Showing papers on "Spectral graph theory published in 2021"

Rotation Averaging with the Chordal Distance: Global Minimizers 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 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 by a large margin and compares favourably to current state-of-the-art. Further, our approach is able to handle the large problem instances commonly occurring in structure from motion settings and it is trivially parallelizable. Experiments are presented for a number of different instances of both synthetic and real-world data.

HERMES: Persistent spectral graph software

Abstract: Persistent homology (PH) is one of the most popular tools in topological data analysis (TDA), while graph theory has had a significant impact on data science. Our earlier work introduced the persistent spectral graph (PSG) theory as a unified multiscale paradigm to encompass TDA and geometric analysis. In PSG theory, families of persistent Laplacian matrices (PLMs) corresponding to various topological dimensions are constructed via a filtration to sample a given dataset at multiple scales. The harmonic spectra from the null spaces of PLMs offer the same topological invariants, namely persistent Betti numbers, at various dimensions as those provided by PH, while the non-harmonic spectra of PLMs give rise to additional geometric analysis of the shape of the data. In this work, we develop an open-source software package, called highly efficient robust multidimensional evolutionary spectra (HERMES), to enable broad applications of PSGs in science, engineering, and technology. To ensure the reliability and robustness of HERMES, we have validated the software with simple geometric shapes and complex datasets from three-dimensional (3D) protein structures. We found that the smallest non-zero eigenvalues are very sensitive to data abnormality.

Partition of Unity Methods for Signal Processing on Graphs

Abstract: Partition of unity methods (PUMs) on graphs are simple and highly adaptive auxiliary tools for graph signal processing. Based on a greedy-type metric clustering and augmentation scheme, we show how a partition of unity can be generated in an efficient way on graphs. We investigate how PUMs can be combined with a local graph basis function (GBF) approximation method in order to obtain low-cost global interpolation or classification schemes. From a theoretical point of view, we study necessary prerequisites for the partition of unity such that global error estimates of the PUM follow from corresponding local ones. Finally, properties of the PUM as cost-efficiency and approximation accuracy are investigated numerically.

Spectral Graph Theory Based Resource Allocation for IRS-Assisted Multi-Hop Edge Computing

Abstract: The performance of mobile edge computing (MEC) depends critically on the quality of the wireless channels. From this viewpoint, the recently advocated intelligent reflecting surface (IRS) technique that can proactively reconfigure wireless channels is anticipated to bring unprecedented performance gain to MEC. In this paper, the problem of network throughput optimization of an IRS-assisted multi-hop MEC network is investigated, in which the phase-shifts of the IRS and the resource allocation of the relays need to be jointly optimized. However, due to the coupling among the transmission links of different hops caused by the utilization of the IRS and the complicated multi-hop network topology, it is difficult to solve the considered problem by directly applying existing optimization techniques. Fortunately, by exploiting the underlying structure of the network topology and spectral graph theory, it is shown that the network throughput can be well approximated by the second smallest eigenvalue of the network Laplacian matrix. This key finding allows us to develop an effective iterative algorithm for solving the considered problem. Numerical simulations are performed to corroborate the effectiveness of the proposed scheme.

Towards Scalable Spectral Embedding and Data Visualization via Spectral Coarsening

Abstract: This paper proposes a scalable multilevel framework for the spectral embedding of large undirected graphs. The proposed method first computes much smaller yet sparse graphs while preserving the key spectral (structural) properties of the original graph, by exploiting a nearly-linear time spectral graph coarsening approach. Then, the resultant spectrally-coarsened graphs are leveraged for the development of much faster algorithms for multilevel spectral graph embedding (clustering) as well as visualization of large data sets. We conducted extensive experiments using a variety of large graphs and datasets and obtained very promising results. For instance, we are able to coarsen the "coPapersCiteseer" graph with 0.43 million nodes and 16 million edges into a much smaller graph with only 13K (32X fewer) nodes and 17K (950X fewer) edges in about 16 seconds; the spectrally-coarsened graphs allow us to achieve up to 1,100X speedup for multilevel spectral graph embedding (clustering) and up to 60X speedup for t-SNE visualization of large data sets.

Quantum walks defined by digraphs and generalized Hermitian adjacency matrices

Abstract: We propose a quantum walk defined by digraphs (mixed graphs). This is like Grover walk that is perturbed by a certain complex-valued function defined by digraphs. The discriminant of this quantum walk is a matrix that is a certain normalization of generalized Hermitian adjacency matrices. Furthermore, we give definitions of the positive and negative supports of the transfer matrix and exhibit explicit formulas of supports of their square. Also, we provide tables on the identification of digraphs by their eigenvalues.

