cellular model of $G$ (see Theorem \[homdim\]). For this reason we indicate the vertex set in our notation for the sink configuration spaces.\
Much of the difficulty with studying configuration spaces of graphs comes from the fact that it is difficult for points on a graph to move past one another. Sink configuration spaces attempt to mitigate this problem by allowing points to collide with one another on vertices. We will find that, in many ways, this change makes sink configuration spaces easier to study.\

$\operatorname{FI}_V$-modules {#fiddef}
-----------------------------

For the remainder of this section, we fix a graph $(G,V,E)$.\
One way we will approach the study of sink configuration spaces is through the language of asymptotic algebra. We will find that Church-Farb representation stability theory [@CF] is not quite general enough to cover these spaces (see, for instance, Theorem \[homdim\]). Instead, we will apply techniques of more general representation stability theories.\

\[freemod\] The category $\operatorname{FI}_V$ is that whose objects are finite sets, and whose morphisms are pairs $(f,g):S \rightarrow T$ of an injection of sets $f:S \hookrightarrow T$ with a map of sets $g:T-f(S) \rightarrow V$. If $(f,g)$ and $(f',g')$ are two composable morphisms, then we set $$(f,g) \circ (f',g') 