John C. Urschel

John C. Urschel, Ludmil T. Zikatanov

Spectral graph bisections are a popular heuristic aimed at approximating the solution of the NP-complete graph bisection problem. This technique, however, does not always provide a robust tool for graph partitioning. Using a special class of graphs, we prove that the standard spectral graph bisection can produce bisections that are far from optimal. In particular, we show that the maximum error in the spectral approximation of the optimal bisection (partition sizes exactly equal) cut for such graphs is bounded below by a constant multiple of the order of the graph squared.

Xiaozhe Hu, John C. Urschel, Ludmil T. Zikatanov

Graph disaggregation is a technique used to address the high cost of computation for power law graphs on parallel processors. The few high-degree vertices are broken into multiple small-degree vertices, in order to allow for more efficient computation in parallel. In particular, we consider computations involving the graph Laplacian, which has significant applications, including diffusion mapping and graph partitioning, among others. We prove results regarding the spectral approximation of the Laplacian of the original graph by the Laplacian of the disaggregated graph. In addition, we construct an alternate disaggregation operator whose eigenvalues interlace those of the original Laplacian. Using this alternate operator, we construct a uniform preconditioner for the original graph Laplacian.

John C. Urschel, Wenfang Xu, Ludmil T. Zikatanov

Analysis of big data has become an increasingly relevant area of research, with data often represented on discrete networks both constructed and organic. While for structured domains, there exist intuitive definitions of signals and frequencies, the definitions are much less obvious for data sets associated with a given network. Often, the eigenvectors of an induced graph Laplacian are used to construct an orthogonal set of low-frequency vectors. For larger graphs, however, the computational cost of creating such structures becomes untenable, and the quality of the approximation is adequate only for signals near the span of the set. We propose a construction of a full basis of frequencies with computational complexity that is near-linear in time and linear in storage. Using this frequency domain, we can compress data sets on unstructured graphs more robustly and accurately than spectral-based constructions.

John C. Urschel, Xiaozhe Hu, Jinchao Xu et al.

In this paper, we develop a cascadic multigrid algorithm for fast computation of the Fiedler vector of a graph Laplacian, namely, the eigenvector corresponding to the second smallest eigenvalue. This vector has been found to have applications in fields such as graph partitioning and graph drawing. The algorithm is a purely algebraic approach based on a heavy edge coarsening scheme and pointwise smoothing for refinement. To gain theoretical insight, we also consider the related cascadic multigrid method in the geometric setting for elliptic eigenvalue problems and show its uniform convergence under certain assumptions. Numerical tests are presented for computing the Fiedler vector of several practical graphs, and numerical results show the efficiency and optimality of our proposed cascadic multigrid algorithm.