |λi(L)− 1|, where λ1(L), . . . , λn(L) are the eigenvalues of L. Over graphs of order n that contain no isolated vertices, we characterize the graphs with minimal L-energy of 2 and maximal L-energy of 2bn/2c. The graphs of maximal L-energy are disconnected, which leads to the question: “What are the connected graphs of order n that have the maximum L-energy?” The technique we use is to first bound the L-energy of a graph G in terms of its general Randic index R−1(G). We highlight known results for R−1(G), most of which assume that G is a tree. We extend an upper bound on R−1(G) from trees to connected graphs, which in turn, provides a bound on the L-energy of a connected graph. We conjecture that the maximal L-energy of a connected graph is equal to n √ 2 asymptotically and provide a class of graphs with this property.

Small Spectral Gap in the Combinatorial Laplacian Implies Hamiltonian

Abstract: We consider the spectral and algorithmic aspects of the problem of finding a Hamil- tonian cycle in a graph. We show that a sufficient condition for a graph being Hamiltonian is that the nontrivial eigenvalues of the combinatorial Laplacian are sufficiently close to the average degree of the graph. An algorithm is given for the problem of finding a Hamiltonian cycle in graphs with bounded spectral gaps which has complexity of order n c ln n .

Random Walks on Digraphs, the Generalized Digraph Laplacian and the Degree of Asymmetry

Abstract: In this paper we extend and generalize the standard random walk theory (or spectral graph theory) on undirected graphs to digraphs In particular, we introduce and define a (normalized) digraph Laplacian matrix, and prove that 1) its Moore-Penrose pseudo-inverse is the (discrete) Green’s function of the digraph Laplacian matrix (as an operator on digraphs), and 2) it is the normalized fundamental matrix of the Markov chain governing random walks on digraphs Using these results, we derive new formula for computing hitting and commute times in terms of the Moore-Penrose pseudo-inverse of the digraph Laplacian, or equivalently, the singular values and vectors of the digraph Laplacian Furthermore, we show that the Cheeger constant defined in [6] is intrinsically a quantity associated with undirected graphs This motivates us to introduce a metric – the largest singular value of \(\Delta:=(\tilde{\cal L}-\tilde{\cal L}^T)/2\) – to quantify and measure the degree of asymmetry in a digraph Using this measure, we establish several new results, such as a tighter bound (than that of Fill’s in [9] and Chung’s in [6]) on the Markov chain mixing rate, and a bound on the second smallest singular value of \(\tilde{\cal L}\)

Network growth and the spectral evolution model

Abstract: We introduce and study the spectral evolution model, which characterizes the growth of large networks in terms of the eigenvalue decomposition of their adjacency matrices: In large networks, changes over time result in a change of a graph's spectrum, leaving the eigenvectors unchanged. We validate this hypothesis for several large social, collaboration, authorship, rating, citation, communication and tagging networks, covering unipartite, bipartite, signed and unsigned graphs. Following these observations, we introduce a link prediction algorithm based on the extrapolation of a network's spectral evolution. This new link prediction method generalizes several common graph kernels that can be expressed as spectral transformations. In contrast to these graph kernels, the spectral extrapolation algorithm does not make assumptions about specific growth patterns beyond the spectral evolution model. We thus show that it performs particularly well for networks with irregular, but spectral, growth patterns.

Least Squares Ranking on Graphs

Abstract: Given a set of alternatives to be ranked, and some pairwise comparison data, ranking is a least squares computation on a graph The vertices are the alternatives, and the edge values comprise the comparison data The basic idea is very simple and old: come up with values on vertices such that their differences match the given edge data Since an exact match will usually be impossible, one settles for matching in a least squares sense This formulation was first described by Leake in 1976 for rankingfootball teams and appears as an example in Professor Gilbert Strang's classic linear algebra textbook If one is willing to look into the residual a little further, then the problem really comes alive, as shown effectively by the remarkable recent paper of Jiang et al With or without this twist, the humble least squares problem on graphs has far-reaching connections with many current areas ofresearch These connections are to theoretical computer science (spectral graph theory, and multilevel methods for graph Laplacian systems); numerical analysis (algebraic multigrid, and finite element exterior calculus); other mathematics (Hodge decomposition, and random clique complexes); and applications (arbitrage, and ranking of sports teams) Not all of these connections are explored in this paper, but many are The underlying ideas are easy to explain, requiring only the four fundamental subspaces from elementary linear algebra One of our aims is to explain these basic ideas and connections, to get researchers in many fields interested in this topic Another aim is to use our numerical experiments for guidance on selecting methods and exposing the need for further development

