Eigenvalue Of Graph Laplacian Research Articles

Cooperative output feedback tracking control of stochastic nonlinear heterogeneous multi‐agent systems

SummaryWe study cooperative output feedback tracking control of stochastic nonlinear heterogeneous leader‐following multi‐agent systems. Each agent has a continuous‐time stochastic nonlinear heterogeneous dynamics with an unmeasurable state, and there are additive and multiplicative noises along with information exchange among agents. We propose admissible distributed observation strategies for estimating the leader's and the followers' states, and admissible cooperative output feedback control strategies based on the certainty equivalence principle. By output regulation theory and stochastic analysis, we show if the dynamics of each agent satisfies the Lipschitz condition, and the product of the leader's Lipschitz coefficient, the intensity of multiplicative measurement noises, and the constant related to the leader's Lipschitz coefficient and dimension is less than the minimum nonzero eigenvalue of graph Laplacian, then there exist admissible distributed observation and cooperative control strategies to ensure mean square bounded output tracking. Finally, the effectiveness of our control strategies is demonstrated by a numerical simulation.

Cooperative Output Feedback Tracking Control of Stochastic Linear Heterogeneous Multiagent Systems

We study cooperative output feedback tracking control of stochastic linear heterogeneous leader- following multiagent systems. Each agent has a conti- nuous-time linear heterogeneous dynamics with incompletely measurable state, and there are additive and multiplicative noises along with information exchange among agents. We propose a set of admissible distributed observation strategies for estimating the leader’s and the followers’ states and a set of admissible cooperative output feedback control strategies based on the certainty equivalence principle. By output regulation theory and stochastic analysis, we show that for observable leader’s dynamics and stabilizable and detectable followers’ dynamics, if the intensity coefficient of multiplicative noises multiplied by the sum of real parts of the leader’s unstable modes is less than one-fourth of the minimum nonzero eigenvalue of graph Laplacian, then there exist admissible distributed observation and cooperative control strategies to ensure mean square bounded output tracking, provided the associated output regulation equations are solvable. Finally, the effectiveness of our control strategies is demonstrated by a numerical simulation.

On the Sum and Spread of Reciprocal Distance Laplacian Eigenvalues of Graphs in Terms of Harary Index

The reciprocal distance Laplacian matrix of a connected graph G is defined as RDL(G)=RT(G)−RD(G), where RT(G) is the diagonal matrix of reciprocal distance degrees and RD(G) is the Harary matrix. Clearly, RDL(G) is a real symmetric matrix, and we denote its eigenvalues as λ1(RDL(G))≥λ2(RDL(G))≥…≥λn(RDL(G)). The largest eigenvalue λ1(RDL(G)) of RDL(G), denoted by λ(G), is called the reciprocal distance Laplacian spectral radius. In this paper, we obtain several upper bounds for the sum of k largest reciprocal distance Laplacian eigenvalues of G in terms of various graph parameters, such as order n, maximum reciprocal distance degree RTmax, minimum reciprocal distance degree RTmin, and Harary index H(G) of G. We determine the extremal cases corresponding to these bounds. As a consequence, we obtain the upper bounds for reciprocal distance Laplacian spectral radius λ(G) in terms of the parameters as mentioned above and characterize the extremal cases. Moreover, we attain several upper and lower bounds for reciprocal distance Laplacian spread RDLS(G)=λ1(RDL(G))−λn−1(RDL(G)) in terms of various graph parameters. We determine the extremal graphs in many cases.

