Showing papers on "Spectral graph theory published in 2018"

RGCNN: Regularized Graph CNN for Point Cloud Segmentation

Abstract: Point cloud, an efficient 3D object representation, has become popular with the development of depth sensing and 3D laser scanning techniques. It has attracted attention in various applications such as 3D tele-presence, navigation for unmanned vehicles and heritage reconstruction. The understanding of point clouds, such as point cloud segmentation, is crucial in exploiting the informative value of point clouds for such applications. Due to the irregularity of the data format, previous deep learning works often convert point clouds to regular 3D voxel grids or collections of images before feeding them into neural networks, which leads to voluminous data and quantization artifacts. In this paper, we instead propose a regularized graph convolutional neural network (RGCNN) that directly consumes point clouds. Leveraging on spectral graph theory, we treat features of points in a point cloud as signals on graph, and define the convolution over graph by Chebyshev polynomial approximation. In particular, we update the graph Laplacian matrix that describes the connectivity of features in each layer according to the corresponding learned features, which adaptively captures the structure of dynamic graphs. Further, we deploy a graph-signal smoothness prior in the loss function, thus regularizing the learning process. Experimental results on the ShapeNet part dataset show that the proposed approach significantly reduces the computational complexity while achieving competitive performance with the state of the art. Also, experiments show RGCNN is much more robust to both noise and point cloud density in comparison with other methods. We further apply RGCNN to point cloud classification and achieve competitive results on ModelNet40 dataset.

RGCNN: Regularized Graph CNN for Point Cloud Segmentation

Abstract: Point cloud, an efficient 3D object representation, has become popular with the development of depth sensing and 3D laser scanning techniques. It has attracted attention in various applications such as 3D tele-presence, navigation for unmanned vehicles and heritage reconstruction. The understanding of point clouds, such as point cloud segmentation, is crucial in exploiting the informative value of point clouds for such applications. Due to the irregularity of the data format, previous deep learning works often convert point clouds to regular 3D voxel grids or collections of images before feeding them into neural networks, which leads to voluminous data and quantization artifacts. In this paper, we instead propose a regularized graph convolutional neural network (RGCNN) that directly consumes point clouds. Leveraging on spectral graph theory, we treat features of points in a point cloud as signals on graph, and define the convolution over graph by Chebyshev polynomial approximation. In particular, we update the graph Laplacian matrix that describes the connectivity of features in each layer according to the corresponding learned features, which adaptively captures the structure of dynamic graphs. Further, we deploy a graph-signal smoothness prior in the loss function, thus regularizing the learning process. Experimental results on the ShapeNet part dataset show that the proposed approach significantly reduces the computational complexity while achieving competitive performance with the state of the art. Also, experiments show RGCNN is much more robust to both noise and point cloud density in comparison with other methods. We further apply RGCNN to point cloud classification and achieve competitive results on ModelNet40 dataset.

Distributed Signal Processing via Chebyshev Polynomial Approximation

Abstract: Unions of graph multiplier operators 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. The proposed method features approximations of the graph multipliers by shifted Chebyshev polynomials, whose recurrence relations make them readily amenable to distributed computation. We demonstrate how the proposed method can be applied to distributed processing tasks such as smoothing, denoising, inverse filtering, and semi-supervised classification, and show that the communication requirements of the method scale gracefully with the size of the network.

Rotation Averaging and Strong Duality

Abstract: In this paper we explore the role of duality principles within the problem of rotation averaging, a fundamental task in a wide range of computer vision applications. In its conventional form, rotation averaging is stated as a minimization over multiple rotation constraints. As these constraints are non-convex, this problem is generally considered challenging to solve globally. We show how to circumvent this difficulty through the use of Lagrangian duality. While such an approach is well-known it is normally not guaranteed to provide a tight relaxation. Based on spectral graph theory, we analytically prove that in many cases there is no duality gap unless the noise levels are severe. This allows us to obtain certifiably global solutions to a class of important non-convex problems in polynomial time. We also propose an efficient, scalable algorithm that outperforms general purpose numerical solvers and is able to handle the large problem instances commonly occurring in structure from motion settings. The potential of this proposed method is demonstrated on a number of different problems, consisting of both synthetic and real-world data.

