In joint work with Rafal Komendarczyk and Ismar Volic, we study the space of braids, that is, the loopspace of the configuration space of points in a Euclidean space.  We relate two different integration-based approaches to its cohomology, both encoded by complexes of graphs.  On the one hand, we can restrict configuration space integrals for spaces of long links to the subspace of braids.  On the other hand, there are integrals for configuration spaces themselves, used in Kontsevich’s proof of the formality of the little disks operad.  Combining the latter integrals with the bar construction and Chen’s iterated integrals yields classes in the space of braids.  We show that these two integration constructions are compatible by relating their respective graph complexes.  As one consequence, we get that the cohomology of the space of long links surjects onto the cohomology of the space of braids.