5 research outputs found
New High Dimensional Expanders from Covers
We present a new construction of high dimensional expanders based on covering
spaces of simplicial complexes. High dimensional expanders (HDXs) are
hypergraph analogues of expander graphs. They have many uses in theoretical
computer science, but unfortunately only few constructions are known which have
arbitrarily small local spectral expansion.
We give a randomized algorithm that takes as input a high dimensional
expander (satisfying some mild assumptions). It outputs a sub-complex that is a high dimensional expander and has infinitely many
simplicial covers. These covers form new families of bounded-degree high
dimensional expanders. The sub-complex inherits 's underlying graph and
its links are sparsifications of the links of . When the size of the links
of is , this algorithm can be made deterministic. Our
algorithm is based on the groups and generating sets discovered by Lubotzky,
Samuels and Vishne (2005), that were used to construct the first discovered
high dimensional expanders. We show these groups give rise to many more
``randomized'' high dimensional expanders.
In addition, our techniques also give a random sparsification algorithm for
high dimensional expanders, that maintains its local spectral properties. This
may be of independent interest
Coboundary and cosystolic expansion without dependence on dimension or degree
We give new bounds on the cosystolic expansion constants of several families
of high dimensional expanders, and the known coboundary expansion constants of
order complexes of homogeneous geometric lattices, including the spherical
building of . The improvement applies to the high dimensional
expanders constructed by Lubotzky, Samuels and Vishne, and by Kaufman and
Oppenheim.
Our new expansion constants do not depend on the degree of the complex nor on
its dimension, nor on the group of coefficients. This implies improved bounds
on Gromov's topological overlap constant, and on Dinur and Meshulam's cover
stability, which may have applications for agreement testing. In comparison,
existing bounds decay exponentially with the ambient dimension (for spherical
buildings) and in addition decay linearly with the degree (for all known
bounded-degree high dimensional expanders). Our results are based on several
new techniques:
* We develop a new "color-restriction" technique which enables proving
dimension-free expansion by restricting a multi-partite complex to small random
subsets of its color classes.
* We give a new "spectral" proof for Evra and Kaufman's local-to-global
theorem, deriving better bounds and getting rid of the dependence on the
degree. This theorem bounds the cosystolic expansion of a complex using
coboundary expansion and spectral expansion of the links.
* We derive absolute bounds on the coboundary expansion of the spherical
building (and any order complex of a homogeneous geometric lattice) by
constructing a novel family of very short cones
The duplicube graph -- a hybrid of structure and randomness
Connect two copies of a given graph by a perfect matching. What are the
properties of the graphs obtained by recursively repeating this procedure? We
show that this construction shares some of the structural properties of the
hypercube, such as a simple routing scheme and small edge expansion. However,
when the matchings are uniformly random, the resultant graph also has
similarities with a random regular graph, including: a smaller diameter and
better vertex expansion than the hypercube; a semicircle law for its
eigenvalues; and no non-trivial automorphisms. We propose a simple
deterministic matching which we believe could provide a derandomization.Comment: 27 pages, 6 figures. Comments welcome
Boolean functions on high-dimensional expanders
We initiate the study of Boolean function analysis on high-dimensional
expanders. We give a random-walk based definition of high-dimensional
expansion, which coincides with the earlier definition in terms of two-sided
link expanders. Using this definition, we describe an analog of the Fourier
expansion and the Fourier levels of the Boolean hypercube for simplicial
complexes. Our analog is a decomposition into approximate eigenspaces of random
walks associated with the simplicial complexes. Our random-walk definition and
the decomposition have the additional advantage that they extend to the more
general setting of posets, encompassing both high-dimensional expanders and the
Grassmann poset, which appears in recent work on the unique games conjecture.
We then use this decomposition to extend the Friedgut-Kalai-Naor theorem to
high-dimensional expanders. Our results demonstrate that a constant-degree
high-dimensional expander can sometimes serve as a sparse model for the Boolean
slice or hypercube, and quite possibly additional results from Boolean function
analysis can be carried over to this sparse model. Therefore, this model can be
viewed as a derandomization of the Boolean slice, containing only
points in contrast to points in the -slice
(which consists of all -bit strings with exactly ones).Comment: 48 pages, Extended version of the prior submission, with more details
of expanding posets (eposets
Boolean Function Analysis on High-Dimensional Expanders
We initiate the study of Boolean function analysis on high-dimensional expanders. We describe an analog of the Fourier expansion and of the Fourier levels on simplicial complexes, and generalize the FKN theorem to high-dimensional expanders.
Our results demonstrate that a high-dimensional expanding complex X can sometimes serve as a sparse model for the Boolean slice or hypercube, and quite possibly additional results from Boolean function analysis can be carried over to this sparse model. Therefore, this model can be viewed as a derandomization of the Boolean slice, containing |X(k)|=O(n) points in comparison to binom{n}{k+1} points in the (k+1)-slice (which consists of all n-bit strings with exactly k+1 ones)