Event-Based Sampled-Data Average Consensus

This study addresses one of the most essential distributed control problems in multiagent systems, called the average consensus issue, using a new event-triggered sampling control perspective. Although the continuous-time sampling for average consensus has provided good results currently, a systematic investigation into the continuous-time agent dynamics with sampled-data control inputs under an event-triggered mechanism is critically lacking. The problem considered in this paper can be formulated into an average consensus problem of hybrid systems. The method considers three types of control schemes, among which periodic sampling is integrant. The first scheme is a classical sampling controller reinvestigated through a lemma. The second scheme realizes aperiodic control update as well as periodic communication, while the third scheme achieves both aspects aperiodically. Corresponding sufficient conditions of the aforementioned three schemes are derived such that the asymptotic stability of systems is ensured by using algebraic graph theory, matrix analysis, and Lyapunov theory. The constraints for the allowed sampling period, event parameter, and maximum eigenvalue of graph Laplacian are explicitly derived. Moreover, the potential Zeno behavior of agents due to the sampling control theory is avoided. Thus, a digitally implementable technique is provided. Finally, some numerical examples are provided to verify the effectiveness of the proposed theoretical analysis.

Summarizing Complex Graphical Models of Multiple Chronic Conditions Using the Second Eigenvalue of Graph Laplacian: Algorithm Development and Validation.

BackgroundIt is important but challenging to understand the interactions of multiple chronic conditions (MCC) and how they develop over time in patients and populations. Clinical data on MCC can now be represented using graphical models to study their interaction and identify the path toward the development of MCC. However, the current graphical models representing MCC are often complex and difficult to analyze. Therefore, it is necessary to develop improved methods for generating these models.ObjectiveThis study aimed to summarize the complex graphical models of MCC interactions to improve comprehension and aid analysis.MethodsWe examined the emergence of 5 chronic medical conditions (ie, traumatic brain injury [TBI], posttraumatic stress disorder [PTSD], depression [Depr], substance abuse [SuAb], and back pain [BaPa]) over 5 years among 257,633 veteran patients. We developed 3 algorithms that utilize the second eigenvalue of the graph Laplacian to summarize the complex graphical models of MCC by removing less significant edges. The first algorithm learns a sparse probabilistic graphical model of MCC interactions directly from the data. The second algorithm summarizes an existing probabilistic graphical model of MCC interactions when a supporting data set is available. The third algorithm, which is a variation of the second algorithm, summarizes the existing graphical model of MCC interactions with no supporting data. Finally, we examined the coappearance of the 100 most common terms in the literature of MCC to validate the performance of the proposed model.ResultsThe proposed summarization algorithms demonstrate considerable performance in extracting major connections among MCC without reducing the predictive accuracy of the resulting graphical models. For the model learned directly from the data, the area under the curve (AUC) performance for predicting TBI, PTSD, BaPa, SuAb, and Depr, respectively, during the next 4 years is as follows—year 2: 79.91%, 84.04%, 78.83%, 82.50%, and 81.47%; year 3: 76.23%, 80.61%, 73.51%, 79.84%, and 77.13%; year 4: 72.38%, 78.22%, 72.96%, 77.92%, and 72.65%; and year 5: 69.51%, 76.15%, 73.04%, 76.72%, and 69.99%, respectively. This demonstrates an overall 12.07% increase in the cumulative sum of AUC in comparison with the classic multilevel temporal Bayesian network.ConclusionsUsing graph summarization can improve the interpretability and the predictive power of the complex graphical models of MCC.

Open Problem on $\sigma$-invariant

Let $G$ be a graph of order $n$ with $m$ edges. Also let $\mu_1 \geq \mu_2 \geq \cdots \geq \mu_{n-1} \geq \mu_n = 0$ be the Laplacian eigenvalues of graph $G$ and let $\sigma = \sigma(G)$ ($1 \leq \sigma \leq n$) be the largest positive integer such that $\mu_{\sigma} \geq 2m/n$. In this paper, we prove that $\mu_2(G) \geq 2m/n$ for almost all graphs. Moreover, we characterize the extremal graphs for any graphs. Finally, we provide the answer to Problem 3 in [8], that is, the characterization of all graphs with $\sigma = 1$.

Nordhaus-Gaddum-type result on the second largest signless Laplacian eigenvalue of a graph

ABSTRACT Let G be a simple graph of order n with m edges. Denote by the diagonal matrix of its vertex degrees and by its adjacency matrix. Then the signless Laplacian matrix of G is . Let be the signless Laplacian eigenvalues of graph G and also let be the largest positive integer such that . Denote by the complement graph of graph G. If , then we prove that . Moreover, if , then .

