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.

Spectral Estimates for Infinite Quantum Graphs

Abstract: We investigate the bottom of the spectra of infinite quantum graphs, i.e., Laplace operators on metric graphs having infinitely many edges and vertices. We introduce a new definition of the isoperimetric constant for quantum graphs and then prove the Cheeger-type estimate. Our definition of the isoperimetric constant is purely combinatorial and thus it establishes connections with the combinatorial isoperimetric constant, one of the central objects in spectral graph theory and in the theory of simple random walks on graphs. The latter enables us to prove a number of criteria for quantum graphs to be uniformly positive or to have purely discrete spectrum. We demonstrate our findings by considering trees, antitrees and Cayley graphs of finitely generated groups.

Spectral-Based Contractible Parallel Coordinates

Abstract: Parallel coordinates is well-known as a popular tool for visualizing the underlying relationships among variables in high-dimension datasets. However, this representation still suffers from visual clutter arising from intersections among poly line plots especially when the number of data samples and their associated dimension become high. This paper presents a method of alleviating such visual clutter by contracting multiple axes through the analysis of correlation between every pair of variables. In this method, we first construct a graph by connecting axis nodes with an edge weighted by data correlation between the corresponding pair of dimensions, and then reorder the multiple axes by projecting the nodes onto the primary axis obtained through the spectral graph analysis. This allows us to compose a dendrogram tree by recursively merging a pair of the closest axes one by one. Our visualization platform helps the visual interpretation of such axis contraction by plotting the principal component of each data sample along the composite axis. Smooth animation of the associated axis contraction and expansion has also been implemented to enhance the visual readability of behavior inherent in the given high-dimensional datasets.

Node Clustering in Graphs: An Empirical Study

Abstract: Modeling networks is an active area of research and is used for many applications ranging from bioinformatics to social network analysis. An important operation that is often performed in the course of graph analysis is node clustering. Popular methods for node clustering such as the normalized cut method have their roots in graph partition optimization and spectral graph theory. Recently, there has been increasing interest in modeling graphs probabilistically using stochastic block models and other approaches that extend it. In this paper, we present an empirical study that compares the node clustering performances of state-of-the-art algorithms from both the probabilistic and spectral families on undirected graphs. Our experiments show that no family dominates over the other and that network characteristics play a significant role in determining the best model to use.

Effective-Resistance Preserving Spectral Reduction of Graphs

Abstract: This paper proposes a scalable algorithmic framework for effective-resistance preserving spectral reduction of large undirected graphs. The proposed method allows computing much smaller graphs while preserving the key spectral (structural) properties of the original graph. Our framework is built upon the following three key components: a spectrum-preserving node aggregation and reduction scheme, a spectral graph sparsification framework with iterative edge weight scaling, as well as effective-resistance preserving post-scaling and iterative solution refinement schemes. By leveraging recent similarity-aware spectral sparsification method and graph-theoretic algebraic multigrid (AMG) Laplacian solver, a novel constrained stochastic gradient descent (SGD) optimization approach has been proposed for achieving truly scalable performance (nearly-linear complexity) for spectral graph reduction. We show that the resultant spectrally-reduced graphs can robustly preserve the first few nontrivial eigenvalues and eigenvectors of the original graph Laplacian and thus allow for developing highly-scalable spectral graph partitioning and circuit simulation algorithms.ACM Reference Format:Zhiqiang Zhao and Zhuo Feng. 2019. Effective-Resistance Preserving Spectral Reduction of Graphs. In The 56th Annual Design Automation Conference 2019 (DAC '19), June 2–6, 2019, Las Vegas, NV, USA. ACM, New York, NY, USA, 6 pages. https://doi.org/10.1145/3316781.3317809

Monotonicity Properties and Spectral Characterization of Power Redistribution in Cascading Failures

Abstract: In this work, we apply spectral graph theory methods to study the monotonicity and structural properties of power redistribution in a cascading failure process. We demonstrate that in contrast to the lack of monotonicity in physical domain, there is a rich collection of monotonicity one can explore in the spectral domain, leading to a systematic way to define topological metrics that are monotonic. It is further shown that many useful quantities in cascading failure analysis can be unified into a spectral inner product, which itself is related to graphical properties of the transmission network. Such graphical interpretations precisely capture the Kirchhoff's law expressed in terms of graph structural properties and gauge the impact of a line when it is tripped.