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.

Concentration of the adjacency matrix and of the Laplacian in random graphs with independent edges

Abstract: Consider any random graph model where potential edges appear independently, with possibly different probabilities, and assume that the minimum expected degree is !(lnn). We prove that the adjacency matrix and the Laplacian of that random graph are concentrated around the corresponding matrices of the weighted graph whose edge weights are the probabilities in the random model. We apply this result to two different settings. In bond percolation, we show that, whenever the minimum expected degree in the random model is not too small, the Laplacian of the percolated graph is typically close to that of the original graph. As a corollary, we improve upon a bound for the spectral gap of the percolated graph due to Chung and Horn.

Towards a sampling theorem for signals on arbitrary graphs

Abstract: In this paper, we extend the Nyquist-Shannon theory of sampling to signals defined on arbitrary graphs. Using spectral graph theory, we establish a cut-off frequency for all bandlimited graph signals that can be perfectly reconstructed from samples on a given subset of nodes. The result is analogous to the concept of Nyquist frequency in traditional signal processing. We consider practical ways of computing this cut-off and show that it is an improvement over previous results. We also propose a greedy algorithm to search for the smallest possible sampling set that guarantees unique recovery for a signal of given bandwidth. The efficacy of these results is verified through simple examples.

Subexponential Algorithms for Unique Games and Related Problems

Abstract: Subexponential time approximation algorithms are presented for the Unique Games and Small-Set Expansion problems. Specifically, for some absolute constant c, the following two algorithms are presented.(1) An exp(kne)-time algorithm that, given as input a k-alphabet unique game on n variables that has an assignment satisfying 1-ec fraction of its constraints, outputs an assignment satisfying 1-e fraction of the constraints.(2) An exp(ne/δ)-time algorithm that, given as input an n-vertex regular graph that has a set S of δn vertices with edge expansion at most ec, outputs a set S' of at most δ n vertices with edge expansion at most e.subexponential algorithm is also presented with improved approximation to Max Cut, Sparsest Cut, and Vertex Cover on some interesting subclasses of instances. These instances are graphs with low threshold rank, an interesting new graph parameter highlighted by this work.Khot's Unique Games Conjecture (UGC) states that it is NP-hard to achieve approximation guarantees such as ours for Unique Games. While the results here stop short of refuting the UGC, they do suggest that Unique Games are significantly easier than NP-hard problems such as Max 3-Sat, Max 3-Lin, Label Cover, and more, which are believed not to have a subexponential algorithm achieving a nontrivial approximation ratio.Of special interest in these algorithms is a new notion of graph decomposition that may have other applications. Namely, it is shown for every e >0 and every regular n-vertex graph G, by changing at most δ fraction of G's edges, one can break G into disjoint parts so that the stochastic adjacency matrix of the induced graph on each part has at most ne eigenvalues larger than 1-η, where η depends polynomially on e. The subexponential algorithm combines this decomposition with previous algorithms for Unique Games on graphs with few large eigenvalues [Kolla and Tulsiani 2007; Kolla 2010].

Articulated Shape Matching Using Laplacian Eigenfunctions and Unsupervised Point Registration

Abstract: Matching articulated shapes represented by voxel-sets reduces to maximal sub-graph isomorphism when each set is described by a weighted graph. Spectral graph theory can be used to map these graphs onto lower dimensional spaces and match shapes by aligning their embeddings in virtue of their invariance to change of pose. Classical graph isomorphism schemes relying on the ordering of the eigenvalues to align the eigenspaces fail when handling large data-sets or noisy data. We derive a new formulation that finds the best alignment between two congruent $K$-dimensional sets of points by selecting the best subset of eigenfunctions of the Laplacian matrix. The selection is done by matching eigenfunction signatures built with histograms, and the retained set provides a smart initialization for the alignment problem with a considerable impact on the overall performance. Dense shape matching casted into graph matching reduces then, to point registration of embeddings under orthogonal transformations; the registration is solved using the framework of unsupervised clustering and the EM algorithm. Maximal subset matching of non identical shapes is handled by defining an appropriate outlier class. Experimental results on challenging examples show how the algorithm naturally treats changes of topology, shape variations and different sampling densities.