Graph Laplacian Matrix Research Articles

Limit theorems for eigenvectors of the normalized Laplacian for random graphs

We prove a central limit theorem for the components of the eigenvectors corresponding to the $d$ largest eigenvalues of the normalized Laplacian matrix of a finite dimensional random dot product graph. As a corollary, we show that for stochastic blockmodel graphs, the rows of the spectral embedding of the normalized Laplacian converge to multivariate normals and furthermore the mean and the covariance matrix of each row are functions of the associated vertex's block membership. Together with prior results for the eigenvectors of the adjacency matrix, we then compare, via the Chernoff information between multivariate normal distributions, how the choice of embedding method impacts subsequent inference. We demonstrate that neither embedding method dominates with respect to the inference task of recovering the latent block assignments.

Optimization of synchronizability in complex spatial networks

Many real-world phenomena can be modelled as spatial networks where nodes have distinct geographical location. Examples include power grids, transportation networks and the Internet. This paper focuses on optimizing the synchronizability of spatial networks. We consider the eigenratio of the Laplacian Matrix of the connection graph as a metric measuring the synchronizability of the network and develop an efficient rewiring mechanism to optimize the topology of the network for synchronizability, i.e., minimizing the eigenratio. The Euclidean distance between two connected nodes is considered as their connection weights, and the sum of all connection weights is defined as the network cost. The proposed optimization algorithm constructs spatial networks with a certain number of nodes and a predefined network cost. We also study the topological properties of the optimized networks. This algorithm can be used to construct spatial networks with optimal synchronization properties.

ISIRA: Integrated shift–invert residual Arnoldi method for graph Laplacian matrices from big data

The eigenvalue problem of a graph Laplacian matrix $L$ arising from a simple, connected and undirected graph has been given more attention due to its extensive applications, such as spectral clustering, community detection, complex network, image processing and so on. The associated graph Laplacian matrix is symmetric, positive semi-definite, and is usually large and sparse. Computing some smallest positive eigenvalues and corresponding eigenvectors is often of interest. However, the singularity of $L$ makes the classical eigensolvers inefficient since we need to factorize $L$ for the purpose of solving large and sparse linear systems exactly. The next difficulty is that it is usually time consuming or even unavailable to factorize a large and sparse matrix arising from real network problems from big data such as social media transactional databases, and sensor systems because there is in general not only local connections. In this paper, we propose an eignsolver based on the inexact residual Arnoldi method together with an implicit remedy of the singularity and an effective deflation for convergent eigenvalues. Numerical experiments reveal that the integrated eigensolver outperforms the classical Arnoldi/Lanczos method for computing some smallest positive eigeninformation provided the LU factorization is not available.

Method of Image Quality Improvement for Atmospheric Turbulence Degradation Sequence Based on Graph Laplacian Filter and Nonrigid Registration

It is challenging to restore a clear image from an atmospheric degraded sequence. The main reason for the image degradation is geometric distortion and blurring caused by turbulence. In this paper, we present a method to eliminate geometric distortion and blur and to recover a single high-quality image from the degraded sequence images. First, we use optical flow technology to register the sequence images, thereby suppressing the geometric deformation of each frame. Next, sequence images are summed by a temporal filter to obtain a single blurred image. Then, the graph Laplacian matrix is used as the cost function to construct the regularization term. The final clear image and point spread function are obtained by iteratively solving the problem. Experiments show that the method can effectively eliminate the distortion and blur, restore the image details, and significantly improve the image quality.

Model reduction for Kuramoto models with complex topologies.

Synchronization of coupled oscillators is a ubiquitous phenomenon, occurring in topics ranging from biology and physics to social networks and technology. A fundamental and long-time goal in the study of synchronization has been to find low-order descriptions of complex oscillator networks and their collective dynamics. However, for the Kuramoto model, the most widely used model of coupled oscillators, this goal has remained surprisingly challenging, in particular for finite-size networks. Here, we propose a model reduction framework that effectively captures synchronization behavior in complex network topologies. This framework generalizes a collective coordinates approach for all-to-all networks [G. A. Gottwald, Chaos 25, 053111 (2015)CHAOEH1054-150010.1063/1.4921295] by incorporating the graph Laplacian matrix in the collective coordinates. We first derive low dimensional evolution equations for both clustered and nonclustered oscillator networks. We then demonstrate in numerical simulations for Erdős-Rényi networks that the collective coordinates capture the synchronization behavior in both finite-size networks as well as in the thermodynamic limit, even in the presence of interacting clusters.

