Laplacian Matrix Research Articles

Modulational instability in photonic Lieb lattice: a graph Laplacian approach

Modulational instability in photonic Lieb lattice: a graph Laplacian approach

Network community detection via neural embeddings

Recent advances in machine learning research have produced powerful neural graph embedding methods, which learn useful, low-dimensional vector representations of network data. These neural methods for graph embedding excel in graph machine learning tasks and are now widely adopted. However, how and why these methods work—particularly how network structure gets encoded in the embedding—remain largely unexplained. Here, we show that node2vec—shallow, linear neural network—encodes communities into separable clusters better than random partitioning down to the information-theoretic detectability limit for the stochastic block models. We show that this is due to the equivalence between the embedding learned by node2vec and the spectral embedding via the eigenvectors of the symmetric normalized Laplacian matrix. Numerical simulations demonstrate that node2vec is capable of learning communities on sparse graphs generated by the stochastic blockmodel, as well as on sparse degree-heterogeneous networks. Our results highlight the features of graph neural networks that enable them to separate communities in the embedding space.

A Laplacian eigenbasis for threshold graphs

Abstract Let G G be a graph on n n vertices. In this article, we prove that an eigenbasis of the Laplacian matrix of a star graph of order n n is also an eigenbasis of G G if and only if G G is a threshold graph. As an application of this spectral characterization, we show an infinite family of threshold graphs that are weakly Hadamard diagonalizable.

Graphs whose Laplacian eigenvalues are almost all 1 or 2

Abstract We explicitly determine all connected graphs whose Laplacian matrices have at most four eigenvalues different from 1 and 2.

Distributed Optimization Control for the System with Second-Order Dynamic

No matter whether with constraint or without constraint, most of the research about distributed optimization is studied for the kind of quadratic performance criteria that does not have an integrator; these optimization problems only concern the state value at the time of the final state, not the whole process of the system change. For this problem, this paper discusses second-order multi-agent systems with a discrete-time dynamic and a continuous-time dynamic, respectively, for distributed optimization control problems, and proposes sufficient conditions to ensure the quadratic performance criteria with an integrator is positive. Specifically, under sufficient conditions, we describe the multi-agent systems that are considered in this paper to be connected topology; all the agents can obtain the information from their neighbors. In addition, the structure of our controller only relies on the Laplace matrix of the system’s topology, and the reaction coefficients in the controller are the parameters in the performance criteria. Finally, the analysis of convergence is given and verified by numerical examples and simulations.

The number of spanning trees in K n -chain and ring graphs

The problem of counting spanning trees of graphs or networks is a fundamental and crucial area of research in combinatorics, while has numerous important applications in statistical physics, network theory and theoretical computer science. Very recently, Kosar, Zaman, Ali and Ullah obtained a nice formula on the number of spanning trees of a K 5-chain network K5ℓ constructed by connecting ℓcopys of complete graphs K 5. They made extensive use of matrix theory and spectral graph theory, especially the normalized Laplacian of graphs. In this paper, by using a rather simple and more physical treatment (the mesh-star transformation in electrical network) without any linear algebra, we generalize their result to K n -chain graphs and K n -ring graphs. The results show that there is a simple relation between the number of spanning trees of the K n -chain graph Lnℓ and the K n -ring graph Cnℓ . We also calculate the corresponding tree entropy (or so called ‘the asymptotic growth constant’) and find that the tree entropy of the corresponding K n -chain graphs and K n -ring graphs are totally the same.

Device placement using Laplacian PCA and graph attention networks

Abstract The exponential growth in data and parameters in modern neural networks has created the need to distribute these models across multiple devices for efficient training, resulting in the device placement problem. Existing graph encoding approaches for device placement suffer from low efficiency when searching for optimal parallel strategies, primarily due to suboptimal positional information retrieval. To address these challenges, we propose the Laplacian Principal Component Analysis-graph attention networks (LPCA-GAT) model. Firstly, we employ GAT to capture complex relationships between nodes and generate node encodings. Secondly, we leverage LPCA on the graph Laplacian matrix to extract crucial low-dimensional positional information. Finally, by integrating these two components, we obtain the final node encodings. This enhances the representation capability of nodes within the graph, enabling efficient device placement. The experimental results demonstrate that LPCA-GAT achieves superior device placement results, specifically accelerating execution and computation time by 13.24% and 96.38%, respectively, leading to significant improvements in both operational efficiency and performance.

