Weighted Laplacian Matrix Research Articles

Overcoming Dimensionality Constraints: A Gershgorin Circle Theorem-Based Feature Extraction for Weighted Laplacian Matrices in Computer Vision Applications.

In graph theory, the weighted Laplacian matrix is the most utilized technique to interpret the local and global properties of a complex graph structure within computer vision applications. However, with increasing graph nodes, the Laplacian matrix's dimensionality also increases accordingly. Therefore, there is always the "curse of dimensionality"; In response to this challenge, this paper introduces a new approach to reducing the dimensionality of the weighted Laplacian matrix by utilizing the Gershgorin circle theorem by transforming the weighted Laplacian matrix into a strictly diagonal domain and then estimating rough eigenvalue inclusion of a matrix. The estimated inclusions are represented as reduced features, termed GC features; The proposed Gershgorin circle feature extraction (GCFE) method was evaluated using three publicly accessible computer vision datasets, varying image patch sizes, and three different graph types. The GCFE method was compared with eight distinct studies. The GCFE demonstrated a notable positive Z-score compared to other feature extraction methods such as I-PCA, kernel PCA, and spectral embedding. Specifically, it achieved an average Z-score of 6.953 with the 2D grid graph type and 4.473 with the pairwise graph type, particularly on the E_Balanced dataset. Furthermore, it was observed that while the accuracy of most major feature extraction methods declined with smaller image patch sizes, the GCFE maintained consistent accuracy across all tested image patch sizes. When the GCFE method was applied to the E_MNSIT dataset using the K-NN graph type, the GCFE method confirmed its consistent accuracy performance, evidenced by a low standard deviation (SD) of 0.305. This performance was notably lower compared to other methods like Isomap, which had an SD of 1.665, and LLE, which had an SD of 1.325; The GCFE outperformed most feature extraction methods in terms of classification accuracy and computational efficiency. The GCFE method also requires fewer training parameters for deep-learning models than the traditional weighted Laplacian method, establishing its potential for more effective and efficient feature extraction in computer vision tasks.

On energy of matrices of a weighted graph

The energy of a square matrix is the sum of the absolute values of deviations of its eigenvalues from their mean. In this paper, we extend the notion of the Laplacian and the signless Laplacian energy of a graph to non-negative real symmetric matrices with zero diagonal. Such a matrix [Formula: see text] can be considered as the adjacency matrix of some weighted graph. Considering the sum of the weights of edges that are incident to a vertex as the weight of the corresponding vertex, we construct the diagonal matrix [Formula: see text] of which the diagonal entries are the weights of the corresponding vertices. We define a weighted version of the Laplacian matrix as [Formula: see text] and that of the signless Laplacian matrix as [Formula: see text]. In this work, we obtain some bounds of the energies of the weighted adjacency matrix, the weighted Laplacian matrix, and the weighted signless Laplacian matrix by studying the corresponding weighted graph. We also obtain some relations interconnecting those energies.

Distribution Grid Modeling Using Smart Meter Data

The knowledge of distribution grid models, including topologies and line impedances, is essential for grid monitoring, control and protection. However, such information is often unavailable, incomplete or outdated. The increasing deployment of smart meters (SMs) provides a unique opportunity to tackle this issue. This paper proposes a two-stage framework for distribution grid modeling using SM data. In the first stage, the network topology is identified by reconstructing a weighted Laplacian matrix of distribution networks. In the second stage, a least absolute deviations (LAD) regression model is developed for estimating line impedance of a single branch based on the nonlinear (inverse) power flow model, wherein a conductor library is leveraged to narrow down the solution space. The LAD regression model is originally a mixed-integer nonlinear program whose continuous relaxation is still non-convex. Thus, we specially address its convex relaxation and discuss the exactness. The modified regression model is then embedded within a bottom-up sweep algorithm to achieve the identification across the network in a branch-wise manner. Numerical results on the IEEE 13-bus, 37-bus and 69-bus test feeders validate the effectiveness of the proposed methods.

