Showing papers on "Spectral graph theory published in 2014"
TL;DR: This work shows that in every graph there are at least at least 2 disjoint sets, and shows that the √log k bound is tight, up to constant factors, for the “noisy hypercube” graphs.
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 zeroIt has been conjectured that an analogous characterization holds for higher multiplicities: 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 positively Our result provides a theoretical justification for clustering algorithms that use the bottom k eigenvectors to embed the vertices into Rk, and then apply geometric considerations to the embeddingWe also show that these techniques yield a nearly optimal quantitative connection between the expansion of sets of size a n/k and λk, the kth smallest eigenvalue of the normalized Laplacian, where n is the number of vertices In particular, we show that in every graph there are at least k/2 disjoint sets (one of which will have size at most 2n/k), each having expansion at most O(√λk log k) Louis, Raghavendra, Tetali, and Vempala have independently proved a slightly weaker version of this last result The √log k bound is tight, up to constant factors, for the “noisy hypercube” graphs
235 citations
TL;DR: The present paper reports on the results related to the distance matrix of a graph and its spectral properties.
233 citations
04 May 2014
TL;DR: Using spectral graph theory, a cut-off frequency is established for all bandlimited graph signals that can be perfectly reconstructed from samples on a given subset of nodes that guarantees unique recovery for a signal of given bandwidth.
Abstract: In this paper, we extend the Nyquist-Shannon theory of sampling to signals defined on arbitrary graphs. Using spectral graph theory, we establish a cut-off frequency for all bandlimited graph signals that can be perfectly reconstructed from samples on a given subset of nodes. The result is analogous to the concept of Nyquist frequency in traditional signal processing. We consider practical ways of computing this cut-off and show that it is an improvement over previous results. We also propose a greedy algorithm to search for the smallest possible sampling set that guarantees unique recovery for a signal of given bandwidth. The efficacy of these results is verified through simple examples.
216 citations
TL;DR: An iterative graph-based framework for image restoration based on a new definition of the normalized graph Laplacian, which comprises of outer and inner iterations, where in each outer iteration, the similarity weights are recomputed using the previous estimate and the updated objective function is minimized using inner conjugate gradient iterations.
Abstract: Any image can be represented as a function defined on a weighted graph, in which the underlying structure of the image is encoded in kernel similarity and associated Laplacian matrices. In this paper, we develop an iterative graph-based framework for image restoration based on a new definition of the normalized graph Laplacian. We propose a cost function, which consists of a new data fidelity term and regularization term derived from the specific definition of the normalized graph Laplacian. The normalizing coefficients used in the definition of the Laplacian and associated regularization term are obtained using fast symmetry preserving matrix balancing. This results in some desired spectral properties for the normalized Laplacian such as being symmetric, positive semidefinite, and returning zero vector when applied to a constant image. Our algorithm comprises of outer and inner iterations, where in each outer iteration, the similarity weights are recomputed using the previous estimate and the updated objective function is minimized using inner conjugate gradient iterations. This procedure improves the performance of the algorithm for image deblurring, where we do not have access to a good initial estimate of the underlying image. In addition, the specific form of the cost function allows us to render the spectral analysis for the solutions of the corresponding linear equations. In addition, the proposed approach is general in the sense that we have shown its effectiveness for different restoration problems, including deblurring, denoising, and sharpening. Experimental results verify the effectiveness of the proposed algorithm on both synthetic and real examples.
124 citations
Proceedings Article•
01 Jan 2014TL;DR: Techniques from spectral graph theory are applied to analyze repeated patterns in musical recordings and produce a low-dimensional encoding of repetition structure, and exposes the hierarchical relationships among structural components at differing levels of granularity.
Abstract: Many approaches to analyzing the structure of a musical recording involve detecting sequential patterns within a selfsimilarity matrix derived from time-series features. Such patterns ideally capture repeated sequences, which then form the building blocks of large-scale structure. In this work, techniques from spectral graph theory are applied to analyze repeated patterns in musical recordings. The proposed method produces a low-dimensional encoding of repetition structure, and exposes the hierarchical relationships among structural components at differing levels of granularity. Finally, we demonstrate how to apply the proposed method to the task of music segmentation.
