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 Sparsification of Graphs

Abstract: We introduce a new notion of graph sparsification based on spectral similarity of graph Laplacians: spectral sparsification requires that the Laplacian quadratic form of the sparsifier approximate that of the original. This is equivalent to saying that the Laplacian of the sparsifier is a good preconditioner for the Laplacian of the original. We prove that every graph has a spectral sparsifier of nearly linear size. Moreover, we present an algorithm that produces spectral sparsifiers in time $O(m\log^{c}m)$, where $m$ is the number of edges in the original graph and $c$ is some absolute constant. This construction is a key component of a nearly linear time algorithm for solving linear equations in diagonally dominant matrices. Our sparsification algorithm makes use of a nearly linear time algorithm for graph partitioning that satisfies a strong guarantee: if the partition it outputs is very unbalanced, then the larger part is contained in a subgraph of high conductance.

Segmentation of 3D meshes through spectral clustering

Abstract: We formulate and apply spectral clustering to 3D mesh segmentation for the first time and report our preliminary findings Given a set of mesh faces, an affinity matrix which encodes the likelihood of each pair of faces belonging to the same group is first constructed Spectral methods then use selected eigenvectors of the affinity matrix or its closely related graph Laplacian to obtain data representations that can be more easily clustered We develop an algorithm that favors segmentation along concave regions, which is inspired by human perception Our algorithm is theoretically sound, efficient, simple to implement, andean achieve high-quality segmentation results on 3D meshes

Graph Convolutional Networks with EigenPooling

Abstract: Graph neural networks, which generalize deep neural network models to graph structured data, have attracted increasing attention in recent years. They usually learn node representations by transforming, propagating and aggregating node features and have been proven to improve the performance of many graph related tasks such as node classification and link prediction. To apply graph neural networks for the graph classification task, approaches to generate thegraph representation from node representations are demanded. A common way is to globally combine the node representations. However, rich structural information is overlooked. Thus a hierarchical pooling procedure is desired to preserve the graph structure during the graph representation learning. There are some recent works on hierarchically learning graph representation analogous to the pooling step in conventional convolutional neural (CNN) networks. However, the local structural information is still largely neglected during the pooling process. In this paper, we introduce a pooling operator $\pooling$ based on graph Fourier transform, which can utilize the node features and local structures during the pooling process. We then design pooling layers based on the pooling operator, which are further combined with traditional GCN convolutional layers to form a graph neural network framework $\m$ for graph classification. Theoretical analysis is provided to understand $\pooling$ from both local and global perspectives. Experimental results of the graph classification task on $6$ commonly used benchmarks demonstrate the effectiveness of the proposed framework.