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.

Random walks on a tree with applications.

Abstract: The various dynamics of many complex systems, both natural and artificial, can be well described as random walks on a newly proposed yet powerful object known as a complex network. Here we consider random walks on a special kind of network, a tree. Using the methods of probability-generating functions, we derive a close connection between the widely studied parameters mean first-passage time M and mean shortest path length A on trees. This suggests that for a given tree, the analytic value for M can be easily obtained by calculating the value A when determining the latter is more convenient and vice versa. As a result, the well-known T graph is selected as one of various applications of our methods, and we then obtain an exact solution to its quantity M. On the one hand, the result addressed here is in perfect agreement with previous ones. On the other hand, our method is easier to manipulate than most preexisting ones, for instance, methods from spectral graph theory, since no complicated techniques are involved.

Network Density of States

Abstract: Spectral analysis connects graph structure to the eigenvalues and eigenvectors of associated matrices. Much of spectral graph theory descends directly from spectral geometry, the study of differentiable manifolds through the spectra of associated differential operators. But the translation from spectral geometry to spectral graph theory has largely focused on results involving only a few extreme eigenvalues and their associated eigenvalues. Unlike in geometry, the study of graphs through the overall distribution of eigenvalues - the spectral density - is largely limited to simple random graph models. The interior of the spectrum of real-world graphs remains largely unexplored, difficult to compute and to interpret. In this paper, we delve into the heart of spectral densities of real-world graphs. We borrow tools developed in condensed matter physics, and add novel adaptations to handle the spectral signatures of common graph motifs. The resulting methods are highly efficient, as we illustrate by computing spectral densities for graphs with over a billion edges on a single compute node. Beyond providing visually compelling fingerprints of graphs, we show how the estimation of spectral densities facilitates the computation of many common centrality measures, and use spectral densities to estimate meaningful information about graph structure that cannot be inferred from the extremal eigenpairs alone.

Spectral Filter Tracking

Abstract: Visual object tracking is a challenging computer vision task with numerous real-world applications. In this paper, we propose a simple but efficient spectral filter tracking (SFT) method from the viewpoint of a graph, where each candidate’s image region is modeled as a pixelwise grid graph. Instead of the conventional graph matching, we formulate the tracking as a plain least square regression problem of learning spectral filters on graphs to predict an optimal vertex, which indicates the center of the target. To bypass computationally expensive eigenvalue decomposition on graph Laplacian $ \mathcal {L}$ , we parameterize spectral graph filters as a polynomial of $ \mathcal {L}$ to aggregate local graph features according to spectral graph theory, in which $ \mathcal {L}^{k}$ exactly encodes a k-hop local neighborhood of each vertex. Thus, different from the holistic regression in those correlation filter-based methods, SFT can operate on localized regions around a pixel (i.e., a vertex), which can effectively reduce the influence of local variations and cluttered backgrounds. Furthermore, we observe that the correlation filter tracking may be viewed as a specific case of our proposed spectral filtering method. The implementation of SFT can simply boil down to only a few line codes, but surprisingly it beats the correlation filter-based model with the same feature input and achieves the state-of-the-art performance on OTB-2015 and VOT2016 under the same feature extraction strategy.

A spectral generative model for graph structure

Abstract: This paper shows how to construct a generative model for graph structure. We commence from a sample of graphs where the correspondences between nodes are unknown ab initio. We also work with graphs where there may be structural differences present, i.e. variations in the number of nodes in each graph and the edge-structure. The idea underpinning the method is to embed the nodes of the graphs into a vector space by performing kernel PCA on the heat kernel. The co-ordinates of the nodes are determined by the eigenvalues and eigenvectors of the Laplacian matrix, together with a time parameter which can be used to scale the embedding. Node correspondences are located by applying Scott and Longuet-Higgins algorithm to the embedded nodes. We capture variations in graph structure using the covariance matrix for corresponding embedded point-positions. We construct a point distribution model for the embedded node positions using the eigenvalues and eigenvectors of the covariance matrix. We show how to use this model to both project individual graphs into the eigenspace of the point-position covariance matrix and how to fit the model to potentially noisy graphs to reconstruct the Laplacian matrix. We illustrate the utility of the resulting method for shape-analysis using data from the COIL database.

Shift-enabled graphs: Graphs where shift-invariant filters are representable as polynomials of shift operations

Abstract: In digital signal processing, shift-invariant filters can be represented as a polynomial expansion of a shift operation,that is, the Z-transform representation. When extended to graph signal processing (GSP), this would mean that a shift-invariant graph filter can be represented as a polynomial of the adjacency (shift) matrix of the graph. However, the characteristic and minimum polynomials of the adjacency matrix must be identical for the property to hold. While it has been suggested that this condition might be ignored as it is always possible to find a polynomial transform to represent the original adjacency matrix by another adjacency matrix that satisfies the condition, this letter shows that a filter that is shift invariant in terms of the original graph may not be shift invariant anymore under the modified graph and vice versa. We introduce the notion of "shift-enabled graph" for graphs that satisfy the aforementioned condition, and present a concrete example of a graph that is not "shift-enabled" and a shift-invariant filter that is not a polynomial of the shift operation matrix. The result provides a deeper understanding of shift-invariant filters when applied in GSP and shows that further investigation of shift-enabled graphs is needed to make it applicable to practical scenarios.