Sparse expanders have negative curvature

TL;DR

This paper proves that bounded-degree expander graphs cannot have non-negative Ollivier-Ricci curvature, resolving a long-standing open problem and extending the result to certain curvature conditions, using limits of random graphs and spectral analysis.

Contribution

It establishes the non-existence of bounded-degree expanders with non-negative Ollivier-Ricci curvature, solving a major open problem in geometric graph theory.

Findings

Non-negative Ollivier-Ricci curvature and spectral expansion are incompatible at infinity.

The result holds even with relaxed degree and eigenvalue conditions.

The approach applies to the Bacry-Emery curvature condition CD(0,∞).

Abstract

We prove that bounded-degree expanders with non-negative Ollivier-Ricci curvature do not exist, thereby solving a long-standing open problem suggested by Naor and Milman and publicized by Ollivier (2010). In fact, this remains true even if we allow for a vanishing proportion of large degrees, large eigenvalues, and negatively-curved edges. To establish this, we work directly at the level of Benjamini-Schramm limits, and exploit the entropic characterization of the Liouville property on stationary random graphs to show that non-negative curvature and spectral expansion are incompatible "at infinity". We then transfer this result to finite graphs via local weak convergence. The same approach also applies to the Bacry-Emery curvature condition CD$(0,\infty)$, thereby settling a recent conjecture of Cushing, Liu and Peyerimhoff (2019).