Identifying multiple influential spreaders in complex networks based on spectral graph theory

One of the hot research topics in propagation dynamics is identifying a set of critical nodes that can influence maximization in a complex network.The importance and dispersion of critical nodes among them are both vital factors that can influence maximization.We therefore propose a multiple influential spreaders identification algorithm based on spectral graph theory.This algorithm first quantifies the role played by the local structure of nodes in the propagation process,then classifies the nodes based on the eigenvectors of the Laplace matrix,and finally selects a set of critical nodes by the constraint that nodes in the same class are not adjacent to each other while different classes of nodes can be adjacent to each other.Experimental results on real and synthetic networks show that our algorithm outperforms the state-of-the-art and classical algorithms in the SIR model.

Clustering in Wireless Multimedia Sensor Networks Using Spectral Graph Partitioning

Wireless multimedia sensor network (WMSN) consists of sensors that can monitor multimedia data from its surrounding, such as capturing image, video and audio. To transmit multimedia information, large energy is required which decreases the lifetime of the network. In this paper we present a clustering approach based on spectral graph partitioning (SGP) for WMSN that increases the lifetime of the network. The efficient strategies for cluster head selection and rotation are also proposed.

On the Spectral Properties of Graphs with Rank 4

Let G be a graph and A(G) the adjacency matrix of G. The spectrum of G is the eigenvalues together with their multiplicities of A(G). Chang et al. (2011) characterized the structures of all graphs with rank 4. Monsalve and Rada (2021) gave the bound of spectral radius of all graphs with rank 4. Based on these results as above, we further investigate the spectral properties of graphs with rank 4. And we give the expressions of the spectral radius and energy of all graphs with rank 4. In particular, we show that some graphs with rank 4 are determined by their spectra.

Automatic 3D Shape Co-Segmentation Using Spectral Graph Method

Co-analyzing a set of 3D shapes is a challenging task considering a large geometrical variability of the shapes. To address this challenge, this paper proposes a new automatic 3D shape co-segmentation algorithm by using spectral graph method. Our method firstly represents input shapes as a set of weighted graphs and extracts multiple geometric features to measure the similarities of faces in each individual shape. Secondly all graphs are embedded into the spectral domain to find meaningful correspondences across the set, After that we build a joint weighted matrix for the graph set and then apply normalized cut criterion to find optimal co-segmentation of the input shapes. Finally we evaluate our approach on different categories of 3D shapes, and the experimental results demonstrate that our method can accurately co-segment a wide variety of shapes, which may have different poses and significant topology changes.

Image registration based on matrix perturbation analysis using spectral graph

We present a novel perspective on characterizing the spectral correspondence between nodes of the weighted graph with application to image registration. It is based on matrix perturbation analysis on the spectral graph. The contribution may be divided into three parts. Firstly, the perturbation matrix is obtained by perturbing the matrix of graph model. Secondly, an orthogonal matrix is obtained based on an optimal parameter, which can better capture correspondence features. Thirdly, the optimal matching matrix is proposed by adjusting signs of orthogonal matrix for image registration. Experiments on both synthetic images and real-world images demonstrate the effectiveness and accuracy of the proposed method.

Nonsmooth critical point theory and applications to the spectral graph theory

Existing critical point theories including metric and topological critical point theories are difficult to be applied directly to some concrete problems in particular polyhedral settings,because the notions of critical sets could be either very vague or too large.To overcome these difficulties,we develop the critical point theory for nonsmooth but Lipschitzian functions defined on convex polyhedrons.This yields natural extensions of classical results in the critical point theory,such as the Liusternik-Schnirelmann multiplicity theorem.More importantly,eigenvectors for some eigenvalue problems involving graph 1-Laplacian coincide with critical points of the corresponding functions on polytopes,which indicates that the critical point theory proposed in the present paper can be applied to study the nonlinear spectral graph theory.

Sharp upper bounds for the adjacency and the signless Laplacian spectral radius of graphs

Let G be a simple graph with n vertices and m edges. In this paper, we present some new upper bounds for the adjacency and the signless Laplacian spectral radius of graphs in which every pair of adjacent vertices has at least one common adjacent vertex. Our results improve some known upper bounds. The main tool we use here is the Lagrange identity.

Bipartite Graphs with the First and Second Largest Laplacian Spectral Radius

Let kn be the class of bipartite graphs with n vertices and k cut edges.The extremal graphs with the first and the second largest Laplacian spectral radius among all graphs in kn are presented.The bounds of the Laplacian spectral radius of these extremal graphs are also obtained.

Some notes on the spectral perturbations of the signless Laplacian of a graph

Let G be a simple graph and let Q（G） be the signless Laplacian matrix of G. In this paper we obtain some results on the spectral perturbation of the matrix Q（G） under an edge addition or an edge contraction.

The Signless Laplacian Spectral Radius of Some Special Bipartite Graphs

This paper mainly researches on the signless laplacian spectral radius of bipartite graphs Dr(m1,m2;n1,n2). We consider how the signless laplacian spectral radius of Dr(m1,m2;n1,n2)?changes under some special cases. As application, we give two upper bounds on the signless laplacian spectral radius of Dr(m1,m2;n1,n2), and determine the graphs that obtain the upper bounds.

The Spectral Radii of Some Adhesive Graphs

The spectral radius of a graph is the maximum eigenvalues of its adjacency matrix. In this paper, using the property of quotient graph, the sharp upper bounds for the spectral radii of some adhesive graphs are determined.