Anchor Selection for SLAM Based on Graph Topology and Submodular Optimization

This article considers simultaneous localization and mapping (SLAM) problem for robots in situations where accurate estimates for some of the robot poses, termed anchors, are available. These may be acquired through external means, for example, by either stopping the robot at some previously known locations or pausing for a sufficient period of time to measure the robot poses with an external measurement system. The main contribution is an efficient algorithm for selecting a fixed number of anchors from a set of potential poses that minimizes estimated error in the SLAM solution. Based on a graph-topological connection between the D-optimality design metric and the tree-connectivity of the pose-graph, the anchor selection problem can be formulated approximately as a submatrix selection problem for reduced weighted Laplacian matrix, leading to a cardinality-constrained submodular maximization problem. Two greedy methods are presented to solve this submodular optimization problem with a performance guarantee. These methods are complemented by Cholesky decomposition, approximate minimum degree permutation, order reuse, and rank-1 update that exploit the sparseness of the weighted Laplacian matrix. We demonstrate the efficiency and effectiveness of the proposed techniques on public-domain datasets, Gazebo simulations, and real-world experiments.

Modeling and Quantifying the Impact of Wind Penetration on Slow Coherency of Power Systems

This paper presents a mathematical analysis of how wind generation impacts the coherency property of power systems. Coherency arises from time-scale separation in the dynamics of synchronous generators, where generator states inside a coherent area synchronize over a fast time-scale due to stronger coupling, while the areas themselves synchronize over a slower time-scale due to weaker coupling. This time-scale separation is reflected in the form of a spectral separation in the weighted Laplacian matrix describing the swing dynamics of the generators. However, when wind farms with doubly-fed induction generators (DFIG) are integrated in the system then this Laplacian matrix changes based on both the level of wind penetration and the location of the wind farms. The modified Laplacian changes the effective slow eigenspace of the generators. Depending on penetration level, this change may result in changing the identities of the coherent areas. We develop a theoretical framework to quantify this modification, and validate our results with numerical simulations of the IEEE 68-bus system with one and multiple wind farms. We compare our model based results on clustering with results using measurement-based principal component analysis to substantiate our derivations.

Cramér–Rao Bounds and Optimal Design Metrics for Pose-Graph SLAM

Two-dimensional (2-D)/3-D pose-graph simultaneous localization and mapping (SLAM) is a problem of estimating a set of poses based on noisy measurements of relative rotations and translations. This article focuses on the relation between the graphical structure of pose-graph SLAM and Fisher information matrix (FIM), Cramer–Rao lower bounds (CRLB), and its optimal design metrics (T-optimality and D-optimality). As a main contribution, based on the assumption of isotropic Langevin noise for rotation and block-isotropic Gaussian noise for translation, the FIM and CRLB are derived and shown to be closely related to the graph structure, in particular, the weighted Laplacian matrix. We also prove that total node degree and weighted number of spanning trees, as two graph connectivity metrics, are, respectively, closely related to the trace and determinant of the FIM. The discussions show that, compared with the D-optimality metric, the T-optimality metric is more easily computed but less effective. We also present upper and lower bounds for the D-optimality metric, which can be efficiently computed and are almost independent of the estimation results. The results are verified with several well-known datasets, such as Intel, KITTI, sphere, and so on.

Effects of dynamical and structural modifications on synchronization.

Synchronization is a phenomenon of the collective behavior of coupled oscillators and involves the detailed interplay of the intrinsic frequencies of the oscillators, the underlying topological features of their interaction network, and external perturbations. In this work we investigate, in the strong coupling regime, the response of a system to external perturbations of its natural frequencies and network modifications. Our investigation relies on two performance measures (one for phases and the other for frequencies) and a spectral perturbation analysis. Given strongly localized perturbations in time, corresponding to the dominant eigenmode of the weighted Laplacian matrix of the network, we present a sufficient condition for the maximization of the system's stability, along with analytical results for the effects of structural perturbations on the system's response. A number of simulations are conducted to illustrate the theory presented.

Speeding up finite-time consensus via minimal polynomial of a weighted graph — A numerical approach

This work proposes an approach to speed up finite-time consensus algorithm using the weights of a weighted Laplacian matrix. It is motivated by the need to reach consensus among states of a multi-agent system in a distributed control/optimization setting. The approach is an iterative procedure that finds a low-order minimal polynomial that is consistent with the topology of the underlying graph. In general, the lowest-order minimal polynomial achievable for a network system is an open research problem. This work proposes a numerical approach that searches for the lowest order minimal polynomial via a rank minimization problem using a two-step approach: the first being an optimization problem involving the nuclear norm and the second a correction step. Convergence of the algorithm is shown and effectiveness of the approach is demonstrated via several examples.

Pseudoinverse of the Laplacian and best spreader node in a network.