On the relationship between the Kirchhoff and the Narumi-Katayama indices

Let G be a simple connected graph with n vertices and m edges, sequence of vertex degrees Δ = d1 ≥ d2 ≥...dn = δ > 0 and diagonal matrix D = diag(d1,d2, ..., dn) of its vertex degrees. Denote by Kf (G) = nΣn-1, i=1 1/μi , where μi are the Laplacian eigenvalues of graph G, the Kirchhoff index of G, and by NK = Πn,i=1 di the Narumi-Katayama index. In this paper we prove some inequalities that exhibit relationship between the Kirchhoff and Narumi-Katayama indices.

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.

Local Spectral Clustering for Overlapping Community Detection

Large graphs arise in a number of contexts and understanding their structure and extracting information from them is an important research area. Early algorithms for mining communities have focused on global graph structure, and often run in time proportional to the size of the entire graph. As we explore networks with millions of vertices and find communities of size in the hundreds, it becomes important to shift our attention from macroscopic structure to microscopic structure in large networks. A growing body of work has been adopting local expansion methods in order to identify communities from a few exemplary seed members. In this article, we propose a novel approach for finding overlapping communities called L emon ( L ocal E xpansion via M inimum O ne N orm). Provided with a few known seeds , the algorithm finds the community by performing a local spectral diffusion. The core idea of L emon is to use short random walks to approximate an invariant subspace near a seed set, which we refer to as local spectra . Local spectra can be viewed as the low-dimensional embedding that captures the nodes’ closeness in the local network structure. We show that L emon ’s performance in detecting communities is competitive with state-of-the-art methods. Moreover, the running time scales with the size of the community rather than that of the entire graph. The algorithm is easy to implement and is highly parallelizable. We further provide theoretical analysis of the local spectral properties, bounding the measure of tightness of extracted community using the eigenvalues of graph Laplacian. We thoroughly evaluate our approach using both synthetic and real-world datasets across different domains, and analyze the empirical variations when applying our method to inherently different networks in practice. In addition, the heuristics on how the seed set quality and quantity would affect the performance are provided.

Classifying the Dimensional Variation in Additive Manufactured Parts From Laser-Scanned Three-Dimensional Point Cloud Data Using Machine Learning Approaches

The objective of this work is to develop and apply a spectral graph theoretic approach for differentiating between (classifying) additive manufactured (AM) parts contingent on the severity of their dimensional variation from laser-scanned coordinate measurements (3D point cloud). The novelty of the approach is in invoking spectral graph Laplacian eigenvalues as an extracted feature from the laser-scanned 3D point cloud data in conjunction with various machine learning techniques. The outcome is a new method that classifies the dimensional variation of an AM part by sampling less than 5% of the 2 million 3D point cloud data acquired (per part). This is a practically important result, because it reduces the measurement burden for postprocess quality assurance in AM—parts can be laser-scanned and their dimensional variation quickly assessed on the shop floor. To realize the research objective, the procedure is as follows. Test parts are made using the fused filament fabrication (FFF) polymer AM process. The FFF process conditions are varied per a phased design of experiments plan to produce parts with distinctive dimensional variations. Subsequently, each test part is laser scanned and 3D point cloud data are acquired. To classify the dimensional variation among parts, Laplacian eigenvalues are extracted from the 3D point cloud data and used as features within different machine learning approaches. Six machine learning approaches are juxtaposed: sparse representation, k-nearest neighbors, neural network, naïve Bayes, support vector machine, and decision tree. Of these, the sparse representation technique provides the highest classification accuracy (F-score > 97%).