76 citations
TL;DR: This paper shows that the there are topological constraints on the index of the Laplacian matrix related to the dimension of a certain homology group, which gives bounds on the number of positive and negative eigenvalues.
Abstract: Many applied problems can be posed as a dynamical system defined on a network with attractive and repulsive interactions. Examples include synchronization of nonlinear oscillator networks; the behavior of groups, or cliques, in social networks; and the study of optimal convergence for consensus algorithm. It is important to determine the index of a matrix, i.e., the number of positive and negative eigenvalues, and the dimension of the kernel. In this paper we consider the common examples where the matrix takes the form of a signed graph Laplacian. We show that the there are topological constraints on the index of the Laplacian matrix related to the dimension of a certain homology group. When the homology group is trivial, the index of the operator is determined only by the topology of the network and is independent of the strengths of the interactions. In general, these constraints give bounds on the number of positive and negative eigenvalues, with the dimension of the homology group counting the number ...
76 citations
TL;DR: In this article, the authors consider a data set X in which the graph associated with it changes depending on some set of parameters, and they analyze this type of data in terms of the diffusion distance and the corresponding diffusion map.
71 citations
01 Feb 2014
TL;DR: This work defines a class of graphs called path-complete graphs, and shows that any such graph gives rise to a method for proving stability of the switched system, which enables several asymptotically tight hierarchies of semidefinite programming relaxations that unify and generalize many existing techniques.
Abstract: We introduce the framework of path-complete graph Lyapunov functions for ap- proximation of the joint spectral radius. The approach is based on the analysis of the underlying switched system via inequalities imposed among multiple Lyapunov functions associated to a labeled directed graph. Inspired by concepts in automata theory and symbolic dynamics, we define a class of graphs called path-complete graphs, and show that any such graph gives rise to a method for proving stability of the switched system. This enables us to derive several asymptotically tight hierarchies of semidefinite programming relaxations that unify and generalize many existing techniques such as common quadratic, common sum of squares, path-dependent quadratic, and maximum/minimum- of-quadratics Lyapunov functions. We compare the quality of approximation obtained by certain classes of path-complete graphs including a family of dual graphs and all path-complete graphs with two nodes on an alphabet of two matrices. We derive approximation guarantees for several families of path-complete graphs, such as the De Bruijn graphs. This provides worst-case performance bounds for path-dependent quadratic Lyapunov functions and a constructive converse Lyapunov theorem for maximum/minimum-of-quadratics Lyapunov functions.
69 citations
TL;DR: In this paper, the spectral properties of the magnetic discrete Laplacian were studied in detail, and the form-domain, absence of essential spectrum and asymptotic eigenvalue distribution were derived.
62 citations
01 Dec 2014
TL;DR: This work shows how a cut-off frequency for the bandlimited graph signals that can be reconstructed from a given set of samples can be computed exactly, and provides efficient algorithms for finding the subset of nodes of a given size with the largest cut-offs frequency.
Abstract: We consider the problem of sampling from data defined on the nodes of a weighted graph, where the edge weights capture the data correlation structure. As shown recently, using spectral graph theory one can define a cut-off frequency for the bandlimited graph signals that can be reconstructed from a given set of samples (i.e., graph nodes). In this work, we show how this cut-off frequency can be computed exactly. Using this characterization, we provide efficient algorithms for finding the subset of nodes of a given size with the largest cut-off frequency and for finding the smallest subset of nodes with a given cut-off frequency. In addition, we study the performance of random uniform sampling when compared to the centralized optimal sampling provided by the proposed algorithms.
60 citations
TL;DR: The spectral graph theory and molecular dynamics simulation method are used to describe both morphological variation of ion aggregates in high salt solutions and ion effects on water hydrogen-bonding network structure, revealing intricate connectivity and topology of ion aggregate structure that can be classified as either ion cluster or ion network.
