Spectrum Of Graph Laplacian Research Articles

Laplacian Change Point Detection for Single and Multi-view Dynamic Graphs

Dynamic graphs are rich data structures that are used to model complex relationships between entities over time. In particular, anomaly detection in temporal graphs is crucial for many real-world applications such as intrusion identification in network systems, detection of ecosystem disturbances, and detection of epidemic outbreaks. In this article, we focus on change point detection in dynamic graphs and address three main challenges associated with this problem: (i) how to compare graph snapshots across time, (ii) how to capture temporal dependencies, and (iii) how to combine different views of a temporal graph. To solve the above challenges, we first propose Laplacian Anomaly Detection (LAD) which uses the spectrum of graph Laplacian as the low dimensional embedding of the graph structure at each snapshot. LAD explicitly models short-term and long-term dependencies by applying two sliding windows. Next, we propose MultiLAD, a simple and effective generalization of LAD to multi-view graphs. MultiLAD provides the first change point detection method for multi-view dynamic graphs. It aggregates the singular values of the normalized graph Laplacian from different views through the scalar power mean operation. Through extensive synthetic experiments, we show that (i) LAD and MultiLAD are accurate and outperforms state-of-the-art baselines and their multi-view extensions by a large margin, (ii) MultiLAD’s advantage over contenders significantly increases when additional views are available, and (iii) MultiLAD is highly robust to noise from individual views. In five real-world dynamic graphs, we demonstrate that LAD and MultiLAD identify significant events as top anomalies such as the implementation of government COVID-19 interventions which impacted the population mobility in multi-view traffic networks.

Impact of Signal-to-Noise Ratio and Bandwidth on Graph Laplacian Spectrum From High-Dimensional Noisy Point Cloud

We systematically study the spectrum of kernel-based graph Laplacian (GL) constructed from high-dimensional and noisy random point cloud in the nonnull setup. The problem is motived by studying the model when the clean signal is sampled from a manifold that is embedded in a low-dimensional Euclidean subspace, and corrupted by high-dimensional noise. We quantify how the signal and noise interact in different regions of signal-to-noise ratio (SNR), and report the resulting peculiar spectral behavior of GL. In addition, we explore the impact of chosen kernel bandwidth on the spectrum of GL over different regions of SNR, which lead to an adaptive choice of kernel bandwidth that coincides with the common practice in real data. This result paves the way to a theoretical understanding of how practitioners apply GL when the dataset is noisy.

On Seidel Laplacian matrix and energy of graphs

Abstract In this work, the Seidel Laplacian spectrum of graphs are determined. Then new bounds are presented for the Seidel Laplacian energy of regular graphs and graphs by using their Seidel Laplacian spectrum and other techniques. Further, the Seidel Laplacian energy of specific graphs are computed.

Network embedding from the line graph: Random walkers and boosted classification

Abstract In this paper, we propose to embed edges instead of nodes using state-of-the-art neural/factorization methods (DeepWalk, node2vec, NetMF). These methods produce latent representations based on co-ocurrence statistics by simulating fixed-length random walks and then taking bags-of-vectors as the input to the Skip Gram Learning with Negative Sampling (SGNS). We commence by expressing commute times embedding as matrix factorization, and thus relating this embedding to those of DeepWalk and node2vec. Recent results showing formal links between all these methods via the spectrum of graph Laplacian, are then extended to understand the results obtained by SGNS when we embed edges instead of nodes. Since embedding edges is equivalent to embedding nodes in the line graph, we proceed to combine both existing formal characterizations of the line graphs and empirical evidence in order to explain why this embedding dramatically outperforms its nodal counterpart in multi-label classification tasks.

A criterion for stability of cluster synchronization in networks with external equitable partitions

Networks of nonlinear dynamical systems may exhibit a collective state, known as cluster synchronization, characterized by the formation of groups, where the units within each cluster converge to the same behavior that is distinct for each cluster. In this brief paper we derive a criterion for local stability in Milnor sense of this synchronous state that is expressed as a simple relationship between the maximum Lyapunov exponent of the uncoupled dynamics, the coupling coefficient, and one eigenvalue of the graph Laplacian spectrum. In the last part of the paper we discuss the application of the criterion to networks design and control.

