Spectral methods on the hydrogen atom

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

Humans routinely solve problems of immense computational complexity by intuitively forming simple, low-dimensional heuristic strategies. Citizen science (or crowd sourcing) is a way of exploiting this ability by presenting scientific research problems to non-experts. 'Gamification'--the application of game elements in a non-game context--is an effective tool with which to enable citizen scientists to provide solutions to research problems. The citizen science games Foldit, EteRNA and EyeWire have been used successfully to study protein and RNA folding and neuron mapping, but so far gamification has not been applied to problems in quantum physics. Here we report on Quantum Moves, an online platform gamifying optimization problems in quantum physics. We show that human players are able to find solutions to difficult problems associated with the task of quantum computing. Players succeed where purely numerical optimization fails, and analyses of their solutions provide insights into the problem of optimization of a more profound and general nature. Using player strategies, we have thus developed a few-parameter heuristic optimization method that efficiently outperforms the most prominent established numerical methods. The numerical complexity associated with time-optimal solutions increases for shorter process durations. To understand this better, we produced a low-dimensional rendering of the optimization landscape. This rendering reveals why traditional optimization methods fail near the quantum speed limit (that is, the shortest process duration with perfect fidelity). Combined analyses of optimization landscapes and heuristic solution strategies may benefit wider classes of optimization problems in quantum physics and beyond.

In this thesis, we focus on two problems in extremal graph theory. Extremal graph theory consists of all problems related to optimizing parameters defined on graphs. The concept of ``editing'' appears in many key results and techniques in extremal graph theory, either as a means to account for error in structural results, or as a quantity to minimize or maximize. A typical problem in spectral extremal graph theory seeks relationships between the extremes of certain graph parameters and the extremes of eigenvalues commonly associated to graphs. The \emph{edit distance problem} asks the following problem: for any fixed ``forbidden'' graph $F$, how many ``edits'' are needed to ensure that any graph on $n$ vertices can be made to contain no induced copies of $F$. If $F$ is a complete graph, then Tur\'{a}n's Theorem, an early fundamental result in extremal graph theory, provides a precise answer. The \emph{edit distance function} plays an essential role in answering this question and relates to the \emph{speed} of a graph hereditary property $\hh$ as well as the $\hh$-chromatic number of a random graph. The main techniques revolve around so-called \emph{colored regularity graphs (CRGs)}. We find an asymptotically almost sure formula for the edit distance function when $F$ is an Erd\H{o}s-R\'{e}nyi random graph whose density lies in $[1-1/\phi, 1/\phi]\approx [0.382¸0.618]$. As an intermediate step, we make several advances on the application of CRGs, such as the introduction and application of \emph{$p$-prohibited CRGs}. %In \emph{spectral graph theory}, we ask: given graph $G$ and some matrix $M$ which may be naturally associated to $G$, what do the eigenvalues of $M$ say about $G$? For any $n$-vertex graph $G$, its adjacency matrix $A = A_G$ is the $\{0,1\}$-valued $n\times n$ matrix whose $(u,v)$ entry indicates whether $uv$ is an edge of $G$. In $1999$, Gregory, Hershkowitz, and Kirkland defined the \emph{(adjacency) spread} of a graph as the difference between the maximum and minimum eigenvalues of its adjacency matrix. In their paper, since cited $68$ times, the authors conjectured that the graph on $n$ vertices which maximizes spread is the join of a complete graph on $\lfloor 2n/3\rfloor$ vertices with an independent set on $\lceil n/3\rceil$ vertices. We prove this claim for all $n$ sufficiently large. As an intermediate step, we prove an analogous result for the eigenvalues of \emph{graphons} (equivalently, kernel operators on symmetric functions $W:[0,1]^2\to [0,1]$).

Spectral graph theory and its applications constitute an important forward step in modern network theory. Its increasing consensus over the last decades fostered the development of innovative tools, allowing network theory to model a variety of different scenarios while answering questions of increasing complexity. Nevertheless, a comprehensive understanding of spectral graph theory’s principles requires a solid technical background which, in many cases, prevents its diffusion through the scientific community. To overcome such an issue, we developed and released an open-source MATLAB toolbox - SPectral graph theory And Random walK (SPARK) toolbox - that combines spectral graph theory and random walk concepts to provide a both static and dynamic characterization of digraphs. Once described the theoretical principles grounding the toolbox, we presented SPARK structure and the list of available indices and measures. SPARK was then tested in a variety of scenarios including: two-toy examples on synthetic networks, an example using public datasets in which SPARK was used as an unsupervised binary classifier and a real data scenario relying on functional brain networks extracted from the EEG data recorded from two stroke patients in resting state condition. Results from both synthetic and real data showed that indices extracted using SPARK toolbox allow to correctly characterize the topology of a bi-compartmental network. Furthermore, they could also be used to find the “optimal” vertex set partition (i.e., the one that minimizes the number of between-cluster links) for the underlying network and compare it to a given a priori partition. Finally, the application to real EEG-based networks provides a practical case study where the SPARK toolbox was used to describe networks’ alterations in stroke patients and put them in relation to their motor impairment.

