3 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
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 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)