Distributed optimisation and control of graph Laplacian eigenvalues for robust consensus via an adaptive multilayer strategy

SummaryFunctions of eigenvalues of the graph Laplacian matrix L, especially the extremal non‐trivial eigenvalues, the algebraic connectivity λ2 and the spectral radius λn, have been shown to be important in determining the performance in a host of consensus and synchronisation applications. In this paper, we focus on formulating an entirely distributed control law for the control of edge weights in an undirected graph to solve a constrained optimisation problem involving these extremal eigenvalues.As an objective for the distributed control law, edge weights must be found that minimise the spectral radius of the graph Laplacian, thereby maximising the robustness of the network to time delays under a simple linear consensus protocol. To constrain the problem, we use both local weight constraints that weights must be non‐negative, and a global connectivity constraint, maintaining a designated minimum algebraic connectivity. This ensures that the network remains sufficiently well connected.The distributed control law is formulated as a multilayer strategy, using three layers of successive distributed estimation. Adequate timescale separation between the layers is of paramount importance for the proper functioning of the system, and we derive conditions under which the distributed system converges as we would expect for the centralised control or optimisation system to converge. © 2017 The Authors International Journal of Robust and Nonlinear Control published by John Wiley & Sons Ltd.

A Spectral Graph Theoretical Approach to Oriented Energy Features

In this work, we propose a novel method for determining oriented energy features of an image. Oriented energy features, useful for many machine vision applications like contour detection, texture segmentation and motion analysis, are determined from the filters whose outputs are enhanced at the edges of the image at a given orientation. We use the eigenvectors and eigenvalues of graph Laplacian for determining the oriented energy features of an image. Our method is based on spectral graph theoretical approach in which a graph is assigned complex-valued edge weights whose phases encode orientation information. These edge weights give rise to a complex-valued Hermitian Laplacian whose spectrum enables us to extract oriented energy features of the image. We perform a set of numerical experiments to determine the efficiency and characteristics of the proposed method. In addition, we apply our feature extraction method to texture segmentation problem. We do this in comparison with other known methods, and show that our method performs better for various test textures.

On Energy and Laplacian Energy of Graphs

Let $G=(V,E)$ be a simple graph of order $n$ with $m$ edges. The energy of a graph $G$, denoted by $\mathcal{E}(G)$, is defined as the sum of the absolute values of all eigenvalues of $G$. The Laplacian energy of the graph $G$ is defined as \[ LE = LE(G)=\sum^n_{i=1}\left|\mu_i-\frac{2m}{n}\right| \] where $\mu_1,\,\mu_2,\,\ldots,\,\mu_{n-1},\,\mu_n=0$ are the Laplacian eigenvalues of graph $G$. In this paper, some lower and upper bounds for $\mathcal{E}(G)$ are presented in terms of number of vertices, number of edges, maximum degree and the first Zagreb index, etc. Moreover, a relation between energy and Laplacian energy of graphs is given.

Interarea Oscillation Revisit

This letter revisits interarea oscillation analysis using networked control analysis techniques. The power system analysis problem is analyzed as a networked control problem, specifically consensus control of homogeneous systems with static output feedback. The power grid is represented by a graph Laplacian matrix. Stability of the entire system can be evaluated by individual system dynamics and graph Laplacian's eigenvalues. Through this technique, the classical large-scale power system analysis problem is decomposed into multiple small-scale system analysis problems. Analysis of the classical two-area four-machine system is conducted by the proposed approach and compared with the small-signal analysis results from Power System Toolbox. The interarea oscillation mode is found to be related to the second smallest eigenvalue of the graph Laplacian matrix, while the local oscillation modes are related to the other eigenvalues of the graph Laplacian matrix.

