300 research outputs found
On The Isoperimetric Spectrum of Graphs and Its Approximations
In this paper we consider higher isoperimetric numbers of a (finite directed)
graph. In this regard we focus on the th mean isoperimetric constant of a
directed graph as the minimum of the mean outgoing normalized flows from a
given set of disjoint subsets of the vertex set of the graph. We show that
the second mean isoperimetric constant in this general setting, coincides with
(the mean version of) the classical Cheeger constant of the graph, while for
the rest of the spectrum we show that there is a fundamental difference between
the th isoperimetric constant and the number obtained by taking the minimum
over all -partitions. In this direction, we show that our definition is the
correct one in the sense that it satisfies a Federer-Fleming-type theorem, and
we also define and present examples for the concept of a supergeometric graph
as a graph whose mean isoperimetric constants are attained on partitions at all
levels. Moreover, considering the -completeness of the isoperimetric
problem on graphs, we address ourselves to the approximation problem where we
prove general spectral inequalities that give rise to a general Cheeger-type
inequality as well. On the other hand, we also consider some algorithmic
aspects of the problem where we show connections to orthogonal representations
of graphs and following J.~Malik and J.~Shi () we study the close
relationships to the well-known -means algorithm and normalized cuts method
Random Walks on Hypergraphs with Edge-Dependent Vertex Weights
Hypergraphs are used in machine learning to model higher-order relationships
in data. While spectral methods for graphs are well-established, spectral
theory for hypergraphs remains an active area of research. In this paper, we
use random walks to develop a spectral theory for hypergraphs with
edge-dependent vertex weights: hypergraphs where every vertex has a weight
for each incident hyperedge that describes the contribution
of to the hyperedge . We derive a random walk-based hypergraph
Laplacian, and bound the mixing time of random walks on such hypergraphs.
Moreover, we give conditions under which random walks on such hypergraphs are
equivalent to random walks on graphs. As a corollary, we show that current
machine learning methods that rely on Laplacians derived from random walks on
hypergraphs with edge-independent vertex weights do not utilize higher-order
relationships in the data. Finally, we demonstrate the advantages of
hypergraphs with edge-dependent vertex weights on ranking applications using
real-world datasets.Comment: Accepted to ICML 201
Non self-adjoint laplacians on a directed graph
We consider a non self-adjoint Laplacian on a directed graph with non
symmetric edge weights. We analyse spectral properties of this Laplacian under
a Kirchhoff assumption. Moreover we establish isoperimet-ric inequalities in
terms of the numerical range to show the absence of the essential spectrum of
the Laplacian on heavy end directed graphs
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