Spectrum Of Laplacian Research Articles

Spectra of the Dirichlet Laplacian in 3-dimensional polyhedral layers

The structure of the spectrum of the three-dimensional Dirichlet Laplacian in a 3D polyhedral layer of fixed width is studied. It turns out that the essential spectrum is determined by the smallest dihedral angle that forms the boundary of the layer while the discrete spectrum is always finite. An example of a layer with empty discrete spectrum is constructed. The spectrum is proved to be nonempty in regular polyhedral layer.

Optimizing network insights: AI-Driven approaches to circulant graph based on Laplacian spectra

The study of Laplacian and signless Laplacian spectra extends across various fields, including theoretical chemistry, computer science, electrical networks, and complex networks, providing critical insights into the structures of real-world networks and enabling the prediction of their structural properties. A key aspect of this study is the spectrum-based analysis of circulant graphs. Through these analyses, important network measures such as mean-first passage time, average path length, spanning trees, and spectral radius are derived. This research enhances our understanding of the relationship between graph spectra and network characteristics, offering a comprehensive perspective on complex networks. Consequently, it supports the ability to make predictions and conduct analyses across a wide range of scientific disciplines.

The spectrum of an operator associated with G 2-instantons with 1-dimensional singularities and Hermitian Yang–Mills connections with isolated singularities

abstract: This is the first step in an attempt at a deformation theory for $G_{2}$-instantons with $1$-dimensional conic singularities. Under a set of model data, the linearization yields a Dirac operator $P$ on a certain bundle over $\mathbb{S}^{5}$, called the \textit{link operator}. As a dimension reduction, the link operator also arises from Hermitian Yang--Mills connections with isolated conic singularities on a Calabi--Yau $3$-fold. Using the quaternion structure in the Sasakian geometry of $\mathbb{S}^{5}$, we describe the set of all eigenvalues of $P$, denoted by $\Spec P$. We show that $\Spec P$ consists of finitely many integers induced by certain sheaf cohomologies on $\mathbb{P}^{2}$, and infinitely many real numbers induced by the spectrum of the rough Laplacian on the pullback endomorphism bundle over $\mathbb{S}^{5}$. The multiplicities and the form of an eigensection can be described fairly explicitly. In particular, there is a relation between the spectrum on $\mathbb{S}^{5}$ to certain sheaf cohomologies on~$\mathbb{P}^{2}$. Moreover, on a Calabi--Yau $3$-fold, the index of the linearized operator for admissible singular Hermitian Yang--Mills connections is also calculated, in terms of these sheaf cohomologies. Using the representation theory of $\SU(3)$ and the subgroup $S[U(1)\times U(2)]$, we show an example in which $\Spec P$ and the multiplicities can be completely determined.

On the Asymptotic Network Indices of Weighted Three-Layered Structures with Multi-Fan Composed Subgraphs

In this paper, three sorts of network indices for the weighted three-layered graph are studied through the methods of graph spectra theory combined with analysis methods. The concept of union of graphs are applied to design two sorts of weighted layered multi-fan composed graphs, and the accurate mathematical expressions of the network indices are obtained through the derivations of Laplacian spectra; furthermore, the asymptotic properties are also derived. We find that when the cardinalities of the vertices on a sector-edge-link tend to infinity, the indices of FONC and EMFPT are irrelevant with the number of copies of the fan-substructure based on the considered graph framework.

A generalization of Fiedler's lemma and its applications

In this article, taking a Fiedler's result on the spectrum of a matrix formed from two symmetric matrices as a motivation, we deduce a more general result on the eigenvalues of a matrix, which form from n symmetric matrices. As an important application, we obtain the adjacency spectra, Laplacian spectra and signless Laplacian spectra of a graph with a particular almost equitable partition.

Model selection for network data based on spectral information

In this work, we explore the extent to which the spectrum of the graph Laplacian can characterize the probability distribution of random graphs for the purpose of model evaluation and model selection for network data applications. Network data, often represented as a graph, consist of a set of pairwise observations between elements of a population of interests. The statistical network analysis literature has developed many different classes of network data models, with notable model classes including stochastic block models, latent node position models, and exponential families of random graph models. We develop a novel methodology which exploits the information contained in the spectrum of the graph Laplacian to predict the data-generating model from a set of candidate models. Through simulation studies, we explore the extent to which network data models can be differentiated by the spectrum of the graph Laplacian. We demonstrate the potential of our method through two applications to well-studied network data sets and validate our findings against existing analyses in the statistical network analysis literature.

Discrete spectrum of the magnetic Laplacian on almost flat magnetic barriers

The magnetic Laplacian with a step magnetic field has been intensively studied during the last years. We adapt the construction introduced by Bonnaillie-Noël et al. [Bull. London Math. Soc. 56, 2529 (2024)] to prove the existence of bound states of a new effective operator involving a magnetic step field on a domain with an almost flat magnetic barrier. This result emphasizes the fact that even a small non-smoothness of the discontinuity region can cause the appearance of eigenvalues below the essential spectrum. We also give an example where this effective operator arises.