On the Signless Laplacian Spectral Radius of C4-free k-cyclic Graphs

A k-cyclic graph is a connected graph of order n and size n + k-1. In this paper, we determine the maximal signless Laplacian spectral radius and the corresponding extremal graph among all C_4-free k-cyclic graphs of order n. Furthermore, we determine the first three unicycles and bicyclic, C_4-free graphs whose spectral radius of the signless Laplacian is maximal. Similar results are obtained for the(combinatorial)

非平衡符号双圈图的拉普拉斯谱半径的排序

研究了非平衡符号双圈图的第一到第六大的拉普拉斯特征值的分布规律,完善了现有结论中一些不准确的情况,推广了现有的结果,并给出了取得极值情况的图例。

图卷积神经网络综述

图卷积神经网络是图论与深度学习的交叉,已成为机器学习领域的研究热点。基于此,介绍了图卷积神经网络的形成,梳理了两大类经典的图卷积神经网络:谱方法和空间方法,详细介绍了这两类图卷积神经网络模型,分析了图卷积操作的核心理论基础,介绍了图卷积神经网络在各领域的应用,总结了图卷积神经网络面临的主要挑战,展望了图卷积神经网络的发展趋势,并分析了图卷积神经网络在野外环境下蝴蝶识别任务中的潜在应用。

关于赋权非正则图的A_(α)特征值和特征向量

设G_(ω)=(G,ω)是一个赋权图,其邻接矩阵和赋权度对角矩阵分别A(G_(ω))和D(G_(ω))。对于α∈[0,1],G_(ω)的A_(α)-矩阵为A_(α)(G_(ω))=αD(G_(ω))+(1-α)A(G_(ω))。对于连通赋权非正则图G_(ω),给出了其关于A_(α)-特征值的一些界,并得到了A_(α)-谱半径对应的特征向量中最大分量与最小分量比值的下界。

面向节点分类任务的节点级自适应图卷积神经网络

图神经网络通过对图中节点的递归采样与聚合以学习节点嵌入,而现有方法中节点采样与聚合的模式较固定,对局部模式的多样性捕获存在不足,从而降低模型性能.因此,文中提出节点级自适应图卷积神经网络(Node-Level Adaptive Graph Convolutional Neural Network,NA-GCN).设计基于节点重要性的采样策略,自适应地确定各节点的邻域规模.同时,提出基于自注意力机制的聚合策略,自适应地融合给定邻域内的节点信息.在多个基准图数据集上的实验表明,NA-GCN在节点分类任务上具有较优性能.

集中式电力市场下基于谱聚类与图注意力网络的电力枢纽节点设计方法研究

电力枢纽节点是现货市场上可以进行统一交易的聚合节点,也是电力金融市场发展与稳定运行的基石,对构建统一电力市场体系、完善市场功能具有十分重要的意义。从一般属性、节点选择、权重设计、时间性的角度分析了枢纽节点的设计方式,并针对性地提出了基于谱聚类的枢纽区域划分方法与基于图注意力网络的权重计算方法。基于双尺度度量的谱聚类分区策略,其核心是建立以关键线路功率传输分布因子、多时段节点电价和节点邻接性为度量的双尺度耦合矩阵,优先保证枢纽区域的地理特性、区内的无阻塞特性和节点连通性。基于图注意力网络建立了定价节点聚合权重的分配模型,通过自注意力机制来对邻域节点进行聚合,实现不同时段节点电价下节点权重的自适应匹配。最后通过IEEE118节点算例对所提出的设计方法进行验证,通过横向对比,证明了所提方法的合理性与有效性,可为实际市场中枢纽节点设定提供理论支撑。

基于加权张量低秩约束的多视图谱聚类

现有基于图的多视图聚类方法通常难以同时考虑不同视图的潜在高阶相关信息和每个视图内的全局几何结构,导致聚类性能受限。为此,提出一种基于加权张量低秩约束的多视图谱聚类方法(WTLR-MSC)。根据多视图数据构建概率转移矩阵,将所有的概率转移矩阵构建为三阶张量,并借助鲁棒主成分分析思想将其分解为目标张量和误差张量。使用加权张量核范数约束目标张量的旋转张量,利用奇异值先验信息准确挖掘多视图数据的潜在高阶相关信息,并利用核范数约束目标张量的每个正切片以刻画每个视图内的全局几何结构。基于此建立数学模型,并设计有效的求解算法。在BBCSport、BBC4View、COIL20、UCI Digits 4个常用数据集上的实验结果表明,WTLR-MSC较ERLRT、MCA~2M、MGL-WTNN等聚类方法的性能有显著提升,准确率、标准化互信息、F1值、精确率、召回率相较于次优方法最高提升约1.3、1.0、1.2、1.6和0.8个百分点,大幅增强了多视图聚类的稳健性。

给定点连通度的图的补图的无符号拉普拉斯谱半径

假设G是一个具有点集V(G)={v_(1),v_(2),…,v_(n)}和边集E(G)的连通简单图,矩阵Q(G)=D(G)+A(G)被称为图G的无符号拉普拉斯矩阵,其中D(G)和A(G)分别是图G的度对角矩阵和邻接矩阵。称矩阵Q(G)的最大特征值为图G的无符号拉普拉斯谱半径。图G的补图记为G^(c)=(V(G^(c))),E(G^(c)),这里V(G^(c))=V(G)和E(G^(c))={xy|x,y∈V(G),xy∉E(G)}.文章在给定点连通度且直径大于3的图的所有补图中,确定了无符号拉普拉斯谱半径达到最小时的唯一图。