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Gromov hyperbolicity of a metric space measures the distance of the spacefrom a perfect tree-like structure. The measure has a "worst-case" aspect toit, in the sense that it detects a region in the space which sees the maximumdeviation from tree-like structure. In this article we introduce an"average-case" version of Gromov hyperbolicity, which detects whether the "mostof the space", with respect to a given probability measure, looks like a tree.The main result of the paper is that if this average hyperbolicity is small,then the space can be approximately embedded in a tree. The proof uses aweighted version of Szemeredi's regularity lemma from graph theory. The resultapplies to Gromov hyperbolic spaces as well, since average hyperbolicity isbounded above by Gromov hyperbolicity. As an application, we give aconstruction of hierarchically organized pure states in any model of a spinglass that satisfies the Parisi ultrametricity ansatz.

Average Gromov hyperbolicity and the Parisi ansatz