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.

Study on the Determination and Application of the Cluster Number of the Images Based on the Spectral Graph Theory

Abstract: According to spectral graph theory,a new method of determining the cluster number of image is proposed for resolving the accuracy problem in this paper.The relationship between the number of zeros eigenvalue of Laplace matrix of weighted graph and the cluster number of image is studied.By analyzing the ideal format,block format and general format of Laplace matrix of weighted graph,we have deduced that the cluster number of image is closed to the number of zeros eigenvalue of Laplace matrix of weighted graph via the matrix theory.In the numerical simulations,the parameter of weighted graph is discussed and it shows that we can get the accurate cluster number of image when the parameter is limited to 0.4 and 0.6.Meanwhile,the numerical result employing our method is better than that of using the number of one eigenvalue of weighted matrix.

Minimum Spectral Connectivity Projection Pursuit

Abstract: We study the problem of determining the optimal low dimensional projection for maximising the separability of a binary partition of an unlabelled dataset, as measured by spectral graph theory. This is achieved by finding projections which minimise the second eigenvalue of the graph Laplacian of the projected data, which corresponds to a non-convex, non-smooth optimisation problem. We show that the optimal univariate projection based on spectral connectivity converges to the vector normal to the maximum margin hyperplane through the data, as the scaling parameter is reduced to zero. This establishes a connection between connectivity as measured by spectral graph theory and maximal Euclidean separation. The computational cost associated with each eigen-problem is quadratic in the number of data. To mitigate this issue, we propose an approximation method using microclusters with provable approximation error bounds. Combining multiple binary partitions within a divisive hierarchical model allows us to construct clustering solutions admitting clusters with varying scales and lying within different subspaces. We evaluate the performance of the proposed method on a large collection of benchmark datasets and find that it compares favourably with existing methods for projection pursuit and dimension reduction for data clustering.

Cheeger-type approximation for sparsest $st$-cut

Abstract: We introduce the $st$-cut version the Sparsest-Cut problem, where the goal is to find a cut of minimum sparsity among those separating two distinguished vertices $s,t\in V$. Clearly, this problem is at least as hard as the usual (non-$st$) version. Our main result is a polynomial-time algorithm for the product-demands setting, that produces a cut of sparsity $O(\sqrt{\OPT})$, where $\OPT$ denotes the optimum, and the total edge capacity and the total demand are assumed (by normalization) to be $1$. Our result generalizes the recent work of Trevisan [arXiv, 2013] for the non-$st$ version of the same problem (Sparsest-Cut with product demands), which in turn generalizes the bound achieved by the discrete Cheeger inequality, a cornerstone of Spectral Graph Theory that has numerous applications. Indeed, Cheeger's inequality handles graph conductance, the special case of product demands that are proportional to the vertex (capacitated) degrees. Along the way, we obtain an $O(\log n)$-approximation, where $n=\card{V}$, for the general-demands setting of Sparsest $st$-Cut.

Generalizing p-Laplacian: spectral hypergraph theory and a partitioning algorithm

Abstract: Abstract For hypergraph clustering, various methods have been proposed to define hypergraph p -Laplacians in the literature. This work proposes a general framework for an abstract class of hypergraph p -Laplacians from a differential-geometric view. This class includes previously proposed hypergraph p -Laplacians and also includes previously unstudied novel generalizations. For this abstract class, we extend current spectral theory by providing an extension of nodal domain theory for the eigenvectors of our hypergraph p -Laplacian. We use this nodal domain theory to provide bounds on the eigenvalues via a higher-order Cheeger inequality. Following our extension of spectral theory, we propose a novel hypergraph partitioning algorithm for our generalized p -Laplacian. Our empirical study shows that our algorithm outperforms spectral methods based on existing p -Laplacians.

Closed Walk Sampler: An Efficient Method for Estimating Eigenvalues of Large Graphs

Abstract: Eigenvalues of a graph are of high interest in graph analytics for Big Data due to their relevance to many important properties of the graph including network resilience, community detection and the speed of viral propagation. Accurate computation of eigenvalues of extremely large graphs is usually not feasible due to the prohibitive computational and storage costs and also because full access to many social network graphs is often restricted to most researchers. In this paper, we present a series of new sampling algorithms which solve both of the above-mentioned problems and estimate the two largest eigenvalues of a large graph efficiently and with high accuracy. Unlike previous methods which try to extract a subgraph with the most influential nodes, our algorithms sample only a small portion of the large graph via a simple random walk, and arrive at estimates of the two largest eigenvalues by estimating the number of closed walks of a certain length. Our experimental results using real graphs show that our algorithms are substantially faster while also achieving significantly better accuracy on most graphs than the current state-of-the-art algorithms.