On distance Laplacian and distance signless Laplacian eigenvalues of graphs

ABSTRACTLet , and be, respectively, the distance matrix, the distance Laplacian matrix and the distance signless Laplacian matrix of graph G, where denotes the diagonal matrix of the vertex transmissions in G. The eigenvalues of and will be denoted by and , respectively. In this paper, we study the properties of the distance Laplacian eigenvalues and the distance signless Laplacian eigenvalues of graph G.

Finding GEMS: Multi-Scale Dictionaries For High-Dimensional Graph Signals

Modern data introduces new challenges to classic signal processing approaches, leading to a growing interest in the field of graph signal processing. A powerful and well established model for real world signals in various domains is sparse representation over a dictionary, combined with the ability to train the dictionary from signal examples. This model has been successfully applied to graph signals as well by integrating the underlying graph topology into the learned dictionary. Nonetheless, dictionary learning methods for graph signals are typically restricted to small dimensions due to the computational constraints that the dictionary learning problem entails, and due to the direct use of the graph Laplacian matrix. In this paper, we propose a dictionary learning algorithm that applies to a broader class of graph signals, and is capable of handling much higher dimensional data. We incorporate the underlying graph topology both implicitly, by forcing the learned dictionary atoms to be sparse combinations of graph-wavelet functions, and explicitly, by adding direct graph constraints to promote smoothness in both the feature and manifold domains. The resulting atoms are thus adapted to the data of interest while adhering to the underlying graph structure and possessing a desired multi-scale property. Experimental results on several datasets, representing both synthetic and real network data of different nature, demonstrate the effectiveness of the proposed algorithm for graph signal processing even in high dimensions.

The Critical Group of the Corona of Trees and Cycles

The critical group of a connected graph is a refinement of the number of spanning trees of the graph. It is a subtle isomorphism invariant of the graph and is closely connected with the graph Laplacian matrix. In this paper, the structure of the critical group of the corona of trees and cycles is determined and the explicit expression of the Smith normal form of critical group on the corona of trees and circle are given.

Sensor-Based Build Condition Monitoring in Laser Powder Bed Fusion Additive Manufacturing Process Using a Spectral Graph Theoretic Approach

The goal of this work is to monitor the laser powder bed fusion (LPBF) process using an array of sensors so that a record may be made of those temporal and spatial build locations where there is a high probability of defect formation. In pursuit of this goal, a commercial LPBF machine at the National Institute of Standards and Technology (NIST) was integrated with three types of sensors, namely, a photodetector, high-speed visible camera, and short wave infrared (SWIR) thermal camera with the following objectives: (1) to develop and apply a spectral graph theoretic approach to monitor the LPBF build condition from the data acquired by the three sensors; (2) to compare results from the three different sensors in terms of their statistical fidelity in distinguishing between different build conditions. The first objective will lead to early identification of incipient defects from in-process sensor data. The second objective will ascertain the monitoring fidelity tradeoff involved in replacing an expensive sensor, such as a thermal camera, with a relatively inexpensive, low resolution sensor, e.g., a photodetector. As a first-step toward detection of defects and process irregularities that occur in practical LPBF scenarios, this work focuses on capturing and differentiating the distinctive thermal signatures that manifest in parts with overhang features. Overhang features can significantly decrease the ability of laser heat to diffuse from the heat source. This constrained heat flux may lead to issues such as poor surface finish, distortion, and microstructure inhomogeneity. In this work, experimental sensor data are acquired during LPBF of a simple test part having an overhang angle of 40.5 deg. Extracting and detecting the difference in sensor signatures for such a simple case is the first-step toward in situ defect detection in additive manufacturing (AM). The proposed approach uses the Eigen spectrum of the spectral graph Laplacian matrix as a derived signature from the three different sensors to discriminate the thermal history of overhang features from that of the bulk areas of the part. The statistical accuracy for isolating the thermal patterns belonging to bulk and overhang features in terms of the F-score is as follows: (a) F-score of 95% from the SWIR thermal camera signatures; (b) 83% with the high-speed visible camera; (c) 79% with the photodetector. In comparison, conventional signal analysis techniques—e.g., neural networks, support vector machines, linear discriminant analysis were evaluated with F-score in the range of 40–60%.

