Normalized Laplacian Matrix Research Articles

Characterization of graphs with some normalized Laplacian eigenvalue of multiplicity n − 3

Graphs with few distinct eigenvalues have been investigated extensively. In this paper, we focus on a related problem: characterizing graphs with some eigenvalue of large multiplicity. Specifically, the normalized Laplacian matrix of a graph is considered here. Let ρn−1(G) and ν(G) be the second smallest normalized Laplacian eigenvalue and the independence number of a graph G, respectively. As the main conclusions, two families of n-vertex connected graphs with some normalized Laplacian eigenvalue of multiplicity n−3 are determined: graphs with ρn−1(G)=1 and graphs with ρn−1(G)≠1 and ν(G)=3. Moreover, it is proved that these graphs are determined by their normalized Laplacian spectrum.

Constructing cospectral bipartite graphs

Given a bipartite graph G=(V,E) with bipartition V=A∪B, where |A|=|B|. We present a new construction of pairs of cospectral bipartite graphs by “rotating” G in two different ways. More precisely, let l and k be two positive integers with l≠k. Take l copies of A (resp. B) and k copies of B (resp. A), say A1,A2,…,Al and B1,B2,…,Bk (resp. B1′,B2′,…,Bl′ and A1′,A2′,…,Ak′). Let Γ (resp. Γ′) be the graph with vertex set V(Γ)=A1∪A2∪⋯∪Al∪B1∪B2∪⋯∪Bk (resp. V(Γ′)=B1′∪B2′∪⋯∪Bl′∪A1′∪A2′∪⋯∪Ak′), such that the vertices of Γ (resp. Γ′) are adjacent if and only if they are adjacent in G. We show that Γ and Γ′ are cospectral with respect to the adjacency matrix, as well as the normalized Laplacian matrix. Moreover, we give a simple necessary and sufficient condition for Γ and Γ′ to be non-isomorphic. The main ingredient of the proof involves a use of Hall’s Marriage Theorem.

Spectral analysis for weighted iterated pentagonal graphs and its applications

Deterministic weighted networks have been widely used to model real-world complex systems. In this paper, we study the weighted iterated pentagonal networks. From the construction of the network, we derive recursive relations of all eigenvalues and their multiplicities of its normalized Laplacian matrix from the two successive generations of the weighted iterated pentagonal networks. As applications of spectra of the normalized Laplacian matrix, we study the Kemeny’s constant, the multiplicative degree-Kirchhoff index, and the number of weighted spanning trees and derive their exact closed-form expressions for the weighted iterated pentagonal networks.

Zeta-equivalent digraphs: Simultaneous cospectrality

We introduce a zeta function of digraphs that determines, and is determined by, the spectra of all linear combinations of the adjacency matrix, its transpose, the out-degree matrix, and the in-degree matrix. In particular, zeta-equivalence of graphs encompasses simultaneous cospectrality with respect to the adjacency, the Laplacian, the signless Laplacian, and the normalized Laplacian matrix. In addition, we express zeta-equivalence in terms of Markov chains and in terms of invasions where each edge is replaced by a fixed digraph. We finish with a method for constructing zeta-equivalent digraphs.

Node proximity preserved dynamic network embedding via matrix perturbation

In recent years, network embedding has attracted extensive interests, which aims at representing nodes of an original network in a low-dimensional vector space while preserving the inherent topological structures of the network. Despite the remarkable advantages of complex networks, most existing network embedding methods are mainly focused on static networks while ignoring the evolving characteristic, which is proved to be an essential property of real-world networks. In this paper, we propose Node Proximity Preserved Dynamic Network Embedding via Matrix Perturbation (NPDNE) to tackle the dilemma. Specifically, our method implements a low-rank transformation on the normalized Laplacian matrix of the given networks and then derives the embedding vectors through generalized SVD. Subsequently, the node proximities are preserved in the embedding vectors by exploiting the eigen-decomposition reweighing theorem, which reveals the intrinsic relationship among different-order proximities. Moreover, a generalized eigen perturbation is adopted to update the embedding vectors so that the evolution of given networks can be captured over time. Finally, we conduct experiments of multi-label classification, link prediction, and visualization on several real-world datasets. The experimental results demonstrate the superiority of the proposed NPDNE model compared with state-of-the-art baselines.