Hyperspectral Image Classification with Localized Graph Convolutional Filtering

Abstract: The nascent graph representation learning has shown superiority for resolving graph data. Compared to conventional convolutional neural networks, graph-based deep learning has the advantages of illustrating class boundaries and modeling feature relationships. Faced with hyperspectral image (HSI) classification, the priority problem might be how to convert hyperspectral data into irregular domains from regular grids. In this regard, we present a novel method that performs the localized graph convolutional filtering on HSIs based on spectral graph theory. First, we conducted principal component analysis (PCA) preprocessing to create localized hyperspectral data cubes with unsupervised feature reduction. These feature cubes combined with localized adjacent matrices were fed into the popular graph convolution network in a standard supervised learning paradigm. Finally, we succeeded in analyzing diversified land covers by considering local graph structure with graph convolutional filtering. Experiments on real hyperspectral datasets demonstrated that the presented method offers promising classification performance compared with other popular competitors.

Shapes of Uncertainty in Spectral Graph Theory

Abstract: We present a flexible framework for uncertainty principles in spectral graph theory. In this framework, general filter functions modeling the spatial and spectral localization of a graph signal can be incorporated. It merges several existing uncertainty relations on graphs, among others the Landau-Pollak principle describing the joint admissibility region of two projection operators, and uncertainty relations based on spectral and spatial spreads. Using theoretical and computational aspects of the numerical range of matrices, we are able to characterize and illustrate the shapes of the uncertainty curves and to study the space-frequency localization of signals inside the admissibility regions.

Bridging the Gap between von Neumann Graph Entropy and Structural Information: Theory and Applications

Abstract: The von Neumann graph entropy (VNGE) is a measure of graph complexity based on the Laplacian spectrum. It has recently found applications in various learning tasks driven by networked data. However, it is computationally demanding and hard to interpret using simple structural patterns. Due to the close relation between Lapalcian spectrum and degree sequence, we conjecture that the structural information, defined as the Shannon entropy of the normalized degree sequence, might approximate VNGE well. In this work, we thereby study the difference between the structural information and VNGE named as entropy gap. Based on the knowledge that the degree sequence is majorized by the Laplacian spectrum, we for the first time prove the entropy gap is between 0 and log 2e in any undirected unweighted graphs. Consequently we certify that the structural information is a good approximation of VNGE that achieves provable accuracy, scalability, and interpretability simultaneously. This approximation is further applied to two entropy-related tasks: network design and graph similarity measure, where novel graph similarity measure and fast algorithms are proposed. Our experimental results on graphs of various scales and types show that the very small entropy gap readily applies to a wide range of graphs and weighted graphs. As an approximation of VNGE, the structural information is the only one that achieves both high efficiency and high accuracy among the prominent methods. It is at least two orders of magnitude faster than SLaQ [40] with comparable accuracy. Our structural information based methods also exhibit superior performance in two entropy-related tasks.

Nonsmooth critical point theory and applications to the spectral graph theory

Abstract: Existing critical point theories including metric and topological critical point theories are difficult to be applied directly to some concrete problems in particular polyhedral settings, because the notions of critical sets could be either very vague or too large. To overcome these difficulties, we develop the critical point theory for nonsmooth but Lipschitzian functions defined on convex polyhedrons. This yields natural extensions of classical results in the critical point theory, such as the Liusternik-Schnirelmann multiplicity theorem. More importantly, eigenvectors for some eigenvalue problems involving graph 1-Laplacian coincide with critical points of the corresponding functions on polytopes, which indicates that the critical point theory proposed in the present paper can be applied to study the nonlinear spectral graph theory.

Positive Consensus of Directed Multi-Agent Systems

Abstract: This paper investigates the positive consensus problem of multi-agent systems with directed communication topologies where all the agents have identical continuous-time positive linear dynamics. Existing works of such problem mainly focus on the case where the networked communication topologies are of either undirected and incomplete graphs, or strongly connected directed graphs. In stark contrast to them, we study the problem in which the communication topologies of the networked system are described by directed graphs each containing a spanning tree which is a more general and new scenario due to the interplay between the eigenvalues of the Laplacian matrix and the controller gains. Based on the existing results in spectral graph theory and positive linear systems theory, several necessary and sufficient conditions on the positive consensus of the directed multi-agent system are derived through using linear matrix inequality techniques. Then a primal-dual iterative algorithm is developed for computation and finally several numerical simulations are provided to illustrate the effectiveness of the proposed theoretical results.

Geary’s c and Spectral Graph Theory

Abstract: Spatial autocorrelation, of which Geary’s c has traditionally been a popular measure, is fundamental to spatial science. This paper provides a new perspective on Geary’s c. We discuss this using concepts from spectral graph theory/linear algebraic graph theory. More precisely, we provide three types of representations for it: (a) graph Laplacian representation, (b) graph Fourier transform representation, and (c) Pearson’s correlation coefficient representation. Subsequently, we illustrate that the spatial autocorrelation measured by Geary’s c is positive (resp. negative) if spatially smoother (resp. less smooth) graph Laplacian eigenvectors are dominant. Finally, based on our analysis, we provide a recommendation for applied studies.

