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.

Accurately Modeling the Resting Brain Functional Correlations Using Wave Equation With Spatiotemporal Varying Hypergraph Laplacian

Abstract: How spontaneous brain neural activities emerge from the underlying anatomical architecture, characterized by structural connectivity (SC), has puzzled researchers for a long time. Over the past decades, much effort has been directed toward the graph modeling of SC, in which the brain SC is generally considered as relatively invariant. However, the graph representation of SC is unable to directly describe the connections between anatomically unconnected brain regions and fail to model the negative functional correlations. Here, we extend the static graph model to a spatiotemporal varying hypergraph Laplacian diffusion (STV-HGLD) model to describe the propagation of the spontaneous neural activity in human brain by incorporating the Laplacian of the hypergraph representation of the structural connectome ( h SC) into the regular wave equation. Theoretical solution shows that the dynamic functional couplings between brain regions fluctuate in the form of an exponential wave regulated by the spatiotemporal varying Laplacian of h SC. Empirical study suggests that the cortical wave might give rise to resonance with SC during the self-organizing interplay between excitation and inhibition among brain regions, which orchestrates the cortical waves propagating with harmonics emanating from the h SC while being bound by the natural frequencies of SC. Besides, the average statistical dependencies between brain regions, normally defined as the functional connectivity (FC), arises just at the moment before the cortical wave reaches the steady state after the wave spreads across all the brain regions. Comprehensive tests on four extensively studied empirical brain connectome datasets with different resolutions confirm our theory and findings.

A Graph Theoretic Approach for Certain Properties of Spectral Null Codes

Abstract: Yeh and Parhami [6] introduced the concept of the index-permutation graph model, which is an extension of the Cayley graph model and applied it to the systematic development of communication-efficient interconnection networks. Inspiring the concept of building a relationship between an index and a permutation symbol, we make use in this chapter of the spectral null equations variables in each grouping by representing only their corresponding indices in a permutation sequence form. In another way, these indices will be presented by a permutation sequence, where the symbols refer to the position of the corresponding variables in the spectral null equation.

GraphFlow: A New Graph Convolutional Network Based on Parallel Flows.

Abstract: In view of the huge success of convolution neural networks (CNN) for image classification and object recognition, there have been attempts to generalize the method to general graph-structured data. One major direction is based on spectral graph theory and graph signal processing. In this paper, we study the problem from a completely different perspective, by introducing parallel flow decomposition of graphs. The essential idea is to decompose a graph into families of non-intersecting one dimensional (1D) paths, after which, we may apply a 1D CNN along each family of paths. We demonstrate that the our method, which we call GraphFlow, is able to transfer CNN architectures to general graphs. To show the effectiveness of our approach, we test our method on the classical MNIST dataset, synthetic datasets on network information propagation and a news article classification dataset.

Analysis of the 2-sum problem and the spectral algorithm

Abstract: This paper presents the analysis of the 2-sum problem and the spectral algorithm. The spectral algorithm was proposed by Barnard, Pothen and Simon in [1]; its heuristic properties have been advocated by George and Pothen in [4] by formulation of the 2-sum problem as a Quadratic Assignment Problem. In contrast to that analysis another approach is proposed: permutations are considered as vectors of Euclidian space. This approach enables one to prove the bound results originally obtained in [4] in an easier way. The geometry of permutations is considered in order to explain what are ‘good’ and ‘pathological’ situations for the spectral algorithm. Upper bounds for approximate solutions generated by the spectral algorithm are proved. The results of numerical computations on (graphs of) large sparse matrices from real-world applications are presented to support the obtained results and illustrate considerations related to the ‘pathological’ cases.

On the Ky Fan $k$-norm of the $LI$- and $QI$-matrix of graphs

Abstract: Let $A(G)$ and $D(G)$ be the adjacency matrix and the degree diagonal matrix of a graph $G$, respectively. Then $L(G)=D(G)-A(G)$ and $Q(G)=D(G)+A(G)$ are called Laplacian matrix and signless Laplacian matrix of the graph $G$, respectively. Let $G$ be a graph with $n$ vertices and $m$ edges. Then the $LI$-matrix and $QI$-matrix of $G$ are defined as $$LI(G)=L(G)-\frac{2m}{n}I_n \quad \text{and} \quad QI(G)=Q(G)-\frac{2m}{n}I_n,$$ where $I_n$ is the identity matrix. In this paper, we are interested in extremal properties of the Ky Fan $k$-norm of the $LI$-matrix and $QI$-matrix of graphs, which are closely related to the well-known problems and results in spectral graph theory, such as the (signless) Laplacian spectral radius, the (signless) Laplacian spread, the sum of the $k$ largest (signless) Laplacian eigenvalues, the (signless) Laplacian energy, and other parameters. Upper and lower bounds on the Ky Fan $k$-norm of the $LI$-matrix and $QI$-matrix of graphs are given, and the extremal graphs are partly characterized. In addition, upper and lower bounds on the Ky Fan $k$-norm of $LI$-matrix and $QI$-matrix of trees, unicyclic graphs and bicyclic graphs are determined, and the corresponding extremal graphs are characterized.