Normalized Laplacian spectrum of complete multipartite graphs

The spectrum of the normalized Laplacian matrix of a graph provides many structural information of the graph, and it has many applications in numerous areas and in different guises. Let G be a complete k-partite graph with k≥3. In this paper, we give the necessary and sufficient condition for G which is determined by their normalized Laplacian spectrum. Moreover, we obtain a majorization theory of normalized Laplacian spectral radius of G, which enables us to find the maximal and minimal normalized Laplacian spectral radii among all complete k-partite graphs with fixed order, respectively.

Spectral analysis for weighted iterated q-triangulations of graphs

Much information about the structural properties and dynamical aspects of a network is measured by the eigenvalues of its normalized Laplacian matrix. In this paper, we aim to present a first study on the spectra of the normalized Laplacian of weighed iterated [Formula: see text]-triangulations of graphs. We analytically obtain all the eigenvalues, as well as their multiplicities from two successive generations. As examples of application of these results, we then derive closed-form expressions for their Kemeny’s constant and multiplicative Kirchhoff index. Simulation example is also provided to demonstrate the effectiveness of the theoretical analysis.

Spectral analysis for weighted iterated q-triangulation networks.

Deterministic weighted networks have been widely used to model real-world complex systems. In this paper, we study the weighted iterated q-triangulation networks, which are generated by iteration operation F(⋅). We add q(q∈N+) new nodes on each old edge and connect them with two endpoints of the old edge. At the same time, the newly linked edges are given weight factor r(0<r≤1). From the construction of the network, we obtain all the eigenvalues and their multiplicities of its normalized Laplacian matrix from the two successive generations of the weighted iterated q-triangulation network. Further, as applications of spectra of the normalized Laplacian matrix, we study the Kemeny constant, the multiplicative degree-Kirchhoff index, and the number of weighted spanning trees and derive their exact closed-form expressions for weighted iterated q-triangulation networks.

Colouring the Normalized Laplacian

We apply Cauchy's interlacing theorem to derive some eigenvalue bounds to the chromatic number using the normalized Laplacian matrix, including a combinatorial characterization of when equality occurs. Further, we introduce some new expansion type of parameters which generalize the Cheeger constant of a graph, and relate them to the colourings which meet our eigenvalue bound with equality. Finally, we point out a connection to the Erdős-Faber-Lovász conjecture.

The normalized Laplacian polynomial of rooted product of graphs

Let [Formula: see text] be a simple graph with [Formula: see text] vertices and [Formula: see text] be a sequence of [Formula: see text] rooted graphs [Formula: see text]. The rooted product [Formula: see text], of [Formula: see text] by [Formula: see text] is constructed by identifying the root vertex of [Formula: see text] with the [Formula: see text]th vertex of [Formula: see text]. In this paper, the characteristic polynomial of the normalized Laplacian matrix of [Formula: see text] is obtained. As an application of our results, we obtain the normalized Laplacian polynomial and spectrum of the generalized Bethe trees.

Thermodynamic edge entropy in Alzheimer’s disease

In this paper, we explore how to decompose the global thermodynamic entropy of a network into components associated with its edges. Commencing from a statistical mechanical picture, in which the normalised Laplacian matrix plays the role of Hamiltonian operator, thermodynamic entropy can be calculated from partition function associated with different energy level occupation distributions arising from Maxwell–Boltzmann statistics. Using the spectral decomposition of the Laplacian, we show how to project the edge-entropy components so that the detailed distribution of entropy across the edges of a network can be achieved. We apply the resulting method to the brain functional connectivity networks using BOLD-fMRI data. The entropic measurement turns out to be an effective tool for the diagnosis of Alzheimer’s disease by finding the most salient functional connectivity features from the corresponding anatomical brain regions.