Balanced portfolio via signed graphs and spectral clustering in the Brazilian stock market

Abstract: A portfolio associated with a balanced signed graph that contains both positive and negative edges is more predictable and risk-averse, and is therefore likely to require less turnover. In this paper, we use a method based on spectral graph theory to determine whether a portfolio is balanced or not. Moreover, we combine this with spectral clustering to propose a strategy to build a balanced portfolio with hedging assets. This is applied to stocks listed on the Brazil Stock Exchange.

Application of the Spectra of Graphs in Network Forensics

Abstract: Network forensics is a discipline of growing importance. The ability to mathematically evaluate network intrusion incidents can substantially improve investigations. Graph theory is a robust mathematical tool that is readily applied to network traffic and has had been used in a limited fashion for network forensics. However, the full scope of graph theory has not previously been applied to network forensics. In particular, spectral graph theory has not been previously utilized for analyzing network forensics. This paper describes the application of spectral graph theory to specific network intrusion issues. This provides a mathematical tool to be utilized in network forensics. A case study is also utilized to demonstrate precisely how the methodology described in this paper should but utilized in an actual case.

Power up! Robust Graph Convolutional Network via Graph Powering.

Abstract: Graph convolutional networks (GCNs) are powerful tools for graph-structured data. However, they have been recently shown to be vulnerable to topological attacks. To enhance adversarial robustness, we go beyond spectral graph theory to robust graph theory. By challenging the classical graph Laplacian, we propose a new convolution operator that is provably robust in the spectral domain and is incorporated in the GCN architecture to improve expressivity and interpretability. By extending the original graph to a sequence of graphs, we also propose a robust training paradigm that encourages transferability across graphs that span a range of spatial and spectral characteristics. The proposed approaches are demonstrated in extensive experiments to simultaneously improve performance in both benign and adversarial situations.

A Spectral Representation of Power Systems with Applications to Adaptive Grid Partitioning and Cascading Failure Localization.

Abstract: Transmission line failures in power systems propagate and cascade non-locally. This well-known yet counter-intuitive feature makes it even more challenging to optimally and reliably operate these complex networks. In this work we present a comprehensive framework based on spectral graph theory that fully and rigorously captures how multiple simultaneous line failures propagate, distinguishing between non-cut and cut set outages. Using this spectral representation of power systems, we identify the crucial graph sub-structure that ensures line failure localization -- the network bridge-block decomposition. Leveraging this theory, we propose an adaptive network topology reconfiguration paradigm that uses a two-stage algorithm where the first stage aims to identify optimal clusters using the notion of network modularity and the second stage refines the clusters by means of optimal line switching actions. Our proposed methodology is illustrated using extensive numerical examples on standard IEEE networks and we discussed several extensions and variants of the proposed algorithm.

Rotation Averaging in a Split Second: A Primal-Dual Method and a Closed-Form for Cycle Graphs

Abstract: A cornerstone of geometric reconstruction, rotation averaging seeks the set of absolute rotations that optimally explains a set of measured relative orientations between them. In spite of being an integral part of bundle adjustment and structure-from-motion, averaging rotations is both a non-convex and high-dimensional optimization problem. In this paper, we address it from a maximum likelihood estimation standpoint and make a twofold contribution. Firstly, we set forth a novel initialization-free primal-dual method which we show empirically to converge to the global optimum. Further, we derive what is to our knowledge, the first optimal closed-form solution for rotation averaging in cycle graphs and contextualize this result within spectral graph theory. Our proposed methods achieve a significant gain both in precision and performance.

Constructing cospectral signed graphs

Abstract: A well-known fact in Spectral Graph Theory is the existence of pairs of cospectral (or isospectral) nonisomorphic graphs, known as PINGS. The work of A.J. Schwenk (in 1973) and of C. Godsil and B. ...

Manifold fitting algorithm of noisy manifold data based on variable-scale spectral graph

Abstract: Manifold fitting is a manifold verification technique for data with noise and manifold structures. By extracting the expected manifold structure, the reliability of the data manifold hypothesis can be determined, and the true structure of the data without noise can conform to a manifold. This paper proposes a manifold fitting algorithm for the variable-scale spectral graph theory and estimates the deviation of the manifold structure caused by noise. Considering the scale variations in frequency-domain analysis based on spectral graph theory, details of the data under the effect of small scale and characteristics of data shape under the effect of large scale are highlighted. This study uses the average calculation and meanshift method to obtain two types of mean vectors from each neighborhood, which are essential in suppressing noise and maintaining shape, respectively. Therefore, manifold fitting is carried out from two aspects, specifically weakening noise and characterizing the manifold shape. To obtain a closer estimate of the deviation caused by noise to the manifold structure, this study also estimates the neighborhood distribution of data under the effect of medium scale, obtains the covariance information of each neighborhood, and uses the variance information to estimate the manifold structure deviations.