Properties of the Laplacian spectra of certain basic and hierarchical graphs

Properties of the Laplacian spectra of certain basic and hierarchical graphs

Laplacian Spectrum and Energy of the Cyclic Order Product Prime Graph of Semi-dihedral Groups

This paper introduces cyclic order product prime graphs to examine the spectral graph properties of semi-dihedral groups, specifically focusing on the Laplacian spectrum and energy. Combining principles from cyclic graphs and order product prime graphs enhances understanding of group algebraic structures. In this context, the cyclic order product prime graph for a finite group is defined as one where two distinct vertices are adjacent if their generated subgroup is cyclic, and the product of their orders is a prime power. Our methodological approach begins with establishing a general presentation for these graphs within semi-dihedral groups. This foundational step is essential for deriving some properties, such as vertex degrees, the number of edges, and Laplacian characteristic polynomials. This information subsequently facilitates the determination of the Laplacian spectrum, characterized by seven eigenvalues of various multiplicities, and the computation of their Laplacian energy. The semi-dihedral group of order 16 is a particular example to illustrate the practicality and generality of our theorems.

Latent spectral regularization for continual learning

While biological intelligence grows organically as new knowledge is gathered throughout life, Artificial Neural Networks forget catastrophically whenever they face a changing training data distribution. Rehearsal-based Continual Learning (CL) approaches have been established as a versatile and reliable solution to overcome this limitation; however, sudden input disruptions and memory constraints are known to alter the consistency of their predictions. We study this phenomenon by investigating the geometric characteristics of the learner’s latent space and find that replayed data points of different classes increasingly mix up, interfering with classification. Hence, we propose a geometric regularizer that enforces weak requirements on the Laplacian spectrum of the latent space, promoting a partitioning behavior. Our proposal, called Continual Spectral Regularizer for Incremental Learning (CaSpeR-IL), can be easily combined with any rehearsal-based CL approach and improves the performance of SOTA methods on standard benchmarks.

Physical networks as network-of-networks

Physical networks are made of nodes and links that are physical objects embedded in a geometric space. Understanding how the mutual volume exclusion between these elements affects the structure and function of physical networks calls for a suitable generalization of network theory. Here, we introduce a network-of-networks framework where we describe the shape of each extended physical node as a network embedded in space and these networks are bound together by physical links. Relying on this representation, we introduce a minimal model of network growth and we show for a general class of physical networks that volume exclusion induces heterogeneity in both node volume and degree, with the two becoming correlated. These emergent properties strongly affect the dynamics on physical networks: by calculating their Laplacian spectrum as a function of the coupling strength between the nodes we show that degree-volume correlations suppress the role of hubs as early spreaders in diffusive dynamics. We apply the network-of-networks framework to describe several real systems and find properties analog to the minimal model networks. The prevalence of these properties points towards general growth mechanisms that do not depend on the specifics of the systems.

Floquet–Bloch Functions on Non-simply Connected Manifolds, the Aharonov–Bohm Fluxes, and Conformal Invariants of Immersed Surfaces

We define spectral (Bloch) varieties of multidimensional differential operators on non-simply connected manifolds. In their terms we give a description of the analytic dependence of the spectra of magnetic Laplacians on non-simply connected manifolds on the values of the Aharonov–Bohm fluxes, construct analogs of spectral curves for two-dimensional Dirac operators on Riemann surfaces, and thereby find new conformal invariants of immersions of surfaces into three- and four-dimensional Euclidean spaces.

Discrete spectrum of the magnetic Laplacian on perturbed half‐planes

AbstractThe existence of bound states for the magnetic Laplacian in unbounded domains can be quite challenging in the case of a homogeneous magnetic field. We provide an affirmative answer for almost flat corners and slightly curved half‐planes when the total curvature of the boundary is positive.

