Spectral graph theory

About: Spectral graph theory is a research topic. Over the lifetime, 1334 publications have been published within this topic receiving 77373 citations.

Scalable Sparse Topologies with Small Spectrum

Abstract: One of the fundamental properties of a graph is the number of distinct eigenvalues of its adjacency or Laplacian matrix. Determining this number is of theoretical interest and also of practical impact. Graphs with small spectra exhibit many symmetry properties and are well suited as interconnection topologies. Especially load balancing can be done on such interconnection topologies in a small number of steps. In this paper we are interested in graphs with maximal degree O(log n), where n is the number of vertices, and with a small number of distinct eigenvalues. Our goal is to find scalable families of such graphs with polylogarithmic spectrum in the number of vertices. We present also the eigenvalues of the Butterfly graph.

Sampling Technique Analysis of Nyström Approximation in Pixel-Wise Affinity Matrix

Abstract: Spectral graph methods are widely employed in image segmentation, and they exhibit excellent performance. However, for high-resolution images, it is impractical to directly calculate the eigenvectors of the affinity matrix owing to the high computational requirements. The Nystrom method provides an efficient way to approximate the large-scale affinity matrix by low-rank approximation. In the machine learning field, previous studies have mainly focused on less data points with high dimensional features. To the best of our knowledge, this is the first study to discuss the performance of sampling methods for Nystrom approximation, in which we focus on the pixel-wise affinity matrix for a single image. In this paper, we propose a mean-shift segmentation-based Nystrom sampling technique for image analysis. The experimental results show that for images with simple compositions and backgrounds, k-means sampling performs better, whereas for images with more complicated compositions and backgrounds, the proposed method can perform better.

Adiabatic Quantum Computation

Abstract: The quantum adiabatic theorem ensures that a slowly changing system, initially prepared in its ground state, will evolve to its final ground state with arbitrary precision. As a first result this thesis extends the original theorem to projection operators keeping the statement valid for Hamiltonians with degenerate ground spaces. Yet the main focus of this work lies in studying the efficiency of quantum circuit simulations by adabatic quantum computation. The standard Hamiltonian construction by Kitaev is based on a path graph reflecting the $L$ computation steps and influencing the scaling of the necessary evolution time by its spectral gap of $\mathcal{O}\left(\frac{1}{L^2}\right)$. Aspiring to an improved running time we generalize Kitaev's Hamiltonian to so-called standard graph Hamiltonians based on graph families with a different spectral gap. In this generalized construction the first two time derivatives of the Hamiltonian and the fraction of initial vertices appear as additional parameters of running time. In a first step the time derivatives can be proven to be constant. Expansion results from spectral graph theory however impose a trade-off between the spectral gap and the fraction of initial vertices as well as the fraction of final vertices which corresponds to the probability for obtaining the correct computational output. Graphs with spectral gap $\mathcal{O}\left(\frac{1}{L^k}\right)$, $k<2$, turn out to contradict very likely, graphs with $k<1$ even for sure at least one of the efficiency criterias for running time, output probability or Hamiltonian implementation. The above results may also be obstacles for a possible quantum PCP-theorem in complexity theory claiming the local Hamiltonian problem with constant gap to be QMA-complete since the very same Kitaev Hamiltonian is constructed in the QMA-hardness proof for the Local Hamiltonian problem.

Spectral Graph Analysis with Apache Spark

Abstract: Graphs are the cornerstone of many algorithms pertaining to various network analyses. When the problem's dimensionality is relatively small, expressed in the number of vertices and edges of a graph, then most methods perform adequately well. As the problem size increases, more compute power is required. Distributed computing is a one viable option to address this issue, but it cannot scale indefinitely. At one point, it is necessary to turn to heuristic approaches. Spectral graph theory is an example of such approximate scheme. In this paper, we combine spectral analysis with distributed computing using Apache Spark. The paper is accompanied with a publicly available proof of concept implementation. The system was extensively performance tested, and the results show a superb fit of Apache Spark to the purpose of spectral graph analysis. Furthermore, the resulting code is straightforward thankfully to Spark's intuitive distributed programming model, and well-designed APIs.