5,847 research outputs found
On the spectrum of hypergraphs
Here we study the spectral properties of an underlying weighted graph of a
non-uniform hypergraph by introducing different connectivity matrices, such as
adjacency, Laplacian and normalized Laplacian matrices. We show that different
structural properties of a hypergrpah, can be well studied using spectral
properties of these matrices. Connectivity of a hypergraph is also investigated
by the eigenvalues of these operators. Spectral radii of the same are bounded
by the degrees of a hypergraph. The diameter of a hypergraph is also bounded by
the eigenvalues of its connectivity matrices. We characterize different
properties of a regular hypergraph characterized by the spectrum. Strong
(vertex) chromatic number of a hypergraph is bounded by the eigenvalues.
Cheeger constant on a hypergraph is defined and we show that it can be bounded
by the smallest nontrivial eigenvalues of Laplacian matrix and normalized
Laplacian matrix, respectively, of a connected hypergraph. We also show an
approach to study random walk on a (non-uniform) hypergraph that can be
performed by analyzing the spectrum of transition probability operator which is
defined on that hypergraph. Ricci curvature on hypergraphs is introduced in two
different ways. We show that if the Laplace operator, , on a hypergraph
satisfies a curvature-dimension type inequality
with and then any non-zero eigenvalue of can be bounded below by . Eigenvalues of a normalized Laplacian operator defined on a connected
hypergraph can be bounded by the Ollivier's Ricci curvature of the hypergraph
Spectral and Dynamical Properties in Classes of Sparse Networks with Mesoscopic Inhomogeneities
We study structure, eigenvalue spectra and diffusion dynamics in a wide class
of networks with subgraphs (modules) at mesoscopic scale. The networks are
grown within the model with three parameters controlling the number of modules,
their internal structure as scale-free and correlated subgraphs, and the
topology of connecting network. Within the exhaustive spectral analysis for
both the adjacency matrix and the normalized Laplacian matrix we identify the
spectral properties which characterize the mesoscopic structure of sparse
cyclic graphs and trees. The minimally connected nodes, clustering, and the
average connectivity affect the central part of the spectrum. The number of
distinct modules leads to an extra peak at the lower part of the Laplacian
spectrum in cyclic graphs. Such a peak does not occur in the case of
topologically distinct tree-subgraphs connected on a tree. Whereas the
associated eigenvectors remain localized on the subgraphs both in trees and
cyclic graphs. We also find a characteristic pattern of periodic localization
along the chains on the tree for the eigenvector components associated with the
largest eigenvalue equal 2 of the Laplacian. We corroborate the results with
simulations of the random walk on several types of networks. Our results for
the distribution of return-time of the walk to the origin (autocorrelator)
agree well with recent analytical solution for trees, and it appear to be
independent on their mesoscopic and global structure. For the cyclic graphs we
find new results with twice larger stretching exponent of the tail of the
distribution, which is virtually independent on the size of cycles. The
modularity and clustering contribute to a power-law decay at short return
times
Incremental eigenpair computation for graph Laplacian matrices: theory and applications
The smallest eigenvalues and the associated eigenvectors (i.e., eigenpairs) of a graph Laplacian matrix have been widely used for spectral clustering and community detection. However, in real-life applications, the number of clusters or communities (say, K) is generally unknown a priori. Consequently, the majority of the existing methods either choose K heuristically or they repeat the clustering method with different choices of K and accept the best clustering result. The first option, more often, yields suboptimal result, while the second option is computationally expensive. In this work, we propose an incremental method for constructing the eigenspectrum of the graph Laplacian matrix. This method leverages the eigenstructure of graph Laplacian matrix to obtain the Kth smallest eigenpair of the Laplacian matrix given a collection of all previously compute
A Multiscale Pyramid Transform for Graph Signals
Multiscale transforms designed to process analog and discrete-time signals
and images cannot be directly applied to analyze high-dimensional data residing
on the vertices of a weighted graph, as they do not capture the intrinsic
geometric structure of the underlying graph data domain. In this paper, we
adapt the Laplacian pyramid transform for signals on Euclidean domains so that
it can be used to analyze high-dimensional data residing on the vertices of a
weighted graph. Our approach is to study existing methods and develop new
methods for the four fundamental operations of graph downsampling, graph
reduction, and filtering and interpolation of signals on graphs. Equipped with
appropriate notions of these operations, we leverage the basic multiscale
constructs and intuitions from classical signal processing to generate a
transform that yields both a multiresolution of graphs and an associated
multiresolution of a graph signal on the underlying sequence of graphs.Comment: 16 pages, 13 figure
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