Abstract: Graph theory in mathematics and computer science is the study of graphs that are structures with pairwise connections between any objects. Here, the spectral graph theory and molecular dynamics simulation method are used to describe both morphological variation of ion aggregates in high salt solutions and ion effects on water hydrogen-bonding network structure. From the characteristic value analysis of the adjacency matrices that are graph theoretical representations of ion clusters, ion networks, and water H-bond structures, we obtained the ensemble average eigenvalue spectra revealing intricate connectivity and topology of ion aggregate structure that can be classified as either ion cluster or ion network. We further show that there is an isospectral relationship between the eigenvalue spectra of ion networks in high KSCN solutions and those of water H-bonding networks. This reveals the isomorphic relationship between water H-bond structure and ion-ion network structure in KSCN solution. On the other hand, the ion clusters formed in high NaCl solutions are shown to be graph-theoretically and morphologically different from the ion network structures in KSCN solutions. These observations support the bifurcation hypothesis on large ion aggregate growth mechanism via either ion cluster or ion network formation. We thus anticipate that the present spectral graph analyses of ion aggregate structures and their effects on water H-bonding network structures in high salt solutions can provide important information on the specific ion effects on water structures and possibly protein stability resulting from protein-water interactions.
TL;DR: A novel, computationally efficient, approach to graph clustering in the evolutionary context called Incremental Approximate Spectral Clustering (IASC) is introduced and a theoretical bound on the quality of the approximate eigenvectors using perturbation theory is presented.
TL;DR: It is proved that every graph has a spectral sparsifier with a number of edges linear in its number of vertices, and an elementary deterministic polynomial time algorithm is given for constructing $H.
Abstract: A sparsifier of a graph is a sparse graph that approximates it. A spectral sparsifier is one that approximates it spectrally, which means that their Laplacian matrices have similar quadratic forms. We prove that every graph has a spectral sparsifier with a number of edges linear in its number of vertices. In particular, we prove that for every $\epsilon \in (0,1)$ and every undirected, weighted graph $G = (V,E,w)$ on $n$ vertices, there exists a weighted graph $H=(V,F,\tilde{w})$ with at most $\lceil (n-1)/\epsilon^2\rceil$ edges such that for every $x \in \R^{V}$, $ (1-\epsilon)^2 \cdot x^T L_G x \leq x^{T} L_{H} x \leq (1+\epsilon)^2 \cdot x^{T} L_{G} x$, where $L_{G}$ and $L_{H}$ are the Laplacian matrices of $G$ and $H$, respectively. We give an elementary deterministic polynomial time algorithm for constructing $H$. This result is a special case of a significantly more general theorem which provides sparse approximations of general positive semidefinite matrices: given any real matrix $B_{n\times m}$...
TL;DR: In this article, a variation on Cheeger numbers related to the coboundary expanders recently defined by Dotterrer and Kahle is considered, and a Cheeger-type inequality is proved, similar to a result on graphs due to Fan Chung.
04 May 2014
TL;DR: It is proved that LFVC can be related to a monotonic submodular set function that guarantees that greedy node or edge removals come within a factor 1-1/e of the optimal non-greedy batch removal strategy.
Abstract: In this paper, a new centrality called local Fiedler vector centrality (LFVC) is proposed to analyze the connectivity structure of a graph. It is associated with the sensitivity of algebraic connectivity to node or edge removals and features distributed computations via the associated graph Laplacian matrix. We prove that LFVC can be related to a monotonic submodular set function that guarantees that greedy node or edge removals come within a factor 1 1=e of the optimal non-greedy batch removal strategy. Due to the close relationship between graph topology and community structure, we use LFVC to detect deep and overlapping communities on real-world social network datasets. The results offer new insights on community detection by discovering new significant communities and key members in the network. Notably, LFVC is also shown to significantly out- perform other well-known centralities for community detection. approach to detect significant communities and key members in the network, which we refer as deep and overlapping com- munity detection. For instance, when our proposed approach is applied to the network scientist coauthorship dataset we show that a zoologist is correctly identified as an outlier node during the detection process since the authors are mostly physicists, thus leading to revelation of new community structures. Local Fiedler vector centrality (LFVC) is proposed to eval- uate the connectivity structure of a graph based on spectral graph theory (13). LFVC is associated with an upper bound on algebraic connectivity (14) when a subset of nodes or edges are removed from a graph. We show that LFVC relates to a monotonic submodular set function such that greedy node or edge removals can be employed with bounded performance loss relative to the optimal non-greedy batch removal strategy. Moreover, LFVC can be computed in a distributed manner and it is applicable to large-scale network analysis. We apply this method to real-world social network datasets and compare to the modularity method and other well-known centralities.