Identifying network structure similarity using spectral graph theory

Abstract: Most real networks are too large or they are not available for real time analysis Therefore, in practice, decisions are made based on partial information about the ground truth network It is of great interest to have metrics to determine if an inferred network (the partial information network) is similar to the ground truth In this paper we develop a test for similarity between the inferred and the true network Our research utilizes a network visualization tool, which systematically discovers a network, producing a sequence of snapshots of the network We introduce and test our metric on the consecutive snapshots of a network, and against the ground truth To test the scalability of our metric we use a random matrix theory approach while discovering Erdos-Renyi graphs This scaling analysis allows us to make predictions about the performance of the discovery process

Approximating the Spectrum of a Graph

Abstract: The spectrum of a network or graph $G=(V,E)$ with adjacency matrix A , consists of the eigenvalues of the normalized Laplacian $L= I - D^-1/2 A D^-1/2 $. This set of eigenvalues encapsulates many aspects of the structure of the graph, including the extent to which the graph posses community structures at multiple scales. We study the problem of approximating the spectrum, $lambda = (lambda_1,\dots,lambda_|V| )$, of G in the regime where the graph is too large to explicitly calculate the spectrum. We present a sublinear time algorithm that, given the ability to query a random node in the graph and select a random neighbor of a given node, computes a succinct representation of an approximation $\widetilde lambda = (\widetilde lambda_1,\dots,\widetilde lambda_|V| )$, such that $\|\widetilde lambda - lambda\|_1 le e |V|$. Our algorithm has query complexity and running time $exp(O(1/\eps))$, which is independent of the size of the graph, $|V|$. We demonstrate the practical viability of our algorithm on synthetically generated graphs, and on 15 different real-world graphs from the Stanford Large Network Dataset Collection, including social networks, academic collaboration graphs, and road networks. For the smallest of these graphs, we are able to validate the accuracy of our algorithm by explicitly calculating the true spectrum; for the larger graphs, such a calculation is computationally prohibitive. The spectra of these real-world networks reveal insights into the structural similarities and differences between them, illustrating the potential value of our algorithm for efficiently approximating the spectrum of large large networks.

A Topological Investigation of Power Flow

Abstract: This paper combines the fundamentals of an electrical grid, such as flow allocation according to Kirchhoff's laws and the effect of transmission line reactances with spectral graph theory, and expresses the linearized power flow behaviour in slack-bus independent weighted graph matrices to assess the relation between the topological structure and the physical behaviour of a power grid. Based on the pseudoinverse of the weighted network Laplacian, the paper further analytically calculates the effective resistance (Thevenin) matrix and the sensitivities of active power flows to the changes in network topology by means of transmission line removal and addition. Numerical results for the IEEE 118-bus power system are demonstrated to identify the critical components to cascading failures, node isolation, and Braess’ paradox in a power grid.

Graph Laplacian Spectrum and Primary Frequency Regulation

Abstract: We present a framework based on spectral graph theory that captures the interplay among network topology, system inertia, and generator and load damping in determining the overall grid behavior and performance. Specifically, we show that the impact of network topology on a power system can be quantified through the network Laplacian eigenvalues, and such eigenvalues determine the grid robustness against low frequency disturbances. Moreover, we can explicitly decompose the frequency signal along scaled Laplacian eigenvectors when damping-inertia ratios are uniform across buses. The insight revealed by this framework partially explains why load-side participation in frequency regulation not only makes the system respond faster, but also helps lower the system nadir after a disturbance. Finally, by presenting a new controller specifically tailored to suppress high frequency disturbances, we demonstrate that our results can provide useful guidelines in the controller design for load-side primary frequency regulation. This improved controller is simulated on the IEEE 39-bus New England interconnection system to illustrate its robustness against high frequency oscillations compared to both the conventional droop control and a recent controller design.