Scalable $M$-Channel Critically Sampled Filter Banks for Graph Signals

We investigate a scalable M-channel critically sampled filter bank for graph signals, where each of the M filters is supported on a different subband of the graph Laplacian spectrum. For analysis, the graph signal is filtered on each subband and downsampled on a corresponding set of vertices. However, the classical synthesis filters are replaced with interpolation operators. For small graphs, we use a full eigendecomposition of the graph Laplacian to partition the graph vertices such that the mth set comprises a uniqueness set for signals supported on the mth subband. The resulting transform is critically sampled, the dictionary atoms are orthogonal to those supported on different bands, and graph signals are perfectly reconstructable from their analysis coefficients. We also investigate fast versions of the proposed transform that scale efficiently for large, sparse graphs. Issues that arise in this context include designing the filter bank to be more amenable to polynomial approximation, estimating the number of samples required for each band, performing nonuniform random sampling for the filtered signals on each band, and using efficient reconstruction methods. We empirically explore the joint vertex-frequency localization of the dictionary atoms, the sparsity of the analysis coefficients for different classes of signals, the reconstruction error resulting from the numerical approximations, and the ability of the proposed transform to compress piecewise-smooth graph signals. The proposed filter bank also yields a fast, approximate graph Fourier transform with a coarse resolution in the spectral domain.

On the Distance Signless Laplacian Spectrum of Graphs

Let G be a simple, connected graph, D(G) be the distance matrix of G, and Tr(G) be the diagonal matrix of vertex transmissions of G. The distance signless Laplacian matrix of G is defined as $$D^Q(G)=Tr(G)+D(G)$$ , and the distance signless Laplacian spectral radius of G is the largest eigenvalue of $$D^{Q}(G)$$ . In this paper, we study Nordhaus–Gaddum-type inequalities for distance signless Laplacian eigenvalues of graphs and present some new upper and lower bounds on the distance signless Laplacian spectral radius of G and of its line graph L(G), based on other graph-theoretic parameters, and characterize the extremal graphs. Further, we study the distance signless Laplacian spectrum of some graphs obtained by operations.

Results on Laplacian spectra of graphs with pockets

Let F,Hv be simple connected graphs on n and m+1 vertices, respectively. Let v be a specified vertex of Hv and u1,…,uk∈F. Then the graph G=G[F,u1,…,uk,Hv] obtained by taking one copy of F and k copies of Hv, and then attaching the ith copy of Hv to the vertex ui, i=1,…,k, at the vertex v of Hv (identify ui with the vertex v of the ith copy) is called a graph with k pockets. In 2008, Barik raised the question that ‘how far can the Laplacian spectrum of G be described by using the Laplacian spectra of F and Hv?’ and discussed the case when deg(v)=m in Hv. In this article, we study the problem for more general cases and describe the Laplacian spectrum. As an application, we construct new nonisomorphic Laplacian cospectral graphs from the known ones.

Some results on the Laplacian spectra of graphs with pockets

Let F, Hv be simple connected graphs on n and m+1 vertices, respectively. Let v be a specified vertex of Hv and u1,…,uk∈F. Then the graph G=G[F,u1,…,uk,Hv] obtained by taking one copy of F and k copies of Hv, and then attaching the i-th copy of Hv to the vertex ui,i=1,…,k, at the vertex v of Hv (identify ui with the vertex v of the i-th copy) is called a graph with k pockets. In [Barik, S., On the Laplacian spectra of graphs with pockets, Linear and Multilinear Algebra, 56 (2008), 481–490], Barik raised the question that ‘how far can the Laplacian spectrum of G be described by using the Laplacian spectra of F and Hv?’ and discussed the case when deg(v) = m in Hv. In this article, we study the problem for more general cases and describe the Laplacian spectrum. As an application, we construct new nonisomorphic Laplacian cospectral graphs from the known ones.

Effect on normalized graph Laplacian spectrum by motif attachment and duplication

To some extent, graph evolutionary mechanisms can be explained by its spectra. Here, we are interested in two graph operations, namely, motif (subgraph) doubling and attachment that are biologically relevant. We investigate how these two processes affect the spectrum of the normalized graph Laplacian. A high (algebraic) multiplicity of the eigenvalues 1,1±0.5,1±0.5 and others have been observed in the spectrum of many real networks. We attempt to explain the production of distinct eigenvalues by motif doubling and attachment. Results on the eigenvalue 1 are discussed separately.

Performance Analysis of Distributed Consensus on Regular Networks

This paper presents the analysis of the performance of the consensus on regular network structures in information engineering. The consensus protocols introduced are distributed in the sense that each agent only needs information from its neighbors, which reduces the complexity of links between agents significantly. By graph Laplacian spectrum and frequency domain analysis, the performance of the consensus, characterized by the convergence rate and robustness to the communication constraints, are investigated in details. We establish direct connections between the system parameters and the performance of the consensus. Specifically, the convergence rates and the maximum fixed time-delays that can be tolerated by the various regular networks are found explicitly.

An edge-grafting theorem on Laplacian spectra of graphs and its application

Let 𝒰(n, d) be the class of unicyclic graphs on n vertices with diameter d. This article presents an edge-grafting theorem on Laplacian spectra of graphs. By applying this theorem, we determine the unique graph with the maximum Laplacian spectral radius in 𝒰(n, d). This extremal graph is different from that for the corresponding problem on the adjacency spectral radius as done by Liu et al. [Q. Liu, M. Lu, and F. Tian, On the spectral radius of unicyclic graphs with fixed diameter, Linear Algebra Appl. 420 (2007), 449–457].