01 Oct 2014
TL;DR: To reduce the computational cost for large datasets, the Nystr¨om extension method is incorporated which efficiently approximates eigenvectors of the graph Laplacian based on a small portion of the weight matrix.
Abstract: We focus on the multi-class segmentation problem using the piecewise constant Mumford-Shah model in a graph setting. After formulating a graph version of the Mumford-Shah energy, we propose an efficient algorithm called the MBO scheme using threshold dynamics. Theoretical analysis is developed and a Lyapunov functional is proven to decrease as the algorithm proceeds. Furthermore, to reduce the computational cost for large datasets, we incorporate the Nystrom extension method which efficiently approximates eigenvectors of the graph Laplacian based on a small portion of the weight matrix. Finally, we implement the proposed method on the problem of chemical plume detection in hyper-spectral video data.
23 Jun 2014
TL;DR: In this article, the authors consider the approximate weighted graph matching problem and introduce stable and informative first and second order compatibility terms suitable for inclusion into the popular integer quadratic program formulation.
Abstract: In this paper, we consider the approximate weighted graph matching problem and introduce stable and informative first and second order compatibility terms suitable for inclusion into the popular integer quadratic program formulation. Our approach relies on a rigorous analysis of stability of spectral signatures based on the graph Laplacian. In the case of the first order term, we derive an objective function that measures both the stability and informativeness of a given spectral signature. By optimizing this objective, we design new spectral node signatures tuned to a specific graph to be matched. We also introduce the pairwise heat kernel distance as a stable second order compatibility term, we justify its plausibility by showing that in a certain limiting case it converges to the classical adjacency matrix-based second order compatibility function. We have tested our approach on a set of synthetic graphs, the widely-used CMU house sequence, and a set of real images. These experiments show the superior performance of our first and second order compatibility terms as compared with the commonly used ones.
TL;DR: Using spectral graph theory and especially its graph comparison techniques, new methodologies to allocate coupling strengths to guarantee global complete synchronization in complex networks are proposed.
Abstract: Using spectral graph theory and especially its graph comparison techniques, we propose new methodologies to allocate coupling strengths to guarantee global complete synchronization in complex networks. The key step is that all the eigenvalues of the Laplacian matrix associated with a given network can be estimated by utilizing flexibly topological features of the network. The proposed methodologies enable the construction of different coupling-strength combinations in response to different knowledge about subnetworks. Adaptive allocation strategies can be carried out as well using only local network topological information. Besides formal analysis, we use simulation examples to demonstrate how to apply the methodologies to typical complex networks.
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TL;DR: In this paper, a nonlinear spectral graph theory was developed, in which the Laplace operator was replaced by the 1-Laplacian, and the eigenvalue problem is to solve a non-linear system involving a set valued function.
Abstract: We develop a nonlinear spectral graph theory, in which the Laplace operator is replaced by the 1-Laplacian ?$\Delta_1$. The eigenvalue problem is to solve a nonlinear system involving a set valued function. In the study, we investigate the structure of the solutions, the minimax characterization of eigenvalues, the multiplicity theorem, etc. The eigenvalues as well as the eigenvectors are computed for several elementary graphs. The graphic feature of eigenvalues are also studied. In particular, Cheeger's constant, which has only some upper and lower bounds in linear spectral theory, equals to the first non-zero ?$1\Delta_$ eigenvalue for connected graphs.
04 May 2014
TL;DR: This paper proposes a novel graph signal coarsening method with spectral invariance, which means both the spectrum of the graph and the spectrumof the graph signal are approximately kept invariant.
Abstract: Signal processing on graphs is an emerging field that attracts increasing attention. For applications such as multiscale transforms on graphs, it is often necessary to get a coarsened version of graph signal with its underlying graph. However, most of the existing methods use only topology information but no property of graph signals to complete the process. In this paper, we propose a novel graph signal coarsening method with spectral invariance, which means both the spectrum of the graph and the spectrum of the graph signal are approximately kept invariant. The problem is formulated into an optimization problem and is solved by projected subgradient method. Experiment results verify the effectiveness of the coarsening method.
