  It is an open question to give a combinatorial interpretation of the Falk
invariant of a hyperplane arrangement, i.e. the third rank of successive
quotients in the lower central series of the fundamental group of the
arrangement. In this article, we give a combinatorial formula for this
invariant in the case of hyperplane arrangements that are complete lift
representation of certain gain graphs. As a corollary, we compute the Falk
invariant for the cone of the braid, Shi, Linial and semiorder arrangements.