A Brief Introduction to Spectral Graph Theory

Abstract: Spectral graph theory starts by associating matrices to graphs, notably, the adjacency matrix and the laplacian matrix. The general theme is then, firstly, to compute or estimate the eigenvalues of such matrices, and secondly, to relate the eigenvalues to structural properties of graphs. As it turns out, the spectral perspective is a powerful tool. Some of its loveliest applications concern facts that are, in principle, purely graph-theoretic or combinatorial. To give just one example, spectral ideas are a key ingredient in the proof of the so-called Friendship Theorem: if, in a group of people, any two persons have exactly one common friend, then there is a person who is everybody’s friend. This text is an introduction to spectral graph theory, but it could also be seen as an invitation to algebraic graph theory. On the one hand, there is, of course, the linear algebra that underlies the spectral ideas in graph theory. On the other hand, most of our examples are graphs of algebraic origin. The two recurring sources are Cayley graphs of groups, and graphs built out of finite fields. In the study of such graphs, some further algebraic ingredients (e.g., characters) naturally come up. The table of contents gives, as it should, a good glimpse of where is this text going. Very broadly, the first half is devoted to graphs, finite fields, and how they come together. This part is meant as an appealing and meaningful motivation. It provides a context that frames and fuels much of the second, spectral, half. Most sections have one or two exercises. Their position within the text is a hint. The exercises are optional, in the sense that virtually nothing in the main body depends on them. But the exercises are often of the non-trivial variety, and they should enhance the text in an interesting way. The hope is that the reader will enjoy them. We assume a basic familiarity with linear algebra, finite fields, and groups, but not necessarily with graph theory. This, again, betrays our algebraic perspective. This text is based on a course I taught in Göttingen, in the Fall of 2015. I would like to thank Jerome Baum for his help with some of the drawings. The present version is preliminary, and comments are welcome (email: bogdan.nica@gmail.com).

Introduction to Chemical Graph Theory

Abstract: Introduction to Chemical Graph Theory is a concise introduction to the main topics and techniques in chemical graph theory, specifically the theory of topological indices. These include distance-based, degree-based, and counting-based indices. The book covers some of the most commonly used mathematical approaches in the subject. It is also written with the knowledge that chemical graph theory has many connections to different branches of graph theory (such as extremal graph theory, spectral graph theory). The authors wrote the book in an appealing way that attracts people to chemical graph theory. In doing so, the book is an excellent playground and general reference text on the subject, especially for young mathematicians with a special interest in graph theory. Key Features: A concise introduction to topological indices of graph theory Appealing to specialists and non-specialists alike Provides many techniques from current research About the Authors: Stephan Wagner grew up in Graz (Austria), where he also received his PhD from Graz University of Technology in 2006. Shortly afterwards, he moved to South Africa, where he started his career at Stellenbosch University as a lecturer in January 2007. His research interests lie mostly in combinatorics and related areas, including connections to other scientific fields such as physics, chemistry and computer science. Hua Wang received his PhD from University of South Carolina in 2005. He held a Visiting Research Assistant Professor position at University of Florida before joining Georgia Southern University in 2008. His research interests include combinatorics and graph theory, elementary number theory, and related problems

Estimating the Spectral Density of Large Implicit Matrices

Abstract: Many important problems are characterized by the eigenvalues of a large matrix For example, the difficulty of many optimization problems, such as those arising from the fitting of large models in statistics and machine learning, can be investigated via the spectrum of the Hessian of the empirical loss function Network data can be understood via the eigenstructure of a graph Laplacian matrix using spectral graph theory Quantum simulations and other many-body problems are often characterized via the eigenvalues of the solution space, as are various dynamic systems However, naive eigenvalue estimation is computationally expensive even when the matrix can be represented; in many of these situations the matrix is so large as to only be available implicitly via products with vectors Even worse, one may only have noisy estimates of such matrix vector products In this work, we combine several different techniques for randomized estimation and show that it is possible to construct unbiased estimators to answer a broad class of questions about the spectra of such implicit matrices, even in the presence of noise We validate these methods on large-scale problems in which graph theory and random matrix theory provide ground truth