Bayesian inference of frequency-specific functional connectivity in MEG imaging using a spectral graph model

Abstract Understanding the relationship between structural connectivity (SC) and functional connectivity (FC) of the human brain is an important goal of neuroscience. Highly detailed mathematical models of neural masses exist that can simulate the interactions between functional activity and structural wiring. These models are often complex and require intensive computation. Most importantly, they do not provide a direct or intuitive interpretation of this structure–function relationship. In this study, we employ the emerging concepts of spectral graph theory to obtain this mapping in terms of graph harmonics, which are eigenvectors of the structural graph’s Laplacian matrix. In order to imbue these harmonics with biophysical underpinnings, we leverage recent advances in parsimonious spectral graph modeling (SGM) of brain activity. Here, we show that such a model can indeed be cast in terms of graph harmonics, and can provide a closed-form prediction of FC in an arbitrary frequency band. The model requires only three global, spatially invariant parameters, yet is capable of generating rich FC patterns in different frequency bands. Only a few harmonics are sufficient to reproduce realistic FC patterns. We applied the method to predict FC obtained from pairwise magnitude coherence of source-reconstructed resting-state magnetoencephalography (MEG) recordings of 36 healthy subjects. To enable efficient model inference, we adopted a deep neural network-based Bayesian procedure called simulation-based inference. Using this tool, we were able to speedily infer not only the single most likely model parameters, but also their full posterior distributions. We also implemented several other benchmark methods relating SC to FC, including graph diffusion and coupled neural mass models. The present method was shown to give the best performance overall. Notably, we discovered that a single biophysical parameterization is capable of fitting FCs from all relevant frequency bands simultaneously, an aspect that did not receive adequate attention in prior computational studies.

Consensus for discrete-time second-order multi-agent systems in the presence of noises and semi-Markovian switching topologies

In this paper, we focus on the control problem of mean square consensus for second-order continuous-time multi-agent systems with multiplicative noises under Markovian switching topologies. A new Lyapunov function based on the Laplacian matrix of the corresponding union topology of all possible topologies is designed for the stochastic stability analysis of consensus. Applying matrix theory and stochastic stability for stochastic differential equations, we can analyze the consensus problem of the considered systems with the designed Lyapunov function. In addition, we present the sufficient conditions of the mean square consensus exponentially for the considered stochastic systems. Finally, we give an simulation example to numerically validate our theoretical results.

Leader-Following Sampled-Data Consensus of Multiagent Systems With Successive Packet Losses and Stochastic Sampling.

In practical applications, sampled-data systems are often affected by unforeseen physical constraints that cause the sampling interval to deviate from the expected value and fluctuate according to a certain probability distribution. This probability distribution can be determined in advance through statistical analysis. Taking into account this stochastic sampling interval, this article focuses on addressing the leader-following sampled-data consensus problem for linear multiagent systems (MASs) with successive packet losses. First, the relationship of the equivalent sampling interval between two successive update instants is established, taking into account the double randomness introduced by both SPLs and stochastic sampling. Then, the equivalent discrete-time MAS is obtained, and the overall leader-following consensus problem is formulated as a stochastic stability problem of the equivalent system by incorporating the sampled-data consensus protocol and properties of the Laplacian matrix. Based on the equivalent the discrete-time system, a consensus criterion is derived under a directed graph by using the Lyapunov theory and leveraging a vectorization technique. By the introduction of a matrix reconstruction approach, the mathematical expectation of a product of three matrices, including the system matrix and its transpose, can be determined. Then, the consensus protocol gain is designed. Finally, an example is provided to validate our theoretical results.

