1,292 research outputs found
New Explicit Constant-Degree Lossless Expanders
We present a new explicit construction of onesided bipartite lossless
expanders of constant degree, with arbitrary constant ratio between the sizes
of the two vertex sets. Our construction is simpler to state and analyze than
the only prior construction of Capalbo, Reingold, Vadhan, and Wigderson (2002),
and achieves improvements in some parameters.
We construct our lossless expanders by imposing the structure of a
constant-sized lossless expander "gadget" within the neighborhoods of a large
bipartite spectral expander; similar constructions were previously used to
obtain the weaker notion of unique-neighbor expansion. Our analysis simply
consists of elementary counting arguments and an application of the expander
mixing lemma.Comment: Edits to expositio
Coboundary and Cosystolic Expansion from Strong Symmetry
Coboundary and cosystolic expansion are notions of expansion that generalize the Cheeger constant or edge expansion of a graph to higher dimensions. The classical Cheeger inequality implies that for graphs edge expansion is equivalent to spectral expansion. In higher dimensions this is not the case: a simplicial complex can be spectrally expanding but not have high dimensional edge-expansion. The phenomenon of high dimensional edge expansion in higher dimensions is much more involved than spectral expansion, and is far from being understood. In particular, prior to this work, the only known bounded degree cosystolic expanders were derived from the theory of buildings that is far from being elementary.
In this work we study high dimensional complexes which are strongly symmetric. Namely, there is a group that acts transitively on top dimensional cells of the simplicial complex [e.g., for graphs it corresponds to a group that acts transitively on the edges]. Using the strong symmetry, we develop a new machinery to prove coboundary and cosystolic expansion.
It was an open question whether the recent elementary construction of bounded degree spectral high dimensional expanders based on coset complexes give rise to bounded degree cosystolic expanders. In this work we answer this question affirmatively. We show that these complexes give rise to bounded degree cosystolic expanders in dimension two, and that their links are (two-dimensional) coboundary expanders. We do so by exploiting the strong symmetry properties of the links of these complexes using a new machinery developed in this work.
Previous works have shown a way to bound the co-boundary expansion using strong symmetry in the special situation of "building like" complexes. Our new machinery shows how to get coboundary expansion for general strongly symmetric coset complexes, which are not necessarily "building like", via studying the (Dehn function of the) presentation of the symmetry group of these complexes
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 Expanders from Expanders
We present an elementary way to transform an expander graph into a simplicial complex where all high order random walks have a constant spectral gap, i.e., they converge rapidly to the stationary distribution. As an upshot, we obtain new constructions, as well as a natural probabilistic model to sample constant degree high-dimensional expanders.
In particular, we show that given an expander graph G, adding self loops to G and taking the tensor product of the modified graph with a high-dimensional expander produces a new high-dimensional expander. Our proof of rapid mixing of high order random walks is based on the decomposable Markov chains framework introduced by [Jerrum et al., 2004]
Explicit expanders with cutoff phenomena
The cutoff phenomenon describes a sharp transition in the convergence of an
ergodic finite Markov chain to equilibrium. Of particular interest is
understanding this convergence for the simple random walk on a bounded-degree
expander graph. The first example of a family of bounded-degree graphs where
the random walk exhibits cutoff in total-variation was provided only very
recently, when the authors showed this for a typical random regular graph.
However, no example was known for an explicit (deterministic) family of
expanders with this phenomenon. Here we construct a family of cubic expanders
where the random walk from a worst case initial position exhibits
total-variation cutoff. Variants of this construction give cubic expanders
without cutoff, as well as cubic graphs with cutoff at any prescribed
time-point.Comment: 17 pages, 2 figure
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