The Normalized Graph Cut and Cheeger Constant: from Discrete to Continuous

Abstract: Let M be a bounded domain of a Euclidian space with smooth boundary. We relate the Cheeger constant of M and the conductance of a neighborhood graph defined on a random sample from M. By restricting the minimization defining the latter over a particular class of subsets, we obtain consistency (after normalization) as the sample size increases, and show that any minimizing sequence of subsets has a subsequence converging to a Cheeger set of M.

Improved group theoretic method using graph products for the analysis of symmetric-regular structures

Abstract: In this paper, a new combined graph-group method is proposed for eigensolution of special graphs. Symmetric regular graphs are the subject of this study. Many structural models can be viewed as the product of two or three simple graphs. Such models are called regular, and usually have symmetric configurations. The proposed method of this paper performs the symmetry analysis of the entire structure via symmetric properties of its simple generators. Here, a graph is considered as the general model of an arbitrary structure. The Laplacian matrix, as one of the most important matrices associated with a graph, is studied in this paper. The characteristic problem of this matrix is investigated using symmetry analysis via group theory enriched by graph theory. The method is developed and decomposition of the Laplacian matrix of such graphs is studied in a step-by-step manner, based on the proposed method. This method focuses on simple paths which generate large networks, and finds the eigenvalues of the network via analysis of the simple generators. Group theory is the main tool, which is improved using the concept of graph products. As a mechanical application of the method, a benchmark problem of group theory in structural mechanics is studied in this paper. Vibration of cable nets is analyzed and the frequencies of the networks are calculated using the combined graph-group method.

Interactive segmentation of clustered cells via geodesic commute distance and constrained density weighted Nyström method

Abstract: An interactive method is proposed for complex cell segmentation, in particular of clustered cells This article has two main contributions: First, we explore a hybrid combination of the random walk and the geodesic graph based methods for image segmentation and propose the novel concept of geodesic commute distance to classify pixels The computation of geodesic commute distance requires an eigenvector decomposition of the weighted Laplacian matrix of a graph constructed from the image to be segmented Second, by incorporating pairwise constraints from seeds into the algorithm, we present a novel method for eigenvector decomposition, namely a constrained density weighted Nystrom method Both visual and quantitative comparison with other semiautomatic algorithms including Voronoi-based segmentation, grow cut, graph cuts, random walk, and geodesic method are given to evaluate the performance of the proposed method, which is a powerful tool for quantitative analysis of clustered cell images in live cell imaging (C) 2010 International Society for Advancement of Cytometry

Some results on the Laplacian spectrum

Abstract: A graph is called a Laplacian integral graph if the spectrum of its Laplacian matrix consists of integers, and a graph G is said to be determined by its Laplacian spectrum if there does not exist other non-isomorphic graph H such that H and G share the same Laplacian spectrum. In this paper, we obtain a sharp upper bound for the algebraic connectivity of a graph, and identify all the Laplacian integral unicyclic, bicyclic graphs. Moreover, we show that all the Laplacian integral unicyclic, bicyclic graphs are determined by their Laplacian spectra.

Stochastic stability of continuous time consensus protocols

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

Isospectral Graph Transformations, Spectral Equivalence, and Global Stability of Dynamical Networks

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.

Node Clustering in Graphs: An Empirical Study

Abstract: Modeling networks is an active area of research and is used for many applications ranging from bioinformatics to social network analysis. An important operation that is often performed in the course of graph analysis is node clustering. Popular methods for node clustering such as the normalized cut method have their roots in graph partition optimization and spectral graph theory. Recently, there has been increasing interest in modeling graphs probabilistically using stochastic block models and other approaches that extend it. In this paper, we present an empirical study that compares the node clustering performances of state-of-the-art algorithms from both the probabilistic and spectral families on undirected graphs. Our experiments show that no family dominates over the other and that network characteristics play a significant role in determining the best model to use.