Approximate Fast Graph Fourier Transforms via Multilayer Sparse Approximations

The Fast Fourier Transform (FFT) is an algorithm of paramount importance in signal processing as it allows to apply the Fourier transform in O(n log n) instead of O(n 2) arithmetic operations. Graph Signal Processing (GSP) is a recent research domain that generalizes classical signal processing tools, such as the Fourier transform, to situations where the signal domain is given by any arbitrary graph instead of a regular grid. Today, there is no method to rapidly apply graph Fourier transforms. We propose in this paper a method to obtain approximate graph Fourier transforms that can be applied rapidly and stored efficiently. It is based on a greedy approximate diagonalization of the graph Laplacian matrix, carried out using a modified version of the famous Jacobi eigenvalues algorithm. The method is described and analyzed in detail, and then applied to both synthetic and real graphs, showing its potential.

Distributed Non-Convex First-Order Optimization and Information Processing: Lower Complexity Bounds and Rate Optimal Algorithms

We consider a class of popular distributed non-convex optimization problems, in which agents connected by a network $\mathcal{G}$ collectively optimize a sum of smooth (possibly non-convex) local objective functions. We address the following question: if the agents can only access the gradients of local functions, what are the fastest rates that any distributed algorithms can achieve, and how to achieve those rates. First, we show that there exist difficult problem instances, such that it takes a class of distributed first-order methods at least $\mathcal{O}(1/\sqrt{\xi(\mathcal{G})} \times \bar{L} /{\epsilon})$ communication rounds to achieve certain $\epsilon$-solution [where $\xi(\mathcal{G})$ denotes the spectral gap of the graph Laplacian matrix, and $\bar{L}$ is some Lipschitz constant]. Second, we propose (near) optimal methods whose rates match the developed lower rate bound (up to a polylog factor). The key in the algorithm design is to properly embed the classical polynomial filtering techniques into modern first-order algorithms. To the best of our knowledge, this is the first time that lower rate bounds and optimal methods have been developed for distributed non-convex optimization problems.

Proof of conjecture involving algebraic connectivity and average degree of graphs

Let G be a simple connected graph of order n with m edges. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the Laplacian matrix of graph G is L(G)=D(G)−A(G). Among all eigenvalues of the Laplacian matrix L(G) of graph G, the most studied is the second smallest, called the algebraic connectivity a(G) of a graph. Let d‾(G) and δ(G) be the average degree and the minimum degree of graph G, respectively. In this paper we characterize all graphs for which (i) a(G)=1 with δ(G)≥⌈n−12⌉, and (ii)a(G)=2 with δ(G)≥n2. In [1], Aouchiche mentioned a conjecture involving the algebraic connectivity a(G) and the average degree d‾(G) of graph G:a(G)−d‾(G)≥4−n−4n with equality holding if and only if G‾≅K1,n−2∪K1 (K1,n−2 is a star of order n−1 and G‾ is the complement of graph G). Here we prove this conjecture.

Optimal Filter Design for Signal Processing on Random Graphs: Accelerated Consensus

In graph signal processing, filters arise from polynomials in shift matrices that respect the graph structure, such as the graph adjacency matrix or the graph Laplacian matrix. Hence, filter design for graph signal processing benefits from knowledge of the spectral decomposition of these matrices. Often, stochastic influences affect the network structure and, consequently, the shift matrix empirical spectral distribution. Although the joint distribution of the shift matrix eigenvalues is typically inaccessible, deterministic functions that asymptotically approximate the matrix empirical spectral distribution can be found for suitable random graph models using tools from random matrix theory. We employ this information regarding the density of eigenvalues to develop criteria for optimal graph filter design. In particular, we consider filter design for distributed average consensus and related problems, leading to improvements in short-term error minimization or in asymptotic convergence rate.

