Related papers: Incremental Eigenpair Computation for Graph Laplac…
Spectral clustering is a popular method for community detection in network graphs: starting from a matrix representation of the graph, the nodes are clustered on a low dimensional projection obtained from a truncated spectral decomposition…
The independence complex of a graph $G=(V,E)$ is the simplicial complex $I(G)$ on vertex set $V$ whose simplices are the independent sets in $G$. We present new lower bounds on the eigenvalues of the $k$-dimensional Laplacian $L_k(I(G))$ in…
In this paper, we detail the improvement of the Cascadic Multigrid algorithm with the addition of the Gauss Seidel algorithm in order to compute the Fiedler vector of a graph Laplacian, which is the eigenvector corresponding to the second…
As an indicator of the stability of spectral clustering of an undirected weighted graph into $k$ clusters, the $k$th spectral gap of the graph Laplacian is often considered. The $k$th spectral gap is characterized in this paper as an…
Constrained clustering has been well-studied for algorithms such as $K$-means and hierarchical clustering. However, how to satisfy many constraints in these algorithmic settings has been shown to be intractable. One alternative to encode…
Dynamical Systems (DS) are fundamental to the modeling and understanding time evolving phenomena, and have application in physics, biology and control. As determining an analytical description of the dynamics is often difficult, data-driven…
Eigenvectors and eigenvalues of discrete graph Laplacians are often used for manifold learning and nonlinear dimensionality reduction. It was previously proved by Belkin and Niyogi that the eigenvectors and eigenvalues of the graph…
Let G=(V,E) be an undirected graph, lambda_k be the k-th smallest eigenvalue of the normalized laplacian matrix of G. There is a basic fact in algebraic graph theory that lambda_k > 0 if and only if G has at most k-1 connected components.…
One of the fundamental problems in network analysis is detecting community structure in multi-layer networks, of which each layer represents one type of edge information among the nodes. We propose integrative spectral clustering approaches…
Based on matrix perturbation theory, closed-form analytic expansions are studied for a Laplacian eigenvalue of an undirected, possibly weighted graph, which is close to a unique degree in that graph. An approximation is presented to provide…
We are interested in multilayer graph clustering, which aims at dividing the graph nodes into categories or communities. To do so, we propose to learn a clustering-friendly embedding of the graph nodes by solving an optimization problem…
In the first part of this paper, we survey results that are associated with three types of Laplacian matrices:difference, normalized, and signless. We derive eigenvalue and eigenvector formulaes for paths and cycles using circulant matrices…
Spectral clustering requires the time-consuming decomposition of the Laplacian matrix of the similarity graph, thus limiting its applicability to large datasets. To improve the efficiency of spectral clustering, a top-down approach was…
How does coarsening affect the spectrum of a general graph? We provide conditions such that the principal eigenvalues and eigenspaces of a coarsened and original graph Laplacian matrices are close. The achieved approximation is shown to…
Large-scale eigenvalue computations on sparse matrices are a key component of graph analytics techniques based on spectral methods. In such applications, an exhaustive computation of all eigenvalues and eigenvectors is impractical and…
Determining the effect of structural perturbations on the eigenvalue spectra of networks is an important problem because the spectra characterize not only their topological structures, but also their dynamical behavior, such as…
Graph-based multi-view spectral clustering methods have achieved notable progress recently, yet they often fall short in either oversimplifying pairwise relationships or struggling with inefficient spectral decompositions in…
While orthogonalization exists in current dimensionality reduction methods in spectral clustering on undirected graphs, it does not scale in parallel computing environments. We propose four orthogonalization-free methods for spectral…
Spectral embedding is a powerful graph embedding technique that has received a lot of attention recently due to its effectiveness on Graph Transformers. However, from a theoretical perspective, the universal expressive power of spectral…
We present a new clustering algorithm that is based on searching for natural gaps in the components of the lowest energy eigenvectors of the Laplacian of a graph. In comparing the performance of the proposed method with a set of other…