Geometric structures and the Laplace spectrum, Part I

Geometric structures and the Laplace spectrum, Part I

Community Deception in Large Networks: Through the Lens of Laplacian Spectrum

Many complex networks in the real world have community structures. Typical examples include online social networks and ecology networks. While the identification of communities bears numerous practical applications, with the increasing awareness of data security and privacy concerns, the need to protect the community affiliations of individuals from disclosing by attackers emerges. This raises the community deception (CD) problem, that is, the opposite of community detection, which asks for ways to minimally perturb the network structures by rewiring nodes so that the target communities maximally hide themself from community detection algorithms. To this end, we investigate the CD problem through a Laplacian spectrum lens and propose a method named <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathtt{ComDeceptor}$</tex-math> </inline-formula> to hide a flexible target set of communities, which is more universal than most existing methods that either focus on hiding the entire communities or a single community. The key idea of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathtt{ComDeceptor}$</tex-math> </inline-formula> is to first allocate the resources of perturbations fairly and effectively. By proving that hiding communities through intercommunity edge addition and intracommunity edge deletion correspond to maximizing the second smallest eigenvalue <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\lambda_2$</tex-math> </inline-formula> and minimizing the largest eigenvalue <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\lambda_n$</tex-math> </inline-formula> of the graph Laplacian, respectively, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathtt{ComDeceptor}$</tex-math> </inline-formula> then incorporates efficient heuristics for approximately solving the problems, thus selecting the appropriate edge to perturb. Experimental results over nine real-world networks and six community detection algorithms not only demonstrate the efficiency of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathtt{ComDeceptor}$</tex-math> </inline-formula> , but also the superior performance on obfuscating community structures over the baselines.

On the computation of Seidel Laplacian eigenvalues for graph-based binary codes

Let G=(V,E) be a simple graph with vertex set as V and edge set as E. Within the framework of graph-based binary coding, a graph G is a threshold graph if and only if it can be generated by a binary code of the type 0s11t1…0sk1tk, where si and ti are positive integers with 1≤i≤k. This type of binary code is called a binary generating code of G. In this paper, we address a problem originally posed by Harary and Schwenk in [10] titled “Which graphs have integral spectra?” in which the main focus of our investigations lies in the Seidel Laplacian spectrum of G generated by 0s11t1…0sk1tk. We establish that all eigenvalues of G are integers and multiplicities of eigenvalues are equal to powers of bits 0 and 1 (sizes of strings of 0's and 1's) involved in a binary generating code of G. We exhibit a necessary and sufficient condition in terms of eigenvalues of the matrix associated with a graph-based G binary code to be Seidel Laplacian integral. Furthermore, for any prime p, we explore the Seidel Laplacian spectrum of zero-divisor graphs associated with the ring Zpm (where m is a positive integer) and the polynomial ring Zp[x]/(xp).

The Laplacian spectrum of weighted composite networks and the applications

The topological properties of the networks can be described by the Laplacian spectra, but resolving the Laplacian spectra of networks poses difficulties. In this study, a novel approach for solving the Laplacian spectrum of weighted composite networks is presented. We first give the definitions of three weighted graph operations, namely, Cartesian product, corona, and join. Second, the Laplacian spectra of these composite networks are calculated. Finally, we use the obtained Laplacian spectrum to deduce some topological properties of the networks, such as network coherence, entire mean first-passage time, and Laplacian energy, which have several applications in physical chemistry.

Eigenvalue-Based Incremental Spectral Clustering

Abstract Our previous experiments demonstrated that subsets of collections of (short) documents (with several hundred entries) share a common, normalized in some way, eigenvalue spectrum of combinatorial Laplacian. Based on this insight, we propose a method of incremental spectral clustering. The method consists of the following steps: (1) split the data into manageable subsets, (2) cluster each of the subsets, (3) merge clusters from different subsets based on the eigenvalue spectrum similarity to form clusters of the entire set. This method can be especially useful for clustering methods of complexity strongly increasing with the size of the data sample, like in case of typical spectral clustering. Experiments were performed showing that in fact the clustering and merging of subsets yield clusters close to clustering of the entire dataset. Our approach differs from other research streams in that we rely on the entire set (spectrum) of eigenvalues, whereas the other researchers concentrate on few eigenvectors related to lowest eigenvalues. Such eigenvectors are considered in the literature as of low reliability.

The Laplace spectrum on conformally compact manifolds

We consider the spectrum of the Laplace operator acting on L p \mathcal {L}^p over a conformally compact manifold for 1 ≤ p ≤ ∞ 1 \leq p \leq \infty . We prove that for p ≠ 2 p \neq 2 this spectrum always contains an open region of the complex plane. We further show that the spectrum is contained within a certain parabolic region of the complex plane. These regions depend on the value of p p , the dimension of the manifold, and the values of the sectional curvatures approaching the boundary.

Cross-diffusion induced instability on networks

Abstract The concept of Turing instability, namely that diffusion can destabilize the homogenous steady state, is well known either in the context of partial differential equations (PDEs) or in networks of dynamical systems. Recently, reaction–diffusion equations with non-linear cross-diffusion terms have been investigated, showing an analogous effect called cross-diffusion induced instability. In this article, we consider non-linear cross-diffusion effects on networks of dynamical systems, showing that also in this framework the spectrum of the graph Laplacian determines the instability appearance, as well as the spectrum of the Laplace operator in reaction–diffusion equations. We extend to network dynamics a particular network model for competing species, coming from the PDEs context, for which the non-linear cross-diffusion terms have been justified, e.g. via a fast-reaction limit. In particular, the influence of different topology structures on the cross-diffusion induced instability is highlighted, considering regular rings and lattices, and also small-world, Erdős–Réyni, and Barabási–Albert networks.