The Laplacian spectra of graphs with a tree structure§

Trees are very common in the theory and applications of combinatorics. In this article, we consider graphs whose underlying structure is a tree, except that its vertices are graphs in their own right and where adjacent graphs (vertices) are linked by taking their join. We study the spectral properties of the Laplacian matrices of such graphs. It turns out that in order to capture known spectral properties of the Laplacian matrices of trees, it is necessary to consider the Laplacians of vertex-weighted graphs. We focus on the second smallest eigenvalue of such Laplacians and on the properties of their corresponding eigenvector. We characterize the second smallest eigenvalue in terms of the Perron branches of a tree. Finally, we show that our results are applicable to advancing the solution to the problem of whether there exists a graph on n vertices whose Laplacian has the integer eigenvalues 0, 1, …, n − 1. § This study was done when S. Pati visited the Department of Mathematics of University of Connecticut during June–Aug 2006.

Signed (total) domination numbers and Laplacian spectrum of graphs

Let G = (V, E) be a graph of order n and size m. The open neighborhood N(v) of a vertex v ∈ V is the set of all vertices adjacent to v. Its closed neighborhood is N[v] = N(v) ∪ {v}. A signed (total) dominating function on G, is a function f : V → {−1, 1} such that ∑ x∈N[v] f (x) ≥ 1 (∑ x∈N(v) f (x) ≥ 1) for each v ∈ V. The weight of a signed (total) dominating function f is ω(f) = ∑ v∈V f (v). The minimum weight taken over all signed (total) dominating function f of G, denoted by γ s (G)(γ st (G)), is called the signed (total) domination number of G. In the paper, we give bounds on the Laplacian spectrum of G involving the signed domination and signed total domination numbers.

On the Laplacian spectra of graphs with pockets

Let F,H v be simple connected graphs. Let v be a specified vertex of Hv and u 1, … ,u k ∈ F. Then the graph obtained by taking one copy of F and k copies of Hv , and then attaching the ith copy of Hv to the vertex u i i=1, … ,k, at the vertex v of Hv (identify ui with the vertex v of the ith copy) is called a graph with k pockets. The copies of the graph Hv that are attached to the vertices u i ,i=1, … ,k, are referred to as pockets. In this article, we prove some results describing the Laplacian spectrum of G using the Laplacian spectra of F and H v . The complete Laplacian spectrum of G is also described in some particular cases.

A Laplacian spectral method for stereo correspondence

This paper presents a novel algorithm of stereo correspondence by using Laplacian spectra of graphs. Firstly, according to the feature points of two images to be matched, a Laplacian matrix with Gaussian-weighted distance is defined and a closed-form solution is given in terms of the matching matrix constructed on the vectors of eigenspace of the Laplacian matrix. Secondly, we introduce a new method to judge correspondences by using doubly stochastic matrix. Thirdly, in order to render our method robust, we describe an approach to embedding the Laplacian spectral method within the framework of iterative correspondence and transformation estimation. Experimental results show the feasibility and comparatively high accuracy of our methods.

Laplacian spectrum analysis and spanning tree algorithm for circuit partitioning problems

The spectrum of a graph is the set of all eigenvalues of the Laplacian matrix of the graph. There is a closed relationship between the Laplacian spectrum of graphs and some properties of graphs such as connectivity. In the recent years Laplacian spectrum of graphs has been widely applied in many fields. The application of Laplacian spectrum of graphs to circuit partitioning problems is reviewed in this paper. A new criterion of circuit partitioning is proposed and the bounds of the partition ratio for weighted graphs are also presented. Moreover, the deficiency of graph-partitioning algorithms by Laplacian eigenvectors is addressed and an algorithm by means of the minimal spanning tree of a graph is proposed. By virtue of taking the graph structure into consideration this algorithm can fulfill general requirements of circuit partitioning.

Coalescence, majorization, edge valuations and the laplacian spectra of graphs

Let G be a graph and denote by L(G) = D( G) — A(G) the difference between the diagonal matrix of vertex degrees and the adjacency matrix. Denote the spectrum of L(G) by X(G). If G 1 and G 2 are disjoint graphs, then λ( G 1.G 2) majorizes λ(G 1∪ G 2), where G 1.G 2 is any coalescence of G 1 and G 2. An application to the algebraic connectivity of a retraction is given that affirms an earlier conjecture of the authors and V. S. Sunder. An important tool used in the proof of majorization involves the factorization L(G) = Q(G) Q 1(G) where Q(G) is the vertex-edge incidence matrix of G. We are able to use Q(G) to introduce edge valuations of G, particularly when G is a tree.