145 research outputs found
Kronecker Product of Tensors and Hypergraphs: Structure and Dynamics
Hypergraphs and graph products extend traditional graph theory by
incorporating multi-way and coupled relationships, which are ubiquitous in
real-world systems. While the Kronecker product, rooted in matrix analysis, has
become a powerful tool in network science, its application has been limited to
pairwise networks. In this paper, we extend the coupling of graph products to
hypergraphs, enabling a system-theoretic analysis of network compositions
formed via the Kronecker product of hypergraphs. We first extend the notion of
the matrix Kronecker product to the tensor Kronecker product from the
perspective of tensor blocks. We present various algebraic and spectral
properties and express different tensor decompositions with the tensor
Kronecker product. Furthermore, we study the structure and dynamics of
Kronecker hypergraphs based on the tensor Kronecker product. We establish
conditions that enable the analysis of the trajectory and stability of a
hypergraph dynamical system by examining the dynamics of its factor
hypergraphs. Finally, we demonstrate the numerical advantage of this framework
for computing various tensor decompositions and spectral properties.Comment: 29 pages, 4 figures, 2 table
NFFT meets Krylov methods: Fast matrix-vector products for the graph Laplacian of fully connected networks
The graph Laplacian is a standard tool in data science, machine learning, and
image processing. The corresponding matrix inherits the complex structure of
the underlying network and is in certain applications densely populated. This
makes computations, in particular matrix-vector products, with the graph
Laplacian a hard task. A typical application is the computation of a number of
its eigenvalues and eigenvectors. Standard methods become infeasible as the
number of nodes in the graph is too large. We propose the use of the fast
summation based on the nonequispaced fast Fourier transform (NFFT) to perform
the dense matrix-vector product with the graph Laplacian fast without ever
forming the whole matrix. The enormous flexibility of the NFFT algorithm allows
us to embed the accelerated multiplication into Lanczos-based eigenvalues
routines or iterative linear system solvers and even consider other than the
standard Gaussian kernels. We illustrate the feasibility of our approach on a
number of test problems from image segmentation to semi-supervised learning
based on graph-based PDEs. In particular, we compare our approach with the
Nystr\"om method. Moreover, we present and test an enhanced, hybrid version of
the Nystr\"om method, which internally uses the NFFT.Comment: 28 pages, 9 figure
On the Complexity of Recognizing S-composite and S-prime Graphs
S-prime graphs are graphs that cannot be represented as nontrivial subgraphs
of nontrivial Cartesian products of graphs, i.e., whenever it is a subgraph of
a nontrivial Cartesian product graph it is a subgraph of one the factors. A
graph is S-composite if it is not S-prime. Although linear time recognition
algorithms for determining whether a graph is prime or not with respect to the
Cartesian product are known, it remained unknown if a similar result holds also
for the recognition of S-prime and S-composite graphs.
In this contribution the computational complexity of recognizing S-composite
and S-prime graphs is considered. Klav{\v{z}}ar \emph{et al.} [\emph{Discr.\
Math.} \textbf{244}: 223-230 (2002)] proved that a graph is S-composite if and
only if it admits a nontrivial path--coloring. The problem of determining
whether there exists a path--coloring for a given graph is shown to be
NP-complete even for . This in turn is utilized to show that determining
whether a graph is S-composite is NP-complete and thus, determining whether a
graph is S-prime is CoNP-complete. Many other problems are shown to be NP-hard,
using the latter results
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