  We study when the mapping class group of an infinite-type surface $S$ admits
an action with unbounded orbits on a connected graph whose vertices are simple
closed curves on $S$. We introduce a topological invariant for infinite-type
surfaces that determines in many cases whether there is such an action. This
allows us to conclude that, as non-locally compact topological groups, many big
mapping class groups have nontrivial coarse geometry in the sense of Rosendal.
