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 Algorithms for Temporal Graph Cuts

Abstract: The sparsest cut problem consists of identifying a small set of edges that breaks the graph into balanced sets of vertices. The normalized cut problem balances the total degree, instead of the size, of the resulting sets. Applications of graph cuts include community detection and computer vision. However, cut problems were originally proposed for static graphs, an assumption that does not hold in many modern applications where graphs are highly dynamic. In this paper, we introduce the sparsest and normalized cut problems in temporal graphs, which generalize their standard definitions by enforcing the smoothness of cuts over time. We propose novel formulations and algorithms for computing temporal cuts using spectral graph theory, multiplex graphs, divide-and-conquer and low-rank matrix approximation. Furthermore, we extend our formulation to dynamic graph signals, where cuts also capture node values, as graph wavelets. Experiments show that our solutions are accurate and scalable, enabling the discovery of dynamic communities and the analysis of dynamic graph processes.

Graph spectral domain watermarking for unstructured data from sensor networks

Abstract: The modern applications like social networks and sensors networks are increasingly used in the recent years. These applications can be represented as a weighted graph using irregular structure. Unfortunately, we cannot apply the techniques of the traditional signal processing on those graphs. In this paper, graph spread spectrum watermarking is proposed for networked sensor data authentication. Firstly, the graph spectrum is computed based on the eigenvector decomposition of the graph Laplacian. Then, graph Fourier coefficients are obtained by projecting the graph signals onto the basis functions which are the eigenvectors of the graph Laplacian. Finally, the watermark bits are embedded in the graph spectral coefficients using a watermark strength parameter varied according to the eigenvector number. We have considered two scenarios: blind and non-blind watermarking. The experimental results show that the proposed methods are robust, high capacity and result in low distortion in data. The proposed algorithms are robust to many types of attacks: noise, data modification, data deletion, rounding and down-sampling.

Minimum Spectral Connectivity Projection Pursuit for Unsupervised Classification

Abstract: We study the problem of determining the optimal univariate subspace for maximising the separability of a binary partition of unlabeled data, as measured by spectral graph theory. This is achieved by ?nding projections which minimise the second eigenvalue of the Laplacian matrices of the projected data, which corresponds to a non-convex, non-smooth optimisation problem. We show that the optimal 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. It also allows us to apply our methodology to the problem of ?nding large margin linear separators. The computational cost associated with each eigen-problem is quadratic in the number of data. To mitigate this problem, we propose an approximation method using microclusters with provable approximation error bounds. We evaluate the performance of the proposed method on simulated and publicly available data sets and ?nd that it compares favourably with existing methods for projection pursuit and dimension reduction for unsupervised data partitioning.

Power up! Robust Graph Convolutional Network via Graph Powering.

Abstract: Graph convolutional networks (GCNs) are powerful tools for graph-structured data. However, they have been recently shown to be vulnerable to topological attacks. To enhance adversarial robustness, we go beyond spectral graph theory to robust graph theory. By challenging the classical graph Laplacian, we propose a new convolution operator that is provably robust in the spectral domain and is incorporated in the GCN architecture to improve expressivity and interpretability. By extending the original graph to a sequence of graphs, we also propose a robust training paradigm that encourages transferability across graphs that span a range of spatial and spectral characteristics. The proposed approaches are demonstrated in extensive experiments to simultaneously improve performance in both benign and adversarial situations.