Image Segmentation Using Component Tree and Normalized Cut

Abstract: Graph partitioning, or graph cut, has been studied by several authors as a way of image segmenting. In the last years, the Normalized Cut has been widely used in order to implement graph partitioning, based on the graph spectra analysis (eigenvalues and eigenvectors). This area is known as Spectral Graph Theory. This work uses a hierarchical structure in order to represent images, the Component Tree. We provide image segmentation based on Normalized Cut, with image representation based on the Component Tree and on its scale-space analysis. Experimental results present a comparison between other image representations, as pixel grids, including multiscale graph decomposition formulation, and Watershed Transform. As the results show, the proposed approach, applied to different images, presents satisfying image segmentation.

Energy gaps of Hamiltonians from graph Laplacians

Abstract: The Cheeger inequalities give an upper and lower bound on the spectral gap of discrete Laplacians defined on a graph in terms of the geometric characteristics of the graph We generalise this approach and we employ it to determine if a given discrete Hamiltonian with non-positive elements is gapped or not in the thermodynamic limit First, we define the graph that corresponds to such a generic Hamiltonian Then we present a suitable generalisation of the Cheeger inequalities that overcomes scaling deficiencies of the original version By employing simple examples we illustrate how the generalised Cheeger inequalities can successfully identify gapped or gapless phases and we comment on the computational complexity of this approach

A spectral graph theoretic approach to quantification and calibration of collective morphological differences in cell images

Abstract: Motivation: High-throughput image-based assay technologies can rapidly produce a large number of cell images for drug screening, but data analysis is still a major bottleneck that limits their utility. Quantifying a wide variety of morphological differences observed in cell images under different drug influences is still a challenging task because the result can be highly sensitive to sampling and noise. Results: We propose a graph-based approach to cell image analysis. We define graph transition energy to quantify morphological differences between image sets. A spectral graph theoretic regularization is applied to transform the feature space based on training examples of extremely different images to calibrate the quantification. Calibration is essential for a practical quantification method because we need to measure the confidence of the quantification. We applied our method to quantify the degree of partial fragmentation of mitochondria in collections of fluorescent cell images. We show that with transformation, the quantification can be more accurate and sensitive than that without transformation. We also show that our method outperforms competing methods, including neighbourhood component analysis and the multi-variate drug profiling method by Loo et al. We illustrate its utility with a study of Annonaceous acetogenins, a family of compounds with drug potential. Our result reveals that squamocin induces more fragmented mitochondria than muricin A. Availability: Mitochondrial cell images, their corresponding feature sets (SSLF and WSLF) and the source code of our proposed method are available at http://aiia.iis.sinica.edu.tw/. Contact: chunnan@iis.sinica.edu.tw Supplementary information:Supplementary data are available at Bioinformatics online.

The Laplacian eigenvalues of graphs

Abstract: The Laplacian matrix of a graph is the difference between the diagonal matrix of its vertex degrees and its adjacency matrix. The Laplacian eigenvalues of a graph are those eigenvalues of the associated Laplacian matrix. Laplacian eigenvalues are closely related to almost all major invariants of a graph, linking one extremal property to another. There is no question that Laplacian eigenvalues play a central role in our fundamental understanding of graphs. In the past decades, the Laplacian eigenvalues of graphs have received more and more attention, since they have been applied to several fields, such as randomized algorithms, combinatorial optimization problems and machine learning. In this thesis, we focus on the study of the relationships between Laplacian eigenvalues and structural properties of a graph. Various interesting results on the Laplacian spectral radius, the kth largest Laplacian eigenvalue and the algebraic connectivity of a graph are presented. In addition, we investigate some other indices related to the Laplacian eigenvalues of a graph, such as the number of spanning trees, the Laplacian Estrada index, the Laplacian separator and the Laplacian spread. Lastly, we propose some possible directions for further investigating graph Laplacian eigenvalues at the end of this thesis.