[Paper Review] Overlap properties of geometric expanders

Jacob Fox, M. Gromov+3 more|Repository of the Academy's Library (Library of the Hungarian Academy of Sciences)|May 9, 2010

Topological and Geometric Data Analysis17 references4 citations

TL;DR

This paper establishes the existence of infinite families of bounded-degree, (d+1)-uniform hypergraphs with uniformly positive overlap numbers in d-dimensional space, using probabilistic and deterministic constructions. It proves that such hypergraphs exhibit strong geometric overlap properties, with overlap numbers asymptotically approaching the limit of complete hypergraphs as n→∞, and provides a novel partitioning theorem for measures in R^d with applications to geometric combinatorics.

ABSTRACT

The {\em overlap number} of a finite $(d+1)$-uniform hypergraph $H$ is defined as the largest constant $c(H)\in (0,1]$ such that no matter how we map the vertices of $H$ into $\R^d$, there is a point covered by at least a $c(H)$-fraction of the simplices induced by the images of its hyperedges. In~\cite{Gro2}, motivated by the search for an analogue of the notion of graph expansion for higher dimensional simplicial complexes, it was asked whether or not there exists a sequence $\{H_n\}_{n=1}^\infty$ of arbitrarily large $(d+1)$-uniform hypergraphs with bounded degree, for which $\inf_{n\ge 1} c(H_n)>0$. Using both random methods and explicit constructions, we answer this question positively by constructing infinite families of $(d+1)$-uniform hypergraphs with bounded degree such that their overlap numbers are bounded from below by a positive constant $c=c(d)$. We also show that, for every $d$, the best value of the constant $c=c(d)$ that can be achieved by such a construction is asymptotically equal to the limit of the overlap numbers of the complete $(d+1)$-uniform hypergraphs with $n$ vertices, as $n ightarrow\infty$. For the proof of the latter statement, we establish the following geometric partitioning result of independent interest. For any $d$ and any $ε>0$, there exists $K=K(ε,d)\ge d+1$ satisfying the following condition. For any $k\ge K$, for any point $q \in \mathbb{R}^d$ and for any finite Borel measure $μ$ on $\mathbb{R}^d$ with respect to which every hyperplane has measure $0$, there is a partition $\mathbb{R}^d=A_1 \cup \ldots \cup A_{k}$ into $k$ measurable parts of equal measure such that all but at most an $ε$-fraction of the $(d+1)$-tuples $A_{i_1},\ldots,A_{i_{d+1}}$ have the property that either all simplices with one vertex in each $A_{i_j}$ contain $q$ or none of these simplices contain $q$.

Motivation & Objective

• To resolve a question posed by Gromov on the existence of highly overlapping, bounded-degree simplicial complexes in higher dimensions.
• To establish that overlap numbers of sparse hypergraphs can be bounded away from zero, even with bounded degree.
• To prove that the optimal overlap constant for such constructions matches the asymptotic limit of complete hypergraphs as n→∞.
• To develop a new measure partitioning result for R^d with applications to geometric overlap and combinatorial geometry.

Proposed method

• Uses the probabilistic method to show that random embeddings of hypergraphs into R^d yield high overlap with positive probability.
• Applies expander graph constructions to generate deterministic, bounded-degree hypergraphs with uniformly positive overlap.
• Introduces a novel partitioning theorem: for any ε>0 and d, there exists K such that R^d can be partitioned into k≥K measurable parts of equal measure, where almost all (d+1)-tuples of parts either all contain a common point in their simplices or none do.
• Employs concentration inequalities (via Hoeffding-type bounds) to control the deviation of overlap fractions under random vertex mappings.
• Leverages results from topological combinatorics and measure theory, including a topological proof of the Boros-Füredi theorem.
• Analyzes the overlap number of complete (d+1)-uniform hypergraphs and shows that the optimal constant c(d) for sparse constructions is asymptotically equal to the limit of these complete hypergraph overlap numbers.

Experimental results

Research questions

• RQ1 Do there exist infinite families of (d+1)-uniform hypergraphs with bounded degree and uniformly positive overlap numbers in R^d for d≥2?
• RQ2 Can the optimal overlap constant c(d) for sparse constructions be asymptotically matched by the limit of overlap numbers of complete (d+1)-uniform hypergraphs as n→∞?
• RQ3 What geometric partitioning properties ensure that most (d+1)-tuples of measurable sets in R^d have uniform simplex containment behavior?
• RQ4 Can the overlap property be preserved under non-affine, continuous mappings of simplices in higher dimensions?
• RQ5 What is the relationship between the overlap number of a hypergraph and the structure of its 1-skeleton, particularly when the skeleton is an expander?

Key findings

• The paper constructs infinite families of (d+1)-uniform hypergraphs with bounded degree and overlap number bounded below by a positive constant c(d) > 0, answering Gromov’s question affirmatively.
• The optimal overlap constant c(d) achievable by such sparse constructions is asymptotically equal to the limit of the overlap numbers of complete (d+1)-uniform hypergraphs as n→∞.
• A new measure partitioning result is proven: for any d and ε>0, there exists K such that R^d can be partitioned into k≥K equal-measure parts where all but an ε-fraction of (d+1)-tuples of parts have uniform simplex containment behavior with respect to a fixed point q.
• For d=2, the construction achieves an overlap number of at least 2/9 - o(1), matching the asymptotic bound of the Boros-Füredi theorem.
• The probabilistic method yields high-probability constructions where random vertex mappings into R^d result in a point covered by at least a c(H)-fraction of simplices, with c(H) bounded away from zero.
• The paper shows that certain constructions based on expander graphs and finite quotients of buildings yield highly overlapping simplicial complexes, but these fail to preserve overlap under non-affine continuous mappings in d≥3.