Applications of Laplacian spectrum for the vertex–vertex graph

Complex networks have attracted a great deal of attention from scientific communities, and have been proven as a useful tool to characterize the topologies and dynamics of real and human-made complex systems. Laplacian spectrum of the considered networks plays an essential role in their network properties, which have a wide range of applications in chemistry and others. Firstly, we define one vertex–vertex graph. Then, we deduce the recursive relationship of its eigenvalues at two successive generations of the normalized Laplacian matrix, and we obtain the Laplacian spectrum for vertex–vertex graph. Finally, we show the applications of the Laplacian spectrum, i.e. first-order network coherence, second-order network coherence, Kirchhoff index, spanning tree, and Laplacian-energy-like.

Hermitian normalized Laplacian matrix for directed networks

In this paper, we extend and generalize the spectral theory of undirected networks towards directed networks by introducing the Hermitian normalized Laplacian matrix for directed networks. In order to start, we discuss the Courant–Fischer theorem for the eigenvalues of Hermitian normalized Laplacian matrix. Based on the Courant–Fischer theorem, we obtain a similar result towards the normalized Laplacian matrix of undirected networks: for each i ∈ {1, 2,…, n}, any eigenvalue of Hermitian normalized Laplacian matrix λi ∈ [0, 2]. Moreover, we prove some special conditions if 0, or 2 is an eigenvalue of the Hermitian normalized Laplacian matrix L(X). On top of that, we investigate the symmetry of the eigenvalues of L(X) and the edge-version for the eigenvalue interlacing result. Finally we present two expressions for the coefficients of the characteristic polynomial of the Hermitian normalized Laplacian matrix. As an outlook, we sketch some novel and intriguing problems to which our apparatus could generally be applied.

Information Propagation Analysis of Social Network Using the Universality of Random Matrix

Spectral graph theory gives an algebraical approach to analyze the dynamics of a network by using the matrix that represents the network structure. However, it is not easy for social networks to apply the spectral graph theory because the matrix elements cannot be given exactly to represent the structure of a social network. The matrix element should be set on the basis of the relationship between persons, but the relationship cannot be quantified accurately from obtainable data (e.g., call history and chat history). To get around this problem, we utilize the universality of random matrix with the feature of social networks. As such random matrix, we use normalized Laplacian matrix for a network where link weights are randomly given. In this paper, we first clarify that the universality (i.e., the Wigner semicircle law) of the normalized Laplacian matrix appears in the eigenvalue frequency distribution regardless of the link weight distribution. Then, we analyze the information propagation speed by using the spectral graph theory and the universality of the normalized Laplacian matrix. As the results, we show that the worst-case speed of the information propagation changes at most 2 if the structure (i.e., relationship among people) of a social network changes.

On the Normalized Laplacian Permanental Polynomial of a Graph

Let M be an $$n \times n$$ matrix with entries $$m_{ij}$$ ( $$i,j=1,2,\ldots ,n$$ ). The permanent of M is defined to be $$\begin{aligned} \mathrm{per}(M)=\sum _{\sigma }\prod _{i=1}^nm_{i\sigma (i)}, \end{aligned}$$ where the sum is taken over all permutations $$\sigma $$ of $$\{1, 2, \ldots , n\}$$ . The permanental polynomial of M is defined by $$\mathrm{per}(xI_n-M),$$ where $$I_n$$ is the identity matrix of size n. Let G be a graph with n vertices. The normalized Laplacian matrix is defined to be $$\mathcal {L}(G)=D(G)^{-1/2}(D(G)-A(G))D(G)^{-1/2}$$ , where A(G) and D(G) denote the adjacency matrix and degree matrix of G, respectively. The permanental polynomial $$\mathrm{per}(xI_n-\mathcal {L}(G))$$ is called the normalized Laplacian permanental polynomial of G. In this paper, we first give the formula to compute the coefficients of the normalized Laplacian permanental polynomial of a graph, and then we give recursive formulas to compute the normalized Laplacian permanental polynomial of a graph.