Laplacian {−1,0,1}- and {−1,1}-diagonalizable graphs

A graph is called Laplacian integral if the eigenvalues of its Laplacian matrix are all integers. We investigate the subset of these graphs whose Laplacian is furthermore diagonalized by a matrix with entries coming from a fixed set, in particular, the sets {−1,0,1} or {−1,1}. Such graphs include as special cases the recently-investigated families of Hadamard-diagonalizable and weakly Hadamard-diagonalizable graphs. As a combinatorial tool to aid in our investigation, we introduce a family of vectors that we call balanced, which generalizes totally balanced partitions, regular sequences, and complete partitions. We show that balanced vectors completely characterize which graph complements and complete multipartite graphs are {−1,0,1}-diagonalizable, and we furthermore prove results on diagonalizability of the Cartesian product, disjoint union, and join of graphs. Particular attention is paid to the {−1,0,1}- and {−1,1}-diagonalizability of the complete graphs and complete multipartite graphs. Finally, we provide a complete list of all simple, connected graphs on nine or fewer vertices that are {−1,0,1}- or {−1,1}-diagonalizable.

Gradient estimates for unbounded Laplacians with ellipticity condition on graphs

In this article, we prove various gradient estimates for unbounded graph Laplacians which satisfy the ellipticity condition. Unlike common assumptions for unbounded Laplacians, i.e. completeness and non-degenerate measure, the ellipticity condition is purely local that is easy to verify on a graph. First, we establish an equivalent semigroup property, namely the gradient estimate of exponential curvature-dimension inequality, which is a modification of the curvature-dimension inequality and can be viewed as a notion of curvature on graphs. Additionally, we use the semigroup method to prove the Li-Yau inequalities and the Hamilton inequality for unbounded Laplacians on graphs with the ellipticity condition.

Similarity-Induced Weighted Consensus Laplacian Matrix Learning for Multiview Clustering

Similarity-Induced Weighted Consensus Laplacian Matrix Learning for Multiview Clustering

Adaptive Distributed Control of Nonlinear Multiagent Systems With Event-Triggered for Communication Faults and Dead-Zone Inputs.

This article studies the containment control problem of nonlinear multiagent systems (MASs) subjected to communication link faults and dead-zone inputs. In case of an unknown fault in the communication link, there is no constant Laplacian matrix anymore and each follower agent cannot be informed of the global information simultaneously. To deal with this problem, an adaptive compensating estimator is constructed to estimate the signal spanned by the leaders. Instead of using the linear filter, a nonlinear filter is employed, which both solves the classical complexity explosion in the traditional backstepping method and flushes out the usefulness of the boundary layer error. Considering the dead zone input, we propose two event-triggered schemes, that is, the update-triggered scheme and the transmit-triggered scheme. In the former, the threshold function involves the tracking errors and additional dynamic variable, which can provide the desirable tradeoff between the containment control performance of the considered MASs and saving communication resources. In the latter, the triggered condition is designed according to the characteristic of dead zone, which makes the communication burden be reduced further. Following the backstepping design framework, an adaptive containment control is constructed, it is shown that the containment error can converge to an adjustable residual set even if MASs are subjected to the unknown and bounded communication link faults and dead-zone inputs. Finally, an example is given to show the effectiveness of the proposed results.

On the eigenvalues of the distance signless Laplacian matrix of graphs

Let G be a connected graph and let DQ(G) be the distance signless Laplacian matrix of G with eigenvalues ρ1≥ ρ2≥…≥ ρn. The spread of the matrix DQ}(G) is defined as s(DQ(G)) := maxi,j| ρi-ρj| = ρ1- ρn. We derive new bounds for the distance signless Laplacian spectral radius ρ1 of G. We establish a relationship between the distance signless Laplacian energy and the spread of DQ(G). For a real number α ≠ 0, the graph invariant mα (G) is the sum of the α -th power of the distance signless Laplacian eigenvalues of G. Finally, we obtain various bounds for the graph invariant mα(G).

On Energy of Prime Ideal Graph of a Commutative Ring Associated with Transmission-Based Matrices

This research explores the energy of the prime ideal graph of a commutative ring. The study demonstrates the energy formula of the graph associated with transmission-based matrices. Through research, the findings highlight the distance, Wiener-Hosoya, and distance signless Laplacian matrices. It should be noted that the distance and Wiener-Hosoya energies are always twice their spectral radius, meanwhile, it does not hold for distance signless Laplacian energy.

