  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.
