  Approximate-message passing (AMP) algorithms have become an important element
of high-dimensional statistical inference, mostly due to their adaptability and
concentration properties, the state evolution (SE) equations. This is
demonstrated by the growing number of new iterations proposed for increasingly
complex problems, ranging from multi-layer inference to low-rank matrix
estimation with elaborate priors. In this paper, we address the following
questions: is there a structure underlying all AMP iterations that unifies them
in a common framework? Can we use such a structure to give a modular proof of
state evolution equations, adaptable to new AMP iterations without reproducing
each time the full argument ? We propose an answer to both questions, showing
that AMP instances can be generically indexed by an oriented graph. This
enables to give a unified interpretation of these iterations, independent from
the problem they solve, and a way of composing them arbitrarily. We then show
that all AMP iterations indexed by such a graph admit rigorous SE equations,
extending the reach of previous proofs, and proving a number of recent
heuristic derivations of those equations. Our proof naturally includes
non-separable functions and we show how existing refinements, such as spatial
coupling or matrix-valued variables, can be combined with our framework.