Determining a set of "important" nodes in a network constitutes a basic endeavor in network science. Inspired by electrical flows in a resistor network, we propose the best conducting node j in a graph G as the minimizer of the diagonal element Q_{jj}^{†} of the pseudoinverse matrix Q^{†} of the weighted Laplacian matrix of the graph G. We propose a new graph metric that complements the effective graph resistance R_{G} and that specifies the heterogeneity of the nodal spreading capacity in a graph. Various formulas and bounds for the diagonal element Q_{jj}^{†} are presented. Finally, we compute the pseudoinverse matrix of the Laplacian of star, path, and cycle graphs and derive an expansion and lower bound of the effective graph resistance R_{G} based on the complement of the graph G.

Self-weighted spectral clustering with parameter-free constraint

The constrained spectral clustering (or known as the semi-supervised spectral clustering) focuses on enhancing the clustering capability by utilizing the side information. In this paper, a novel constrained spectral clustering method is proposed based on deriving a sparse parameter-free similarity. Different from other works, the proposed method transforms the given pairwise constraints into the intrinsic graph similarity and the penalty graph similarity respectively instead of incorporating them into one single similarity. Besides, the optimal weight can be automatically achieved to balance the graph optimization problems between the intrinsic graph and the penalty graph. Equipped with a general framework of efficiently unraveling the bi-objective optimization, the proposed method could obtain both ratio cut and normalized cut clusterings via updating the weighted Laplacian matrix until convergence. Moreover, the proposed method is equivalent to the spectral clustering, when no side information is provided. Consequently, the effectiveness and the superiority of the proposed method are further verified both analytically and empirically.

A model reduction method for biochemical reaction networks.

BackgroundIn this paper we propose a model reduction method for biochemical reaction networks governed by a variety of reversible and irreversible enzyme kinetic rate laws, including reversible Michaelis-Menten and Hill kinetics. The method proceeds by a stepwise reduction in the number of complexes, defined as the left and right-hand sides of the reactions in the network. It is based on the Kron reduction of the weighted Laplacian matrix, which describes the graph structure of the complexes and reactions in the network. It does not rely on prior knowledge of the dynamic behaviour of the network and hence can be automated, as we demonstrate. The reduced network has fewer complexes, reactions, variables and parameters as compared to the original network, and yet the behaviour of a preselected set of significant metabolites in the reduced network resembles that of the original network. Moreover the reduced network largely retains the structure and kinetics of the original model.ResultsWe apply our method to a yeast glycolysis model and a rat liver fatty acid beta-oxidation model. When the number of state variables in the yeast model is reduced from 12 to 7, the difference between metabolite concentrations in the reduced and the full model, averaged over time and species, is only 8%. Likewise, when the number of state variables in the rat-liver beta-oxidation model is reduced from 42 to 29, the difference between the reduced model and the full model is 7.5%.ConclusionsThe method has improved our understanding of the dynamics of the two networks. We found that, contrary to the general disposition, the first few metabolites which were deleted from the network during our stepwise reduction approach, are not those with the shortest convergence times. It shows that our reduction approach performs differently from other approaches that are based on time-scale separation. The method can be used to facilitate fitting of the parameters or to embed a detailed model of interest in a more coarse-grained yet realistic environment.

Hamiltonian perspective on compartmental reaction–diffusion networks

Inspired by the recent developments in modeling and analysis of reaction networks, we provide a geometric formulation of the reversible reaction networks under the influence of diffusion. Using the graph knowledge of the underlying reaction network, the obtained reaction–diffusion system is a distributed-parameter port-Hamiltonian system on a compact spatial domain. Motivated by the need for computer-based design, we offer a spatially consistent discretization of the PDE system and, in a systematic manner, recover a compartmental ODE model on a simplicial triangulation of the spatial domain. Exploring the properties of a balanced weighted Laplacian matrix of the reaction network and the Laplacian of the simplicial complex, we characterize the space of equilibrium points and provide a simple stability analysis on the state space modulo the space of equilibrium points. The paper rules out the possibility of the persistence of spatial patterns for the compartmental balanced reaction–diffusion networks.

