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Register a new userTransitive bounded-degree 2-expanders from regular 2-expanders
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A two-dimensional simplicial complex is called $d$-{em regular} if every edge of it is contained in exactly $d$ distinct triangles. It is called $epsilon$-expanding if its up-down two-dimensional random walk has a normalized maximal eigenvalue which is at most $1-epsilon$. In this work, we present a class of bounded degree 2-dimensional expanders, which is the result of a small 2-complex action on a vertex set. The resulted complexes are fully transitive, meaning the automorphism group acts transitively on their faces. Such two-dimensional expanders are rare! Known constructions of such bounded degree two-dimensional expander families are obtained from deep algebraic reasonings (e.g. coset geometries). We show that given a small $d$-regular two-dimensional $epsilon$-expander, there exists an $epsilon=epsilon(epsilon)$ and a family of bounded degree two-dimensional simplicial complexes with a number of vertices goes to infinity, such that each complex in the family satisfies the following properties: * It is $4d$-regular. * The link of each vertex in the complex is the same regular graph (up to isomorphism). * It is $epsilon$ expanding. * It is transitive. The family of expanders that we get is explicit if the one-skeleton of the small complex is a complete multipartite graph, and it is random in the case of (almost) general $d$-regular complex. For the randomized construction, we use results on expanding generators in a product of simple Lie groups. This construction is inspired by ideas that occur in the zig-zag product for graphs. It can be seen as a loose two-dimensional analog of the replacement product.
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In this work we present a new local to global criterion for proving a form of high dimensional expansion, which we term cosystolic expansion. Applying this criterion on Ramanujan complexes, yields for every dimension, an infinite family of bounded degree complexes with the topological overlapping property. This answer affirmatively an open question raised by Gromov.
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.
A graph $X$ is defined inductively to be $(a_0,dots,a_{n-1})$-regular if $X$ is $a_0$-regular and for every vertex $v$ of $X$, the sphere of radius $1$ around $v$ is an $(a_1,dots,a_{n-1})$-regular graph. Such a graph $X$ is said to be highly regular (HR) of level $n$ if $a_{n-1} eq 0$. Chapman, Linial and Peled studied HR-graphs of level 2 and provided several methods to construct families of graphs which are expanders globally and locally. They ask whether such HR-graphs of level 3 exist. In this paper we show how the theory of Coxeter groups, and abstract regular polytopes and their generalisations, can lead to such graphs. Given a Coxeter system $(W,S)$ and a subset $M$ of $S$, we construct highly regular quotients of the 1-skeleton of the associated Wythoffian polytope $mathcal{P}_{W,M}$, which form an infinite family of expander graphs when $(W,S)$ is indefinite and $mathcal{P}_{W,M}$ has finite vertex links. The regularity of the graphs in this family can be deduced from the Coxeter diagram of $(W,S)$. The expansion stems from applying superapproximation to the congruence subgroups of the linear group $W$. This machinery gives a rich collection of families of HR-graphs, with various interesting properties, and in particular answers affirmatively the question asked by Chapman, Linial and Peled.
We study regular graphs in which the random walks starting from a positive fraction of vertices have small mixing time. We prove that any such graph is virtually an expander and has no small separator. This answers a question of Pak [SODA, 2002]. As a corollary, it shows that sparse (constant degree) regular graphs with many well-mixing vertices have a long cycle, improving a result of Pak. Furthermore, such cycle can be found in polynomial time. Secondly, we show that if the random walks from a positive fraction of vertices are well-mixing, then the random walks from almost all vertices are well-mixing (with a slightly worse mixing time).
Deterministic constructions of expander graphs have been an important topic of research in computer science and mathematics, with many well-studied constructions of infinite families of expanders. In some applications, though, an infinite family is not enough: we need expanders which are close to each other. We study the following question: Construct an an infinite sequence of expanders $G_0,G_1,dots$, such that for every two consecutive graphs $G_i$ and $G_{i+1}$, $G_{i+1}$ can be obtained from $G_i$ by adding a single vertex and inserting/removing a small number of edges, which we call the expansion cost of transitioning from $G_i$ to $G_{i+1}$. This question is very natural, e.g., in the context of datacenter networks, where the vertices represent racks of servers, and the expansion cost captures the amount of rewiring needed when adding another rack to the network. We present an explicit construction of $d$-regular expanders with expansion cost at most $5d/2$, for any $dgeq 6$. Our construction leverages the notion of a 2-lift of a graph. This operation was first analyzed by Bilu and Linial, who repeatedly applied 2-lifts to construct an infinite family of expanders which double in size from one expander to the next. Our construction can be viewed as a way to interpolate between Bilu-Linial expanders with low expansion cost while preserving good edge expansion throughout. While our main motivation is centralized (datacenter networks), we also get the best-known distributed expander construction in the self-healing model.
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