Towards a spectral theory of graphs based on the signless Laplacian, II

Online social networks (such as Facebook, Twitter, VKontakte, etc.) being an important channel for disseminating information are often used to arrange an impact on the social consciousness for various purposes - from advertising products or services to the full-scale information war thereby making them to be a very relevant object of research. The paper reviewed the analysis methods of social networks (primarily, online), based on the spectral theory of graphs. Such methods use the spectrum of the social graph, i.e. a set of eigenvalues of its adjacency matrix, and also the eigenvectors of the adjacency matrix. Described measures of centrality (in particular, centrality based on the eigenvector and PageRank), which reflect a degree of impact one or another user of the social network has. A very popular PageRank measure uses, as a measure of centrality, the graph vertices, the final probabilities of the Markov chain, whose matrix of transition probabilities is calculated on the basis of the adjacency matrix of the social graph. The vector of final probabilities is an eigenvector of the matrix of transition probabilities. Presented a method of dividing the graph vertices into two groups. It is based on maximizing the network modularity by computing the eigenvector of the modularity matrix. Considered a method for detecting bots based on the non-randomness measure of a graph to be computed using the spectral coordinates of vertices - sets of eigenvector components of the adjacency matrix of a social graph. In general, there are a number of algorithms to analyse social networks based on the spectral theory of graphs. These algorithms show very good results, but their disadvantage is the relatively high (albeit polynomial) computational complexity for large graphs. At the same time it is obvious that the practical application capacity of the spectral graph theory methods is still underestimated, and it may be used as a basis to develop new methods. The work was carried out with the support from the RFBR grant No. 16-29-09517.

Humans are better than computers at performing certain tasks because of their intuition and superior visual processing. Video games are now being used to channel these abilities to solve problems in quantum physics. See Letter p.210 This paper from a team at Aarhus University, Denmark, describes the development of Quantum Moves, an online platform that brings the power of citizen science and game-playing to optimization problems in quantum physics. Jacob Sherson and colleagues have designed a game in which players are asked to find optimal ways of moving optical tweezers in a quantum computing architecture. While brute-force numerical optimization of this problem fails, the players' solutions provide a basis for an optimization method superior to traditional methods. Quantum physics has the reputation of being difficult and unintuitive, but this study shows that even here player intuition can lead to new scientific insights.

This paper describes a hierarchical spectral method for the correspondence matching of point-sets. Conventional spectral methods for correspondence matching are notoriously susceptible to differences in the relational structure of the point-sets under consideration. In this paper we demonstrate how the method can be rendered robust to structural differences by adopting a hierarchical approach. We show how the point-clusters associated with the most significant spectral modes can be used to locate correspondences when significant contamination is present. Spectral graph theory is a term applied to a family of techniques that aim to characterise the global structural properties of graphs using the eigenvalues and eigenvectors of the adjacency matrix [1]. Although the subject has found widespread use in a number of areas including structural chemistry and routeing theory, there have been relatively few applications in the computer vision literature. The reason for this is that although elegant, spectral graph representations are notoriously susceptible to the effect of structural error. In other words, spectral graph theory can furnish very efficient methods for characterising exact relational structures, but soon breaks down when there are spurious nodes and edges in the graphs under study. There are several concrete examples in the pattern analysis literature. Umeyama has an eigendecomposition method that recovers the permutation matrix that maximises the correlation or overlap of the adjacency matrices for graphs of the same size [13]. Horaud and Sossa [5] have adopted a purely structural approach to the recognition of linedrawings. Their representation is based on the immanantal polynomials for the Laplacian matrix of the line-connectivity graph. By comparing the coefficients of the polynomials, they are able to index into a large data-base of line-drawings. Shapiro and Brady [11] have developed a method which draws on a representation which uses weighted edges. They commence from a weighted adjacency matrix (or proximity matrix) which is obtained using a Gaussian function of the distances between pairs of points. The eigen-vectors of the adjacency matrix can be viewed as the basis vectors of an orthogonal transformation on the original point identities. In other words, the components of the eigenvectors represent mixing angles for the transformed points. Matching between different point-sets is effected by comparing the pattern of eigenvectors in different images. Finally, a number of authors have used spectral methods to perform pairwise clustering on image data. Shi and Malik [12] use the second eigenvalue to segment grey-scale images by performing an eigen-decomposition on a matrix of pairwise attribute differences. Inoue and Urahama [6] have shown how the sequential extraction of eigen-modes can be used to cluster pairwise

A spectral graph theory is a theory in which graphs are studied by means of eigenvalues of a matrix M which is in a prescribed way defined for any graph. This theory is called M-theory. We outline a spectral theory of graphs based on the signless Laplacians Q and compare it with other spectral theories, in particular with those based on the adjacency matrix A and the Laplacian L. The Q-theory can be composed using various connections to other theories: equivalency with A-theory and L-theory for regular graphs, or with L-theory for bipartite graphs, general analogies with A-theory and analogies with A-theory via line graphs and subdivision graphs. We present results on graph operations, inequalities for eigenvalues and reconstruction problems.