Prediction of partial synchronization in delay-coupled nonlinear oscillators, with application to Hindmarsh–Rose neurons

The full synchronization of coupled nonlinear oscillators has been widely studied. In this paper we investigate conditions for which partial synchronization of time-delayed diffusively coupled systems arises. The coupling configuration of the systems is described by a directed graph. As a novel quantitative result we first give necessary and sufficient conditions for the presence of forward invariant sets characterized by partially synchronous motion. These conditions can easily be checked from the eigenvalues and eigenvectors of the graph Laplacian. Second, we perform stability analysis of the synchronized equilibria in a (gain,delay) parameter space. For this analysis the coupled nonlinear systems are linearized around the synchronized equilibria and then the resulting characteristic function is factorized. By such a factorization, it is shown that the relation between the behaviour of different agents at the zero of the characteristic function depends on the structure of the eigenvectors of the weighted Laplacian matrix. By determining the structure of the solutions in the unstable manifold, combined with the characterization of invariant sets, we predict which partially synchronous regimes occur and estimate the corresponding coupling gain and delay values. We apply the obtained results to networks of coupled Hindmarsh–Rose neurons and verify the occurrence of the expected partially synchronous regimes by using a numerical simulation. We also make a comparison with an existing approach based on Lyapunov functionals.

A graph-theoretical approach for the analysis and model reduction of complex-balanced chemical reaction networks

In this paper we derive a compact mathematical formulation describing the dynamics of chemical reaction networks that are complex-balanced and are governed by mass action kinetics. The formulation is based on the graph of (substrate and product) complexes and the stoichiometric information of these complexes, and crucially uses a balanced weighted Laplacian matrix. It is shown that this formulation leads to elegant methods for characterizing the space of all equilibria for complex-balanced networks and for deriving stability properties of such networks. We propose a method for model reduction of complex-balanced networks, which is similar to the Kron reduction method for electrical networks and involves the computation of Schur complements of the balanced weighted Laplacian matrix.

Maximum Allowable Loss Probability for Consensus of Multi-Agent Systems Over Random Weighted Lossy Networks

This note studies the consensus seeking problem for a team of general linear dynamical agents that communicate via a weighted random lossy network. Linear state feedback consensus protocols are applied and both the weights and feedback gain are treated as control parameters in the protocol. It is shown that the weights and the link loss probabilities of a network have non-negligible effects on the consensus seeking ability of multi-agent systems. Firstly, a weight condition characterized by the eigenvalues of the weighted Laplacian matrix is given for systems over ideal communication networks without packet losses. Secondly, based on stochastic stability analysis a maximum allowable loss probability bound is proposed for systems over random lossy networks. As long as the link loss probabilities of the network are less than this bound and the mean topology has spanning trees, there exist linear protocols solving the mean-square consensus problem of the system.

Quantized Consensus by Means of Gossip Algorithm

This paper deals with the distributed averaging problem over a connected network of agents, subject to a quantization constraint. It is assumed that at each time update, only a pair of agents can update their own states in terms of the quantized data being exchanged. The agents are also required to communicate with one another in a stochastic fashion. It is shown that a quantized consensus is reached for an arbitrary quantizer by means of the stochastic gossip algorithm proposed in a recent paper. The expected value of the time at which a quantized consensus is reached is lower and upper bounded in terms of the topology of the graph for a uniform quantizer. In particular, it is shown that these bounds are related to the principal submatrices of the weighted Laplacian matrix. A convex optimization is also proposed to determine a set of probabilities used to pick a pair of agents that leads to a fast convergence of the gossip algorithm.

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

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

Characterization and partial synthesis of the behavior of resistive circuits at their terminals

The external behavior of linear resistive circuits with terminals is characterized as a linear input–output map given by a weighted Laplacian matrix. Conditions are derived for shaping the external behavior of the circuit by interconnection with an additional resistive circuit.

Robust consensus of multi-agent systems with noise

The consensus problem of multi-agent systems has attracted wide attention from researchers in recent years, following the initial work of Jadbabaie et al. on the analysis of a simplified Vicsek model. While the original Vicsek model contains noise effects, almost all the existing theoretical results on consensus problem, however, do not take the noise effects into account. The purpose of this paper is to initiate a study of the consensus problems under noise disturbances. First, the class of multi-agent systems under study is transformed into a general time-varying system with noise. Then, for such a system, the equivalent relationships are established among (i) robust consensus, (ii) the positivity of the second smallest eigenvalue of a weighted Laplacian matrix, and (iii) the joint connectivity of the associated dynamical neighbor graphs. Finally, this basic equivalence result is shown to be applicable to several classes of concrete multi-agent models with noise.

On Cheeger‐type inequalities for weighted graphs

AbstractWe give several bounds on the second smallest eigenvalue of the weighted Laplacian matrix of a finite graph and on the second largest eigenvalue of its weighted adjacency matrix. We establish relations between the given Cheeger‐type bounds here and the known bounds in the literature. We show that one of our bounds is the best Cheeger‐type bound available. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 1–17, 2002