Graph Coarsening with Preserved Spectral Properties

Abstract: Large-scale graphs are widely used to represent object relationships in many real world applications. The occurrence of large-scale graphs presents significant computational challenges to process, analyze, and extract information. Graph coarsening techniques are commonly used to reduce the computational load while attempting to maintain the basic structural properties of the original graph. As there is no consensus on the specific graph properties preserved by coarse graphs, how to measure the differences between original and coarse graphs remains a key challenge. In this work, we introduce a new perspective regarding the graph coarsening based on concepts from spectral graph theory. We propose and justify new distance functions that characterize the differences between original and coarse graphs. We show that the proposed spectral distance naturally captures the structural differences in the graph coarsening process. In addition, we provide efficient graph coarsening algorithms to generate graphs which provably preserve the spectral properties from original graphs. Experiments show that our proposed algorithms consistently achieve better results compared to previous graph coarsening methods on graph classification and block recovery tasks.

Similarity-aware spectral sparsification by edge filtering

Abstract: In recent years, spectral graph sparsification techniques that can compute ultra-sparse graph proxies have been extensively studied for accelerating various numerical and graph-related applications. Prior nearly-linear-time spectral sparsification methods first extract low-stretch spanning tree from the original graph to form the backbone of the sparsifier, and then recover small portions of spectrally-critical off-tree edges to the spanning tree to significantly improve the approximation quality. However, it is not clear how many off-tree edges should be recovered for achieving a desired spectral similarity level within the sparsifier. Motivated by recent graph signal processing techniques, this paper proposes a similarity-aware spectral graph sparsification framework that leverages efficient spectral off-tree edge embedding and filtering schemes to construct spectral sparsifiers with guaranteed spectral similarity (relative condition number) level. An iterative graph densification scheme is introduced to facilitate efficient and effective filtering of off-tree edges for highly ill-conditioned problems. The proposed method has been validated using various kinds of graphs obtained from public domain sparse matrix collections relevant to VLSI CAD, finite element analysis, as well as social and data networks frequently studied in many machine learning and data mining applications.

Spectral dimension reduction of complex dynamical networks

Abstract: Dynamical networks are powerful tools for modeling a broad range of complex systems, including financial markets, brains, and ecosystems They encode how the basic elements (nodes) of these systems interact altogether (via links) and evolve (nodes' dynamics) Despite substantial progress, little is known about why some subtle changes in the network structure, at the so-called critical points, can provoke drastic shifts in its dynamics We tackle this challenging problem by introducing a method that reduces any network to a simplified low-dimensional version It can then be used to describe the collective dynamics of the original system This dimension reduction method relies on spectral graph theory and, more specifically, on the dominant eigenvalues and eigenvectors of the network adjacency matrix Contrary to previous approaches, our method is able to predict the multiple activation of modular networks as well as the critical points of random networks with arbitrary degree distributions Our results are of both fundamental and practical interest, as they offer a novel framework to relate the structure of networks to their dynamics and to study the resilience of complex systems

Functional Control of Network Dynamics Using Designed Laplacian Spectra

Abstract: Complex real-world phenomena across a wide range of scales, from aviation and Internet traffic to signal propagation in electronic and gene regulatory circuits, can be efficiently described through dynamic network models. In many such systems, the spectrum of the underlying graph Laplacian plays a key role in controlling the matter or information flow. Spectral graph theory has traditionally prioritized analyzing unweighted networks with specified adjacency properties. Here, we introduce a complementary framework, providing a mathematically rigorous weighted graph construction that exactly realizes any desired spectrum. We illustrate the broad applicability of this approach by showing how designer spectra can be used to control the dynamics of various archetypal physical systems. Specifically, we demonstrate that a strategically placed gap induces generalized chimera states in Kuramoto-type oscillator networks, tunes or suppresses pattern formation in a generic Swift-Hohenberg model, and leads to persistent localization in a discrete Gross-Pitaevskii quantum network. Our approach can be generalized to design continuous band gaps through periodic extensions of finite networks.

