We introduce a novel type of stabilization map on the configuration spaces of a graph
which increases the number of particles occupying an edge. There is an induced
action on homology by the polynomial ring generated by the set of edges, and we
show that this homology module is finitely generated. An analogue of classical
homological and representation stability for manifolds, this result implies eventual
polynomial growth of Betti numbers. We calculate the exact degree of this
polynomial, in particular verifying an upper bound conjectured by Ramos. Because
the action arises from a family of continuous maps, it lifts to an action at the level of
singular chains which contains strictly more information than the homology-level
action. We show that the resulting differential graded module is almost never formal
over the ring of edges.
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