Distributed-Graph-Based Statistical Approach for Intrusion Detection in Cyber-Physical Systems

Cyber-physical systems have recently emerged in several practical engineering applications where security and privacy are of paramount importance. This motivated the paper and a recent surge of interest in development of innovative and novel anomaly and intrusion detection technologies. This paper proposes a novel distributed blind intrusion detection framework by modeling sensor measurements as the target graph-signal and utilizing the statistical properties of the graph-signal for intrusion detection. To fully take into account the underlying network structure, the graph similarity matrix is constructed using both the data measured by the sensors and sensors' proximity resulting in a data-adaptive and structure-aware monitoring solution. In the proposed supervised detection framework, the magnitude of the captured data is modeled by Gaussian Markov random field and the corresponding precision matrix is estimated by learning a graph Laplacian matrix from sensor measurements adaptively. The proposed intrusion detection methodology is designed based on a modified Bayesian likelihood ratio test and the closed-form expressions are derived for the test statistic. Finally, temporal analysis of the network behavior is established by computing the Bhattacharyya distance between the measurement distributions at the consecutive time instants. Experiments are conducted to evaluate the performance of the proposed method and to compare it with that of the state-of-the-art methods. The results show that the proposed intrusion detection framework provides a detection performance superior to those provided by the other existing schemes.

Estimating the Laplacian matrix of Gaussian mixtures for signal processing on graphs

Recent works in signal processing on graphs have been driven to estimate the precision matrix and to use it as the graph Laplacian matrix. The normalized elements of the precision matrix are the partial correlation coefficients which measure the pairwise conditional linear dependencies of the graph. However, the non-linear dependencies inherent in any non-Gaussian model cannot be captured. We propose in this paper a generalized partial correlation coefficient which is derived by assuming an underlying multivariate Gaussian Mixture Model of the observations. Exact and approximate methods are proposed to estimate the generalized partial correlation coefficients from estimates of the Gaussian Mixture Model parameters. Thus it may find application in any non-Gaussian scenario where the Laplacian matrix is to be learned from training signals.

Large-Scale Robust Semisupervised Classification.

Semisupervised learning aims to leverage both labeled and unlabeled data to improve performance, where most of them are graph-based methods. However, the graph-based semisupervised methods are not capable for large-scale data since the computational consumption on the construction of graph Laplacian matrix is huge. On the other hand, the substantial unlabeled data in training stage of semisupervised learning could cause large uncertainties and potential threats. Therefore, it is crucial to enhance the robustness of semisupervised classification. In this paper, a novel large-scale robust semisupervised learning method is proposed in the framework of capped l2,p -norm. This strategy is superior not only in computational cost because it makes the graph Laplacian matrix unnecessary, but also in robustness to outliers since the capped l2,p -norm used for loss measurement. An efficient optimization algorithm is exploited to solve the nonconvex and nonsmooth challenging problem. The complexity of the proposed algorithm is analyzed and discussed in theory detailedly. Finally, extensive experiments are conducted over six benchmark data sets to demonstrate the effectiveness and superiority of the proposed method.

Optimal Cell-to-Cell Balancing Topology Design for Serially Connected Lithium-Ion Battery Packs

A battery pack can see energy imbalance among its cells resulting from cell-to-cell variation in capacity, internal resistance, and other parameters. Its successful and safe operation thus necessitates dynamic energy equalizing to adjust each cell's state-of-charge to the same level. The cell equalizing system for a serially connected battery pack is modeled as a multiagent system here. A consensus-based state-of-charge equalizing algorithm is proposed with its convergence proved through theoretical analysis. Following this development, an interesting and important problem is investigated: how to add extra individual cell equalizers or edges to the original cell equalizing system's topology to maximally accelerate the equalizing process. It is found that the balancing time depends on the algebraic connectivity of the topology graph, which is measured by the second smallest eigenvalue of the graph's Laplacian matrix. Then, a combinatorial 0-1 optimization problem is formulated and addressed to optimally choose added edges that lead to the maximum increase in the algebraic connectivity. The proposed results are validated through simulation and experiments, which demonstrate that the balancing time can be significantly reduced by just adding one optimal individual cell equalizer to the original cell equalizing system.