Thermodynamic Entropy in Quantum Statistics for Stock Market Networks

The stock market is a dynamical system composed of intricate relationships between financial entities, such as banks, corporations, and institutions. Such a complex interactive system can be represented by the network structure. The underlying mechanism of stock exchange establishes a time‐evolving network among companies and individuals, which characterise the correlations of stock prices in the time sequential trades. Here, we develop a novel technique in quantum statistics to analyse the financial market evolution. We commence from heat bath analogy where the normalised Laplacian matrix plays the role of the Hamiltonian operator of the network. The eigenvalues of the Hamiltonian specify energy states of the network. These states are occupied by either indistinguishable bosons or fermions with corresponding Bose‐Einstein and Fermi‐Dirac statistics. Using the relevant partition functions, we develop the thermodynamic entropy to explore dynamic network characterisations. We conduct the experiments to apply this novel method to identify the significant variance in network structure during the financial crisis. The thermodynamic entropy provides an excellent framework to represent the variations taking place in the stock market.

Applications of Laplacian spectrum for the weighted scale-free network with a weight factor

Laplacian spectrum gives a lot of useful information about complex structural properties and relevant dynamical aspects, which has attracted the attention of mathematicians. We introduced the weighted scale-free network inspired by the binary scale-free network. First, the weighted scale-free network with a weight factor is constructed by an iterative way. In the next step, we use the definition of eigenvalue and eigenvector to obtain the recursive relationship of its eigenvalues and multiplicities at two successive generations. Through analysis of eigenvalues of transition weight matrix we find that multiplicities of eigenvalues 0 of transition matrix are different for the binary scale-free network and the weighted scale-free network. Then, we obtain the eigenvalues for the normalized Laplacian matrix of the weighted scale-free network by using the obtained eigenvalues of transition weight matrix. Finally, we show some applications of the Laplacian spectrum in calculating eigentime identity and Kirchhoff index. The leading term of these indexes are completely different for the binary and the weighted scale-free network.

Spectral analysis for weighted iterated quadrilateral graphs

Much information about the structural properties and dynamical aspects of a network is measured by the eigenvalues of its normalized Laplacian matrix. In this paper, we aim to present a first study on the spectra of the normalized Laplacian matrix of weighted iterated quadrilateral graphs. We analytically obtain all the eigenvalues, as well as their multiplicities from two successive generations. As an example of application of these results, we then derive closed-form expressions for the multiplicative degree Kirchhoff index and the Kemeny’s constant, as well as the number of weighted spanning trees.

Limit theorems for eigenvectors of the normalized Laplacian for random graphs

We prove a central limit theorem for the components of the eigenvectors corresponding to the $d$ largest eigenvalues of the normalized Laplacian matrix of a finite dimensional random dot product graph. As a corollary, we show that for stochastic blockmodel graphs, the rows of the spectral embedding of the normalized Laplacian converge to multivariate normals and furthermore the mean and the covariance matrix of each row are functions of the associated vertex's block membership. Together with prior results for the eigenvectors of the adjacency matrix, we then compare, via the Chernoff information between multivariate normal distributions, how the choice of embedding method impacts subsequent inference. We demonstrate that neither embedding method dominates with respect to the inference task of recovering the latent block assignments.

Eigentime identity of the weighted scale-free triangulation networks for weight-dependent walk

The eigenvalues of the normalized Laplacian matrix of a network provide information on its structural properties and some relevant dynamical aspects, in particular for weight-dependent walk. In order to get the eigentime identity for weight-dependent walk, we need to obtain the eigenvalues and their multiplicities of the Laplacian matrix. Firstly, the model of the weighted scale-free triangulation networks is constructed. Then, the eigenvalues and their multiplicities of transition weight matrix are presented, after the recursive relationship of those eigenvalues at two successive generations are given. Consequently, the Laplacian spectrum is obtained. Finally, the analytical expression of the eigentime identity, indicating that the eigentime identity grows sublinearly with the network order, is deduced.