707 research outputs found
Overlap properties of geometric expanders
The {\em overlap number} of a finite -uniform hypergraph is
defined as the largest constant such that no matter how we map
the vertices of into , there is a point covered by at least a
-fraction of the simplices induced by the images of its hyperedges.
In~\cite{Gro2}, motivated by the search for an analogue of the notion of graph
expansion for higher dimensional simplicial complexes, it was asked whether or
not there exists a sequence of arbitrarily large
-uniform hypergraphs with bounded degree, for which . Using both random methods and explicit constructions, we answer this
question positively by constructing infinite families of -uniform
hypergraphs with bounded degree such that their overlap numbers are bounded
from below by a positive constant . We also show that, for every ,
the best value of the constant that can be achieved by such a
construction is asymptotically equal to the limit of the overlap numbers of the
complete -uniform hypergraphs with vertices, as
. For the proof of the latter statement, we establish the
following geometric partitioning result of independent interest. For any
and any , there exists satisfying the
following condition. For any , for any point and
for any finite Borel measure on with respect to which
every hyperplane has measure , there is a partition into measurable parts of equal measure such that all but
at most an -fraction of the -tuples
have the property that either all simplices with
one vertex in each contain or none of these simplices contain
Overlap properties of geometric expanders
The overlap number of a finite (d + 1)-uniform hypergraph H is the largest constant c(H) ∈ (0, 1] such that no matter how we map the vertices of H into ℝd, there is a point covered by at least a c(H)-fraction of the simplices induced by the images of its hyperedges. Motivated by the search for an analogue of the notion of graph expansion for higher dimensional simplicial complexes, we address the question whether or not there exists a sequence of arbitrarily large (d + 1)-uniform hypergraphs with bounded degree for which . Using both random methods and explicit constructions, we answer this question positively by constructing infinite families of (d + 1)-uniform hypergraphs with bounded degree such that their overlap numbers are bounded from below by a positive constant c = c(d). We also show that, for every d, the best value of the constant c = c(d) that can be achieved by such a construction is asymptotically equal to the limit of the overlap numbers of the complete (d + 1)-uniform hypergraphs with n vertices, as n → ∞. For the proof of the latter statement, we establish the following geometric partitioning result of independent interest. For any h, s and any ɛ > 0, there exists K = K(ɛ, h, s) satisfying the following condition. For any k ≧ K and for any semi-algebraic relation R on h-tuples of points in a Euclidean space ℝd with description complexity at most s, every finite set P ⫅ ℝd has a partition P = P1 ∪ ⋯ ∪ Pk into k parts of sizes as equal as possible such that all but at most an ɛ-fraction of the h-tuples (Pi1, , Pih) have the property that either all h-tuples of points with one element in each Pij are related with respect to R or none of them ar
Hypergraph expanders from Cayley graphs
We present a simple mechanism, which can be randomised, for constructing
sparse -uniform hypergraphs with strong expansion properties. These
hypergraphs are constructed using Cayley graphs over and have
vertex degree which is polylogarithmic in the number of vertices. Their
expansion properties, which are derived from the underlying Cayley graphs,
include analogues of vertex and edge expansion in graphs, rapid mixing of the
random walk on the edges of the skeleton graph, uniform distribution of edges
on large vertex subsets and the geometric overlap property.Comment: 13 page
Isoperimetric Inequalities in Simplicial Complexes
In graph theory there are intimate connections between the expansion
properties of a graph and the spectrum of its Laplacian. In this paper we
define a notion of combinatorial expansion for simplicial complexes of general
dimension, and prove that similar connections exist between the combinatorial
expansion of a complex, and the spectrum of the high dimensional Laplacian
defined by Eckmann. In particular, we present a Cheeger-type inequality, and a
high-dimensional Expander Mixing Lemma. As a corollary, using the work of Pach,
we obtain a connection between spectral properties of complexes and Gromov's
notion of geometric overlap. Using the work of Gunder and Wagner, we give an
estimate for the combinatorial expansion and geometric overlap of random
Linial-Meshulam complexes
Ramanujan Complexes and bounded degree topological expanders
Expander graphs have been a focus of attention in computer science in the
last four decades. In recent years a high dimensional theory of expanders is
emerging. There are several possible generalizations of the theory of expansion
to simplicial complexes, among them stand out coboundary expansion and
topological expanders. It is known that for every d there are unbounded degree
simplicial complexes of dimension d with these properties. However, a major
open problem, formulated by Gromov, is whether bounded degree high dimensional
expanders, according to these definitions, exist for d >= 2. We present an
explicit construction of bounded degree complexes of dimension d = 2 which are
high dimensional expanders. More precisely, our main result says that the
2-skeletons of the 3-dimensional Ramanujan complexes are topological expanders.
Assuming a conjecture of Serre on the congruence subgroup property, infinitely
many of them are also coboundary expanders.Comment: To appear in FOCS 201
Coboundary expanders
We describe a natural topological generalization of edge expansion for graphs
to regular CW complexes and prove that this property holds with high
probability for certain random complexes.Comment: Version 2: significant rewrite. 18 pages, title changed, and main
theorem extended to more general random complexe
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