Algorithmic techniques for finding resistance distances on structured graphs

Abstract: In this paper we give a survey of methods used to calculate values of resistance distance (also known as effective resistance) in graphs. Resistance distance has played a prominent role not only in circuit theory and chemistry, but also in combinatorial matrix theory and spectral graph theory. Moreover resistance distance has applications ranging from quantifying biological structures, distributed control systems, network analysis, and power grid systems. In this paper we discuss both exact techniques and approximate techniques and for each method discussed we provide an illustrative example of the technique. We also present some open questions and conjectures.

Smith Normal Form and the Generalized Spectral Characterization of Graphs

Abstract: Spectral characterization of graphs is an important topic in spectral graph theory, which has received a lot of attention from researchers in recent years. It is generally very hard to show a given graph to be determined by its spectrum. Recently, Wang [10] gave a simple arithmetic condition for graphs being determined by their generalized spectra. Let $G$ be a graph with adjacency matrix $A$ on $n$ vertices, and $W=[e,Ae,\ldots,A^{n-1}e]$ ($e$ is the all-one vector) be the walk-matrix of $G$. A theorem of Wang [10] states that if $2^{-\lfloor n/2\rfloor}\det W$ (which is always an integer) is odd and square-free, then $G$ is determined by the generalized spectrum. In this paper, we find a new and short route which leads to a stronger version of the above theorem. The result is achieved by using the Smith Normal Form of the walk-matrix of $G$. The proposed method gives a new insight in dealing with the problem of generalized spectral characterization of graphs.

Applications of rational difference equations to spectral graph theory

Abstract: We study a general class of recurrence relations that appear in the application of a matrix diagonalization procedure. We find a general closed formula and determine the analytical properties of th...

Network Cluster-Robust Inference

Abstract: Since network data commonly consists of observations on a single large network, researchers often partition the network into clusters in order to apply cluster-robust inference methods. All existing such methods require clusters to be asymptotically independent. We prove under mild conditions that, in order for this requirement to hold for network-dependent data, it is necessary and sufficient for clusters to have low conductance, the ratio of edge boundary size to volume. This yields a simple measure of cluster quality. We find in simulations that, when clusters have low conductance, cluster-robust methods outperform HAC estimators in terms of size control. However, for important classes of networks lacking low-conductance clusters, the methods can exhibit substantial size distortion. To assess the existence of low-conductance clusters and construct them, we draw on results in spectral graph theory that connect conductance to the spectrum of the graph Laplacian. Based on these results, we propose to use the spectrum to determine the number of low-conductance clusters and spectral clustering to construct them.

Network Cluster-Robust Inference

Abstract: Network data commonly consists of observations on a single large network. Accordingly, researchers often partition the network into clusters in order to apply cluster-robust inference methods. All existing such methods require clusters to be asymptotically independent. We show that for this requirement to hold, under certain conditions, it is necessary and sufficient for clusters to have small "conductance," which is the ratio of edge boundary size to volume. This yields a quantitative measure of cluster quality. Unfortunately, there are important classes of networks for which small-conductance clusters appear not to exist. Our simulation results show that for such networks, cluster-robust methods can exhibit substantial size distortion. Based on well-known results in spectral graph theory, we suggest using the eigenvalues of the graph Laplacian to determine the existence and number of small-conductance clusters. We also discuss the use of spectral clustering for constructing clusters in practice.

The distance matrix and its variants for graphs and digraphs

Abstract: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv CHAPTER 1. GENERAL INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Definitions for graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Digraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 CHAPTER 2. THE NORMALIZED DISTANCE LAPLACIAN . . . . . . . . . . . . . . . . 7 2.

Deep Learning and Spectral Embedding for Graph Partitioning.

Abstract: We present a graph bisection and partitioning algorithm based on graph neural networks. For each node in the graph, the network outputs probabilities for each of the partitions. The graph neural network consists of two modules: an embedding phase and a partitioning phase. The embedding phase is trained first by minimizing a loss function inspired by spectral graph theory. The partitioning module is trained through a loss function that corresponds to the expected value of the normalized cut. Both parts of the neural network rely on SAGE convolutional layers and graph coarsening using heavy edge matching. The multilevel structure of the neural network is inspired by the multigrid algorithm. Our approach generalizes very well to bigger graphs and has partition quality comparable to METIS, Scotch and spectral partitioning, with shorter runtime compared to METIS and spectral partitioning.