271 research outputs found
On the Expansion of Group-Based Lifts
A -lift of an -vertex base graph is a graph on
vertices, where each vertex of is replaced by vertices
and each edge in is replaced by a matching
representing a bijection so that the edges of are of the form
. Lifts have been studied as a means to efficiently
construct expanders. In this work, we study lifts obtained from groups and
group actions. We derive the spectrum of such lifts via the representation
theory principles of the underlying group. Our main results are:
(1) There is a constant such that for every , there
does not exist an abelian -lift of any -vertex -regular base graph
with being almost Ramanujan (nontrivial eigenvalues of the adjacency matrix
at most in magnitude). This can be viewed as an analogue of the
well-known no-expansion result for abelian Cayley graphs.
(2) A uniform random lift in a cyclic group of order of any -vertex
-regular base graph , with the nontrivial eigenvalues of the adjacency
matrix of bounded by in magnitude, has the new nontrivial
eigenvalues also bounded by in magnitude with probability
. In particular, there is a constant such that for
every , there exists a lift of every Ramanujan graph in
a cyclic group of order with being almost Ramanujan. We use this to
design a quasi-polynomial time algorithm to construct almost Ramanujan
expanders deterministically.
The existence of expanding lifts in cyclic groups of order
can be viewed as a lower bound on the order of the largest abelian group
that produces expanding lifts. Our results show that the lower bound matches
the upper bound for (upto in the exponent)
Spectral Expanding Expanders
Dinitz, Schapira, and Valadarsky [Dinitz et al., 2017] introduced the intriguing notion of expanding expanders - a family of expander graphs with the property that every two consecutive graphs in the family differ only on a small number of edges. Such a family allows one to add and remove vertices with only few edge updates, making them useful in dynamic settings such as for datacenter network topologies and for the design of distributed algorithms for self-healing expanders. [Dinitz et al., 2017] constructed explicit expanding-expanders based on the Bilu-Linial construction of spectral expanders [Bilu and Linial, 2006]. The construction of expanding expanders, however, ends up being of edge expanders, thus, an open problem raised by [Dinitz et al., 2017] is to construct spectral expanding expanders (SEE).
In this work, we resolve this question by constructing SEE with spectral expansion which, like [Bilu and Linial, 2006], is optimal up to a poly-logarithmic factor, and the number of edge updates is optimal up to a constant. We further give a simple proof for the existence of SEE that are close to Ramanujan up to a small additive term. As in [Dinitz et al., 2017], our construction is based on interpolating between a graph and its lift. However, to establish spectral expansion, we carefully weigh the interpolated graphs, dubbed partial lifts, in a way that enables us to conduct a delicate analysis of their spectrum. In particular, at a crucial point in the analysis, we consider the eigenvectors structure of the partial lifts
Minors in expanding graphs
Extending several previous results we obtained nearly tight estimates on the
maximum size of a clique-minor in various classes of expanding graphs. These
results can be used to show that graphs without short cycles and other H-free
graphs contain large clique-minors, resolving some open questions in this area
Explicit Abelian Lifts and Quantum LDPC Codes
For an abelian group H acting on the set [?], an (H,?)-lift of a graph G? is a graph obtained by replacing each vertex by ? copies, and each edge by a matching corresponding to the action of an element of H.
Expanding graphs obtained via abelian lifts, form a key ingredient in the recent breakthrough constructions of quantum LDPC codes, (implicitly) in the fiber bundle codes by Hastings, Haah and O\u27Donnell [STOC 2021] achieving distance ??(N^{3/5}), and in those by Panteleev and Kalachev [IEEE Trans. Inf. Theory 2021] of distance ?(N/log(N)). However, both these constructions are non-explicit. In particular, the latter relies on a randomized construction of expander graphs via abelian lifts by Agarwal et al. [SIAM J. Discrete Math 2019].
In this work, we show the following explicit constructions of expanders obtained via abelian lifts. For every (transitive) abelian group H ? Sym(?), constant degree d ? 3 and ? > 0, we construct explicit d-regular expander graphs G obtained from an (H,?)-lift of a (suitable) base n-vertex expander G? with the following parameters:
ii) ?(G) ? 2?{d-1} + ?, for any lift size ? ? 2^{n^{?}} where ? = ?(d,?),
iii) ?(G) ? ? ? d, for any lift size ? ? 2^{n^{??}} for a fixed ?? > 0, when d ? d?(?), or
iv) ?(G) ? O?(?d), for lift size "exactly" ? = 2^{?(n)}. As corollaries, we obtain explicit quantum lifted product codes of Panteleev and Kalachev of almost linear distance (and also in a wide range of parameters) and explicit classical quasi-cyclic LDPC codes with wide range of circulant sizes.
Items (i) and (ii) above are obtained by extending the techniques of Mohanty, O\u27Donnell and Paredes [STOC 2020] for 2-lifts to much larger abelian lift sizes (as a byproduct simplifying their construction). This is done by providing a new encoding of special walks arising in the trace power method, carefully "compressing" depth-first search traversals. Result (iii) is via a simpler proof of Agarwal et al. [SIAM J. Discrete Math 2019] at the expense of polylog factors in the expansion
Spectra of lifted Ramanujan graphs
A random -lift of a base graph is its cover graph on the vertices
, where for each edge in there is an independent
uniform bijection , and has all edges of the form .
A main motivation for studying lifts is understanding Ramanujan graphs, and
namely whether typical covers of such a graph are also Ramanujan.
Let be a graph with largest eigenvalue and let be the
spectral radius of its universal cover. Friedman (2003) proved that every "new"
eigenvalue of a random lift of is with high
probability, and conjectured a bound of , which would be tight by
results of Lubotzky and Greenberg (1995). Linial and Puder (2008) improved
Friedman's bound to . For -regular graphs,
where and , this translates to a bound of
, compared to the conjectured .
Here we analyze the spectrum of a random -lift of a -regular graph
whose nontrivial eigenvalues are all at most in absolute value. We
show that with high probability the absolute value of every nontrivial
eigenvalue of the lift is . This result is
tight up to a logarithmic factor, and for it
substantially improves the above upper bounds of Friedman and of Linial and
Puder. In particular, it implies that a typical -lift of a Ramanujan graph
is nearly Ramanujan.Comment: 34 pages, 4 figure
Entropy in Dimension One
This paper completely classifies which numbers arise as the topological
entropy associated to postcritically finite self-maps of the unit interval.
Specifically, a positive real number h is the topological entropy of a
postcritically finite self-map of the unit interval if and only if exp(h) is an
algebraic integer that is at least as large as the absolute value of any of the
conjugates of exp(h); that is, if exp(h) is a weak Perron number. The
postcritically finite map may be chosen to be a polynomial all of whose
critical points are in the interval (0,1). This paper also proves that the weak
Perron numbers are precisely the numbers that arise as exp(h), where h is the
topological entropy associated to ergodic train track representatives of outer
automorphisms of a free group.Comment: 38 pages, 15 figures. This paper was completed by the author before
his death, and was uploaded by Dylan Thurston. A version including endnotes
by John Milnor will appear in the proceedings of the Banff conference on
Frontiers in Complex Dynamic
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