Graph Laplacian Spectrum and Primary Frequency Regulation

Abstract: We present a framework based on spectral graph theory that captures the interplay among network topology, system inertia, and generator and load damping in determining the overall grid behavior and performance. Specifically, we show that the impact of network topology on a power system can be quantified through the network Laplacian eigenvalues, and such eigenvalues determine the grid robustness against low frequency disturbances. Moreover, we can explicitly decompose the frequency signal along scaled Laplacian eigenvectors when damping-inertia ratios are uniform across buses. The insight revealed by this framework partially explains why load-side participation in frequency regulation not only makes the system respond faster, but also helps lower the system nadir after a disturbance. Finally, by presenting a new controller specifically tailored to suppress high frequency disturbances, we demonstrate that our results can provide useful guidelines in the controller design for load-side primary frequency regulation. This improved controller is simulated on the IEEE 39-bus New England interconnection system to illustrate its robustness against high frequency oscillations compared to both the conventional droop control and a recent controller design.

Network simulation tools and spectral graph theory in teaching computer network

Abstract: In this work, we described software surrounding's for protocol analysis and learning about computer networks by using spectral graph theory. Software surrounding's consists of the computer network simulators and software surrounding‘s for spectral graph analysis which allows students get to know Internetworking technologies. System made in Java programming language, allows editing of the arbitrary topology computer network with switches, hubs, routers, and work stations. Configuration of elements of the network is done by using standard Windows or line interface. The visual tracking of IP packages, and also the content of the IP packages and frames is allowed. Basic component of this system is software package for generating corresponding graph and calculating the basic parameters from spectral graph theory and analysis of computer network of the given topology and type of protocol.

Spectral Algorithms for Temporal Graph Cuts

Abstract: The sparsest cut problem consists of identifying a small set of edges that breaks the graph into balanced sets of vertices. The normalized cut problem balances the total degree, instead of the size, of the resulting sets. Applications of graph cuts include community detection and computer vision. However, cut problems were originally proposed for static graphs, an assumption that does not hold in many modern applications where graphs are highly dynamic. In this paper, we introduce sparsest and normalized cuts in temporal graphs, which generalize their standard definitions by enforcing the smoothness of cuts over time. We propose novel formulations and algorithms for computing temporal cuts using spectral graph theory, divide-and-conquer and low-rank matrix approximation. Furthermore, we extend temporal cuts to dynamic graph signals, where vertices have attributes. Experiments show that our solutions are accurate and scalable, enabling the discovery of dynamic communities and the analysis of dynamic graph processes.

Discrete Harmonic Analysis: Representations, Number Theory, Expanders, and the Fourier Transform

Abstract: This self-contained book introduces readers to discrete harmonic analysis with an emphasis on the Discrete Fourier Transform and the Fast Fourier Transform on finite groups and finite fields, as well as their noncommutative versions. It also features applications to number theory, graph theory, and representation theory of finite groups. Beginning with elementary material on algebra and number theory, the book then delves into advanced topics from the frontiers of current research, including spectral analysis of the DFT, spectral graph theory and expanders, representation theory of finite groups and multiplicity-free triples, Tao's uncertainty principle for cyclic groups, harmonic analysis on GL(2,Fq), and applications of the Heisenberg group to DFT and FFT. With numerous examples, figures, and over 160 exercises to aid understanding, this book will be a valuable reference for graduate students and researchers in mathematics, engineering, and computer science.

