1,538 research outputs found
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
Finite Simple Groups as Expanders
We prove that there exist and such that every
non-abelian finite simple group , which is not a Suzuki group, has a set of
generators for which the Cayley graph \Cay(G; S) is an
-expander.Comment: 10 page
High Dimensional Random Walks and Colorful Expansion
Random walks on bounded degree expander graphs have numerous applications,
both in theoretical and practical computational problems. A key property of
these walks is that they converge rapidly to their stationary distribution.
In this work we {\em define high order random walks}: These are
generalizations of random walks on graphs to high dimensional simplicial
complexes, which are the high dimensional analogues of graphs. A simplicial
complex of dimension has vertices, edges, triangles, pyramids, up to
-dimensional cells. For any , a high order random walk on
dimension moves between neighboring -faces (e.g., edges) of the complex,
where two -faces are considered neighbors if they share a common
-face (e.g., a triangle). The case of recovers the well studied
random walk on graphs.
We provide a {\em local-to-global criterion} on a complex which implies {\em
rapid convergence of all high order random walks} on it. Specifically, we prove
that if the -dimensional skeletons of all the links of a complex are
spectral expanders, then for {\em all} the high order random walk
on dimension converges rapidly to its stationary distribution.
We derive our result through a new notion of high dimensional combinatorial
expansion of complexes which we term {\em colorful expansion}. This notion is a
natural generalization of combinatorial expansion of graphs and is strongly
related to the convergence rate of the high order random walks.
We further show an explicit family of {\em bounded degree} complexes which
satisfy this criterion. Specifically, we show that Ramanujan complexes meet
this criterion, and thus form an explicit family of bounded degree high
dimensional simplicial complexes in which all of the high order random walks
converge rapidly to their stationary distribution.Comment: 27 page
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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Expanders with Symmetry: Constructions and Applications
Expanders are sparse yet well-connected graphs with numerous theoretical and practical uses. Symmetry is a valuable structure for expanders as it enables efficient algorithms and a richer set of applications. This thesis studies expanders with symmetry, giving new constructions and applications. We extend expander construction techniques to work with symmetry and give explicit constructions of expanders with varying quality of expansion and symmetries of various groups. In particular, we construct graphs with large Abelian group symmetries via the technique of \textit{graph lifts}. We also give a generic amplification procedure that converts a weak expander to an almost optimal one while preserving symmetries. This procedure is obtained by generalizing prior amplification techniques that work for Cayley graphs over Abelian groups to Cayley graphs over any finite group. In particular, we obtain almost-Ramanujan expanders over every non-abelian finite simple group. We then explore the utility of having both symmetry and expansion simultaneously. We obtain explicit quantum LDPC codes of almost linear distance and \textit{good} classical quasi-cyclic codes with varying circulant sizes using prior results and our constructions of graphs with Abelian symmetries. We show how our generic amplification machinery boosts various structured expander-like objects: \textit{quantum expanders}, \textit{dimension expanders}, and \textit{monotone expanders}. Finally, we prove a structural result about expanding Cayley graphs, showing that they satisfy a \enquote{degree-2} variant of the \textit{expander mixing lemma}. As an application of this, we give a randomness-efficient query algorithm for \textit{homomorphism testing} of unitary-valued functions on finite groups and a derandomized version of the celebrated Babai--Nikolov--Pyber (BNP) lemma
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