Extreme values of the Fiedler vector on trees

Let G be a tree on n vertices and let L=D−A denote the Laplacian matrix on G. The second-smallest eigenvalue λ2(G)>0, also known as the algebraic connectivity, as well as the associated eigenvector have been of substantial interest. We investigate the question of when the maxima and minima of an associated eigenvector are assumed at the endpoints of the longest path in G. Our results also apply to more general graphs that ‘behave globally’ like a tree but can exhibit more complicated local structure. The crucial new ingredient is a reproducing formula for eigenvectors of graphs.

Reachable set control of singular semi-Markov jump multi-agent systems with input delay via state decomposition method

This paper addresses the challenge of reachable set estimation in singular multi-agent systems with time-delay and semi-Markov switching topologies. The primary goal is to construct an ellipsoid that encompasses all states originating from the origin. Furthermore, criteria for reachable sets with reduced conservatism are developed. The singular multi-agent systems are reduced by the Laplace matrix, then the Lyapunov–Krasovskii functional is constructed by using decomposed state vector. In addition, the sufficient conditions for the reachable set of singular multi-agent systems bounded by ellipsoids are given by matrix theory and linear matrix inequality. Numerical examples show that our results is effective.

Recover then aggregate: unified cross-modal deep clustering with global structural information for single-cell data.

Single-cell cross-modal joint clustering has been extensively utilized to investigate the tumor microenvironment. Although numerous approaches have been suggested, accurate clustering remains the main challenge. First, the gene expression matrix frequently contains numerous missing values due to measurement limitations. The majority of existing clustering methods treat it as a typical multi-modal dataset without further processing. Few methods conduct recovery before clustering and do not sufficiently engage with the underlying research, leading to suboptimal outcomes. Additionally, the existing cross-modal information fusion strategy does not ensure consistency of representations across different modes, potentially leading to the integration of conflicting information, which could degrade performance. To address these challenges, we propose the 'Recover then Aggregate' strategy and introduce the Unified Cross-Modal Deep Clustering model. Specifically, we have developed a data augmentation technique based on neighborhood similarity, iteratively imposing rank constraints on the Laplacian matrix, thus updating the similarity matrix and recovering dropout events. Concurrently, we integrate cross-modal features and employ contrastive learning to align modality-specific representations with consistent ones, enhancing the effective integration of diverse modal information. Comprehensive experiments on five real-world multi-modal datasets have demonstrated this method's superior effectiveness in single-cell clustering tasks.

Bi-stochastically normalized graph Laplacian: convergence to manifold Laplacian and robustness to outlier noise.

Bi-stochastic normalization provides an alternative normalization of graph Laplacians in graph-based data analysis and can be computed efficiently by Sinkhorn-Knopp (SK) iterations. This paper proves the convergence of bi-stochastically normalized graph Laplacian to manifold (weighted-)Laplacian with rates, when [Formula: see text] data points are i.i.d. sampled from a general [Formula: see text]-dimensional manifold embedded in a possibly high-dimensional space. Under certain joint limit of [Formula: see text] and kernel bandwidth [Formula: see text], the point-wise convergence rate of the graph Laplacian operator (under 2-norm) is proved to be [Formula: see text] at finite large [Formula: see text] up to log factors, achieved at the scaling of [Formula: see text]. When the manifold data are corrupted by outlier noise, we theoretically prove the graph Laplacian point-wise consistency which matches the rate for clean manifold data plus an additional term proportional to the boundedness of the inner-products of the noise vectors among themselves and with data vectors. Motivated by our analysis, which suggests that not exact bi-stochastic normalization but an approximate one will achieve the same consistency rate, we propose an approximate and constrained matrix scaling problem that can be solved by SK iterations with early termination. Numerical experiments support our theoretical results and show the robustness of bi-stochastically normalized graph Laplacian to high-dimensional outlier noise.