The spectral determinations of the connected multicone graphs $ K_w\bigtriangledown mP_{17} $ and $ K_w\bigtriangledown mS $

Abstract: Finding and discovering any class of graphs which are determined by their spectra is always an important and interesting problem in the spectral graph theory. The main aim of this study is to characterize two classes of multicone graphs which are determined by both their adjacency and Laplacian spectra. A multicone graph is defined to be the join of a clique and a regular graph. Let Kw denote a complete graph on w vertices, and let m be a positive integer number. In A.Z.Abdian (2016) it has been shown that multicone graphs Kw ▽ P17 and Kw ▽ S are determined by both their adjacency and Laplacian spectra, where P17 and S denote the Paley graph of order 17 and the Schlafli graph, respectively. In this paper, we generalize these results and we prove that multicone graphs Kw ▽mP17 and Kw▽mS are determined by their adjacency spectra as well as their Laplacian spectra.

Network Summarization with Preserved Spectral Properties.

Abstract: Large-scale networks are widely used to represent object relationships in many real world applications. The occurrence of large-scale networks presents significant computational challenges to process, analyze, and extract information from such networks. Network summarization techniques are commonly used to reduce the computational load while attempting to maintain the basic structural properties of the original network. Previous works have primarily focused on some type of network partitioning strategies with application-dependent regularizations, most often resulting in strongly connected clusters. In this paper, we introduce a novel perspective regarding the network summarization problem based on concepts from spectral graph theory. We propose a new distance measurement to characterize the spectral differences between the original and coarsened networks. We rigorously justify the spectral distance with the interlacing theorem as well the results from the stochastic block model. We provide an efficient algorithm to generate the coarsened networks that maximally preserves the spectral properties of the original network. Our proposed network summarization framework allows the flexibility to generate a set of coarsened networks with significantly different structures preserved from different aspects of the original network, which distinguishes our work from others. We conduct extensive experimental tests on a variety of large-scale networks, both from real-world applications and the random graph model. We show that our proposed algorithms consistently perform better results in terms of the spectral measurements and running time compared to previous network summarization algorithms.

A Novel Method for Sampling Bandlimited Graph Signals

Abstract: In this paper we propose a novel vertex based sampling method for k-bandlimited signals lying on arbitrary graphs, that has a reasonable computational complexity and results in low reconstruction error. Our goal is to find the smallest set of vertices that can guarantee a perfect reconstruction of any k-bandlimited signal on any connected graph. We propose to iteratively search for the vertices that yield the minimum reconstruction error, by minimizing the maximum eigenvalue of the error covariance matrix using a linear solver. We compare the performance of our method with state-of-the-art sampling strategies and random sampling on graphs. Experimental results show that our method successfully computes the smallest sample sets on arbitrary graphs without any parameter tuning. It provides a small reconstruction error, and is robust to noise.

Nonparametric High-dimensional K-sample Comparison

Abstract: High-dimensional k-sample comparison is a common applied problem. We construct a class of easy-to-implement nonparametric distribution-free tests based on new tools and unexplored connections with spectral graph theory. The test is shown to possess various desirable properties along with a characteristic exploratory flavor that has practical consequences. The numerical examples show that our method works surprisingly well under a broad range of realistic situations.

Construction of cospectral graphs

Abstract: Construction of non-isomorphic cospectral graphs is a nontrivial problem in spectral graph theory specially for large graphs. In this paper, we establish that graph theoretical partial transpose of a graph is a potential tool to create non-isomorphic cospectral graphs by considering a graph as a clustered graph.

Toward a Spectral Theory of Cellular Sheaves

Abstract: This paper outlines a program in what one might call spectral sheaf theory --- an extension of spectral graph theory to cellular sheaves. By lifting the combinatorial graph Laplacian to the Hodge Laplacian on a cellular sheaf of vector spaces over a regular cell complex, one can relate spectral data to the sheaf cohomology and cell structure in a manner reminiscent of spectral graph theory. This work gives an exploratory introduction, and includes results on eigenvalue interlacing, sparsification, effective resistance, and sheaf approximation. These results and subsequent applications are prefaced by an introduction to cellular sheaves and Laplacians.