Configuration spaces of points in graphs are nonsmooth analogs of braid arrangements, appearing in robotics applications and in theory of moduli spaces of tropical curves. While their cohomology is extremely difficult to understand, and depends on the combinatorics of the underlying graph, their one-point compactification is a fascinating homotopy invariant. We'll discuss some mysteriously familiar representations of the symmetric group appearing in their cohomology, how they got there, and what they have to do with cohomology of moduli spaces of tropical curves.