  For any self-similar measure $\mu$ in $\mathbb{R}$, we show that the
distribution of $\mu$ is controlled by products of non-negative matrices
governed by a finite or countable graph depending only on the IFS. This
generalizes the net interval construction of Feng from the equicontractive
finite type case. When the measure satisfies the weak separation condition, we
prove that this directed graph has a unique attractor. This allows us to verify
the multifractal formalism for restrictions of $\mu$ to certain compact subsets
of $\mathbb{R}$, determined by the directed graph. When the measure satisfies
the generalized finite type condition with respect to an open interval, the
directed graph is finite and we prove that if the multifractal formalism fails
at some $q\in\mathbb{R}$, there must be a cycle with no vertices in the
attractor. As a direct application, we verify the complete multifractal
formalism for an uncountable family of IFSs with exact overlaps and without
logarithmically commensurable contraction ratios.