01 Jan 2014
TL;DR: This BSc thesis presents the basic concepts and some basic results from linear algebra and a short introduction to a graph theory, and introduces the concepts of adjacency matrices, eigenvalues and the spectrum of a given graph.
Abstract: In this BSc thesis we deal with matrix graph theory. We are interested primarily in the eigenvalues of the so-called adjacency matrix of a given graph. Because of that, we present the basic concepts and some basic results from linear algebra and a short introduction to a graph theory. We introduce the concepts of adjacency matrices, eigenvalues and the spectrum of a given graph. We investigate how the properties of a given graph reflect on its spectrum. For the well-known families of graphs we calculated their spectra.
TL;DR: Based on spectral graph theory, two new spectral quantities, single and multi-weighted Cheeger constants and corresponding eigenvalue variants, are constructed to direct motions of the defender and the attacker in this dynamic adaptive competition.
Abstract: In this paper, the mobility of network nodes is explored as a new promising approach for jamming defense. To fulfill it, properly designed node motion that can intelligently adapt to the jammer's action is crucial. In our study, anti-jamming mobility control is investigated in the context of the single and multiple commodity flow problems, in the presence of intelligent mobile jammers which can respond to the evasion of legitimate nodes as well. Based on spectral graph theory, two new spectral quantities, single- and multi-weighted Cheeger constants and corresponding eigenvalue variants, are constructed to direct motions of the defender and the attacker in this dynamic adaptive competition. Both analytical and simulation results are presented to justify the effectiveness of the proposed approach. Furthermore, the proposed scheme can also be applied in cognitive radio networks to reconfigure the secondary users in the presence of mobile primary users.
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TL;DR: In this paper, the problem of sampling from data defined on the nodes of a weighted graph, where the edge weights capture the data correlation structure was considered, and algorithms for finding the subset of nodes with the largest cut-off frequency and the smallest subset with a given cut-OFF frequency were proposed.
Abstract: We consider the problem of sampling from data defined on the nodes of a weighted graph, where the edge weights capture the data correlation structure. As shown recently, using spectral graph theory one can define a cut-off frequency for the bandlimited graph signals that can be reconstructed from a given set of samples (i.e., graph nodes). In this work, we show how this cut-off frequency can be computed exactly. Using this characterization, we provide efficient algorithms for finding the subset of nodes of a given size with the largest cut-off frequency and for finding the smallest subset of nodes with a given cut-off frequency. In addition, we study the performance of random uniform sampling when compared to the centralized optimal sampling provided by the proposed algorithms.
01 Dec 2014
TL;DR: A theory relating the rank of the minimum-rank solution of the LMI problem to the sparsity of its underlying graph is developed and two graph-theoretic convex programs are proposed to obtain a low- rank solution.
Abstract: This paper is concerned with the problem of finding a low-rank solution of an arbitrary sparse linear matrix inequality (LMI). To this end, we map the sparsity of the LMI problem into a graph. We develop a theory relating the rank of the minimum-rank solution of the LMI problem to the sparsity of its underlying graph. Furthermore, we propose two graph-theoretic convex programs to obtain a low-rank solution. The first convex optimization needs a tree decomposition of the sparsity graph. The second one does not rely on any computationally-expensive graph analysis and is always polynomial-time solvable. The results of this work can be readily applied to three separate problems of minimumrank matrix completion, conic relaxation for polynomial optimization, and affine rank minimization. The results are finally illustrated on two applications of optimal distributed control and nonlinear optimization for electrical networks.
18 Dec 2014
TL;DR: In this paper, the authors study the consensus problem of third-order multi-agent systems for the case of undirected graphs and show the relationship between scaling strengths and the eigenvalues of the involved Laplacian matrix.
Abstract: This paper studies the consensus problem of third-order multi-agent systems for the case of undirected graph. Necessary and sufficient conditions for third-order consensus have been established under three different protocols. Unlike most of existing papers, we here focus on illustrating the relationship between the scaling strengths and the eigenvalues of the involved Laplacian matrix, which guarantees the third-order consensus for three different protocols.