Maximizing Algebraic Connectivity via Minimum Degree and Maximum Distance

Algebraic connectivity, the second smallest eigenvalue of the graph Laplacian matrix, is a fundamental performance measure in various network systems, such as multi-agent networked systems. Here, we focus on how to add an edge to a network to increase network connectivity and robustness by maximizing the algebraic connectivity. Most efficient algorithms for maximizing algebraic connectivity need to calculate it directly, which results in high time complexity, especially for large networks. We present a heuristic algorithm, the minimum degree and maximum distance algorithm, based on the analysis of the Fiedler vector, which does not need to compute the algebraic connectivity. The proposed algorithm is tested in large random networks and networks of autonomous systems peering information. The results show that it is effective and can achieve shorter running times than other algorithms. Hence, it can be applied to very large networks, especially to large sparse networks.

Revealing Regional Associations of Cortical Folding Alterations with In Utero Ventricular Dilation Using Joint Spectral Embedding.

Fetal ventriculomegaly (VM) is a condition with dilation of one or both lateral ventricles, and is diagnosed as an atrial diameter larger than 10 mm. Evidence of altered cortical folding associated with VM has been shown in the literature. However, existing studies use a holistic approach (i.e., ventricle as a whole) based on diagnosis or ventricular volume, thus failing to reveal the spatially-heterogeneous association patterns between cortex and ventricle. To address this issue, we develop a novel method to identify spatially fine-scaled association maps between cortical development and VM by leveraging vertex-wise correlations between the growth patterns of both ventricular and cortical surfaces in terms of area expansion and curvature information. Our approach comprises multiple steps. In the first step, we define a joint graph Laplacian matrix using cortex-to-ventricle correlations. Next, we propose a spectral embedding of the cortex-to-ventricle graph into a common underlying space where their joint growth patterns are projected. More importantly, in the joint ventricle-cortex space, the vertices of associated regions from both cortical and ventricular surfaces would lie close to each other. In the final step, we perform clustering in the joint embedded space to identify associated sub-regions between cortex and ventricle. Using a dataset of 25 healthy fetuses and 23 fetuses with isolated non-severe VM within the age range of 26-29 gestational weeks, our results show that the proposed approach is able to reveal clinically relevant and meaningful regional associations.

Iterative Reconstrained Low-Rank Representation via Weighted Nonconvex Regularizer

Benefiting from the joint consideration of geometric structures and low-rank constraint, graph low-rank representation (GLRR) method has led to the state-of-the-art results in many applications. However, it faces the limitations that the structure of errors should be known a prior, the isolated construction of graph Laplacian matrix, and the over shrinkage of the leading rank components. To improve GLRR in these regards, this paper proposes a new LRR model, namely iterative reconstrained LRR via weighted nonconvex regularization, using three distinguished properties on the concerned representation matrix. The first characterizes various distributions of the errors into an adaptively learned weight factor for more flexibility of noise suppression. The second generates an accurate graph matrix from weighted observations for less afflicted by noisy features. The third employs a parameterized rational function to reveal the importance of different rank components for better approximation to the intrinsic subspace structure. Following a deep exploration of automatic thresholding, parallel update, and partial SVD operation, we derive a computationally efficient low-rank representation algorithm using an iterative reconstrained framework and accelerated proximal gradient method. Comprehensive experiments are conducted on synthetic data, image clustering, and background subtraction to achieve several quantitative benchmarks as clustering accuracy, normalized mutual information, and execution time. Results demonstrate the robustness and efficiency of IRWNR compared with other state-of-the-art models.