Holey graphs: very large Betti numbers are testable
Dániel Szabó, Simon Apers
TL;DR
This work addresses the problem of testing whether a graph’s clique complex has an extremely large $k$-th Betti number $\\beta_k$ in the dense graph model. It develops a reduction framework that ties $\\beta_k$ to the number of independent $k$-faces via the $k$-simplicial matroid, showing that a large $\\beta_k$ forces few independent $(k+1)$-faces and thus near $K_{k+2}$-freeness; by the graph removal lemma this becomes a tolerant testable property. Concretely, for fixed $k$ and $\\varepsilon>0$, there exists $\\delta(\\varepsilon,k)$ such that testing $\\beta_k\\ge (1-\\delta)d_k$ reduces to tolerantly testing $K_{k+2}$-freeness, implying constant-query testability in the dense model. The results leverage Euler characteristic, matroids, and standard graph-removal machinery, and extend prior bounded-degree findings by Elek (2010) into the dense regime, with implications for understanding when topological features can be inferred from large graphs.
Abstract
We show that the graph property of having a (very) large $k$-th Betti number $β_k$ for constant $k$ is testable with a constant number of queries in the dense graph model. More specifically, we consider a clique complex defined by an underlying graph and prove that for any $\varepsilon>0$, there exists $δ(\varepsilon,k)>0$ such that testing whether $β_k \geq (1-δ) d_k$ for $δ\leq δ(\varepsilon,k)$ reduces to tolerantly testing $(k+2)$-clique-freeness, which is known to be testable. This complements a result by Elek (2010) showing that Betti numbers are testable in the bounded-degree model. Our result combines the Euler characteristic, matroid theory and the graph removal lemma.