TL;DR: In this paper, the Laplacian-energy-like invariant of a connected graph G is defined as LEL = LEL (G ) = ∑ i = 1 n - 1 μ i.
01 Dec 2014
TL;DR: A smoothing method for the spatial filter using spectral graph theory based on an assumption that the electrodes installed in nearby locations observe the electrical activities of the same source to enhance robustness against low SNR and small samples.
Abstract: Spatial filtering is useful for extracting features from multichannel EEG signals. In order to enhance robustness of the spatial filter against low SNR and small samples, we propose a smoothing method for the spatial filter using spectral graph theory. This method is based on an assumption that the electrodes installed in nearby locations observe the electrical activities of the same source. Therefore the spatial filter's coefficients corresponding to the nearby electrodes are supposed to be taken similar values, that is, the coefficients should be spatially smooth. To introduce the smoothness, we define a graph whose edge weights represent the physical distances between the electrodes. The spatial filter spatially smoothed is found out in the subspace that is spanned by the smooth basis of the graph Fourier transform. We evaluate the method with artificial signals and a dataset of motor imagery brain computer interface. The smoothness of the spatial filter given by the method provides robustness of the spatial filter in the condition that the small amount of the samples is available.
TL;DR: In this article, the authors studied the stability characteristics of the normalised Laplacian spectrum associated with the interactive growth mechanism for the increasing of nodes in the generation of large-scale autonomous systems (AS) graphs.
Abstract: In terms of spectral graph theory, the graph spectrums represent more accurate structural properties and robust characteristics of the network topologies. In this study, the authors study the stability characteristics of the normalised Laplacian spectrum associated with the interactive growth mechanism for the increasing of nodes in the generation of large-scale autonomous systems (AS) graphs. With consecutive snapshots of the AS-level Internet topologies, the authors obtain the stable models of the spectrum. In addition, the authors investigate the stability conditions of the spectrum based on the interactive growth mechanism and the comparisons with other metrics and study how to influence the spectrum by the inputs of the interactive growth mechanism.
04 May 2014
TL;DR: The spectral properties of small-world random graphs were studied in this article, focusing on the spectrum of the normalized graph Laplacian, which influences the extent to which a signal supported on the vertices of the graph can be simultaneously localized on the graph and in the spectral domain.
Abstract: We study properties of the family of small-world random graphs introduced in Watts & Strogatz (1998), focusing on the spectrum of the normalized graph Laplacian. This spectrum influences the extent to which a signal supported on the vertices of the graph can be simultaneously localized on the graph and in the spectral domain (the surrogate of the frequency domain for signals supported on a graph). This characterization has implications for inferring or interpolating functions supported on such graphs when observations are only available at a subset of nodes.
TL;DR: In this paper, the authors extend several algebraic graph analysis methods to bipartite networks and show methods for clustering, visualization and link prediction, and introduce new algebraic methods for measuring the bipartivity in near-bipartite graphs.
Abstract: In this article, we extend several algebraic graph analysis methods to bipartite networks. In various areas of science, engineering and commerce, many types of information can be represented as networks, and thus the discipline of network analysis plays an important role in these domains. A powerful and widespread class of network analysis methods is based on algebraic graph theory, i.e., representing graphs as square adjacency matrices. However, many networks are of a very specific form that clashes with that representation: They are bipartite. That is, they consist of two node types, with each edge connecting a node of one type with a node of the other type. Examples of bipartite networks (also called \emph{two-mode networks}) are persons and the social groups they belong to, musical artists and the musical genres they play, and text documents and the words they contain. In fact, any type of feature that can be represented by a categorical variable can be interpreted as a bipartite network. Although bipartite networks are widespread, most literature in the area of network analysis focuses on unipartite networks, i.e., those networks with only a single type of node. The purpose of this article is to extend a selection of important algebraic network analysis methods to bipartite networks, showing that many methods from algebraic graph theory can be applied to bipartite networks with only minor modifications. We show methods for clustering, visualization and link prediction. Additionally, we introduce new algebraic methods for measuring the bipartivity in near-bipartite graphs.