Spectra of uniform hypergraphs

currently those algorithms to mine the alarm association rules are limited to the minimal support, so that they can only obtain the association rules among the frequently occurring alarms. This paper proposes a new mining algorithm based on spectral graph theory. The algorithms firstly sets up alarm association model with time series; Secondly, it regards alarms database as a high-dimensional structure and treats alarms with associated characteristics as part of it. The algorithm discovers the underlying mapping low-dimensional structure embedding in high-dimensional space based on spectral graph theory. Experimental results based on synthetic and real datasets demonstrates that this algorithm not only discoveries association among alarms, but also acquires the fault in the telecommunications network based on the spectral graph transformation.

This paper presents a novel data processing algorithm. This algorithm is used to solve the problem of incomplete and misaligned of point cloud data due to the complexity of nuclear power containment cone-cylinder forgings and the limitation of laser scanner. Based on spectral graph theory and Hungarian matching, this paper first introduces the lazy random walk, and point cloud state vector is calculated during the walk to judge the local information, thereby eliminate the influence of noise. Then, characteristic edges are extracted using spectral graph theory. Additionally, the feature descriptors are calculated and the cost matrix is constructed using the feature descriptors. The Hungarian algorithm is applied for feature matching, facilitating a coarse registration of the point clouds. Finally, the improved point-to-plane iteration closest point is used for fine registration to ensure accurate alignment between point clouds. The experimental results demonstrate the algorithm's effectiveness in the registration of point clouds for nuclear power containment cone-cylinder forgings.

Recent progress in quantum computing shows the need to incorporate many branches of mathematics (graph theory, matrix theory, optimization, theory of orthogonal polynomials and more) into physics, computer science and chemistry. At the 2024 SIAM Quantum Intersections Convening, Bert de Jong (Lawrence Berkeley National Laboratory) gave a talk entitled ‘Quantum Science Needs Mathematicians’ (Report of the SIAM Quantum Intersections Convening. Integrating Mathematical Scientists into Quantum Research, 7–9 October 2024, Tysons, Virginia (doi:10.11337/25M1741017)), since despite the growing demand for research in these domains, the mathematical sciences community has remained largely disengaged from quantum research, with only a few isolated areas of active involvement. This issue brings together researchers from different areas of mathematics to show the relation between spectral graph theory, the theory of orthogonal polynomials and numerical analysis. This interconnectedness highlights the versatility and importance of these areas of mathematics in the context of quantum computing.This article is part of the theme issue ‘Numerical analysis, spectral graph theory, orthogonal polynomials and quantum algorithms’.

Spectral Graph Theory is the study of the spectra of certain matrices defined from a given graph, including the adjacency matrix, the Laplacian matrix and other related matrices. Graphspectra have been studied extensively for more than fifty years. In the last fifteen years, interest has developed in the study of generalized Laplacian matrices of a graph, that is, real symmetric matrices with negative off-diagonal entries in the positions described by the edges of the graph (and zero in every other off-diagonal position). The set of all real symmetric matrices having nonzero off-diagonal entries exactly where the graphG has edges is denoted by S(G). Given a graph G, the problem of characterizing the possible spectra of B, such that B ∈ S(G), has been referred to as the Inverse Eigenvalue Problem of a Graph. In the last fifteen years a number of papers on this problem have appeared, primarily concerning trees. The adjacency matrix and Laplacian matrix of G and their normalized forms are all in S(G).Recent work on generalized Laplacians and Colin de Verdi`ere matrices is bringing the two areascloser together. This paper surveys results in Spectral Graph Theory and the Inverse EigenvalueProblem of a Graph, examines the connections between these problems, and presents some newresults on construction of a matrix of minimum rank for a given graph having a special form suchas a 0,1-matrix or a generalized Laplacian.

In this work, we described software surrounding's for protocol analysis and learning about computer networks by using spectral graph theory. Software surrounding's consists of the computer network simulators and software surrounding‘s for spectral graph analysis which allows students get to know Internetworking technologies. System made in Java programming language, allows editing of the arbitrary topology computer network with switches, hubs, routers, and work stations. Configuration of elements of the network is done by using standard Windows or line interface. The visual tracking of IP packages, and also the content of the IP packages and frames is allowed. Basic component of this system is software package for generating corresponding graph and calculating the basic parameters from spectral graph theory and analysis of computer network of the given topology and type of protocol.