Matched signal detection on graphs: Theory and application to brain imaging data classification

Motivated by recent progress in signal processing on graphs, we have developed a matched signal detection (MSD) theory for signals with intrinsic structures described by weighted graphs. First, we regard graph Laplacian eigenvalues as frequencies of graph-signals and assume that the signal is in a subspace spanned by the first few graph Laplacian eigenvectors associated with lower eigenvalues. The conventional matched subspace detector can be applied to this case. Furthermore, we study signals that may not merely live in a subspace. Concretely, we consider signals with bounded variation on graphs and more general signals that are randomly drawn from a prior distribution. For bounded variation signals, the test is a weighted energy detector. For the random signals, the test statistic is the difference of signal variations on associated graphs, if a degenerate Gaussian distribution specified by the graph Laplacian is adopted. We evaluate the effectiveness of the MSD on graphs both with simulated and real data sets. Specifically, we apply MSD to the brain imaging data classification problem of Alzheimer's disease (AD) based on two independent data sets: 1) positron emission tomography data with Pittsburgh compound-B tracer of 30 AD and 40 normal control (NC) subjects, and 2) resting-state functional magnetic resonance imaging (R-fMRI) data of 30 early mild cognitive impairment and 20 NC subjects. Our results demonstrate that the MSD approach is able to outperform the traditional methods and help detect AD at an early stage, probably due to the success of exploiting the manifold structure of the data.

Tree simplification and the ‘plateaux’ phenomenon of graph Laplacian eigenvalues

We developed a procedure of reducing the number of vertices and edges of a given tree, which we call the “tree simplification procedure,” without changing its topological information. Our motivation for developing this procedure was to reduce computational costs of graph Laplacian eigenvalues of such trees. When we applied this procedure to a set of trees representing dendritic structures of retinal ganglion cells of a mouse and computed their graph Laplacian eigenvalues, we observed two “plateaux” (i.e., two sets of multiple eigenvalues) in the eigenvalue distribution of each such simplified tree. In this article, after describing our tree simplification procedure, we analyze why such eigenvalue plateaux occur in a simplified tree; explain such plateaux can occur in a more general graph if it satisfies a certain condition; and identify these two eigenvalues specifically as well as the lower bound to their multiplicity.

On limit points of the third largest Laplacian eigenvalues of graphs

On limit points of the third largest Laplacian eigenvalues of graphs

Spectrum-Adapted Tight Graph Wavelet and Vertex-Frequency Frames

We consider the problem of designing spectral graph filters for the construction of dictionaries of atoms that can be used to efficiently represent signals residing on weighted graphs. While the filters used in previous spectral graph wavelet constructions are only adapted to the length of the spectrum, the filters proposed in this paper are adapted to the distribution of graph Laplacian eigenvalues, and therefore lead to atoms with better discriminatory power. Our approach is to first characterize a family of systems of uniformly translated kernels in the graph spectral domain that give rise to tight frames of atoms generated via generalized translation on the graph. We then warp the uniform translates with a function that approximates the cumulative spectral density function of the graph Laplacian eigenvalues. We use this approach to construct computationally efficient, spectrum-adapted, tight vertex-frequency and graph wavelet frames. We give numerous examples of the resulting spectrum-adapted graph filters, and also present an illustrative example of vertex-frequency analysis using the proposed construction.

Upper bounds on the (signless) Laplacian eigenvalues of graphs

Let G be a simple graph of order n with vertex set V={v1,v2,…,vn}. Also let μ1(G)≥μ2(G)≥⋯≥μn−1(G)≥μn(G)=0 and q1(G)≥q2(G)≥⋯≥qn(G)≥0 be the Laplacian eigenvalues and signless Laplacian eigenvalues of G, respectively. In this paper we obtain μi(G)≤i−1+minUi⁡max⁡{|NH(vk)∪NH(vj)|:vkvj∈E(H)}, where NH(vk) is the set of neighbors of vertex vk in V(H)=V(G)\\Ui, Ui is any (i−1)-subset of V(G) (here, we agree that i∈{1,…,n−1} and μi(G)≤i−1 if E(H)=∅). For any graph G, this bound does not exceed the order of G. Moreover, we prove thatmax⁡{μi(G),qi(G)}≤maxi≤k≤n⁡{dG(vk)+∑vj∈NG(vk)∩NdG(vj)dG(vk)}≤2dG(vi), where dG(vi) is the i-th largest degree of G and N={vi,vi+1,…,vn}.