  Following the work of Rosendal and Mann and Rafi, we try to answer the
following question: when is the mapping class group of an infinite-type surface
quasi-isometric to a graph whose vertices are curves on that surface? With the
assumption of tameness as defined by Mann and Rafi, we describe a necessary and
sufficient condition, called translatability, for a geometrically nontrivial
big mapping class group to admit such a quasi-isometry. In addition, we show
that the mapping class group of the plane minus a Cantor set is quasi-isometric
to the loop graph defined by Bavard, which we believe represents the first
example of a mapping class group known to be non-elementary hyperbolic.
