  Warning Propagation is a combinatorial message passing algorithm that unifies
and generalises a wide variety of recursive combinatorial procedures. Special
cases include the Unit Clause Propagation and Pure Literal algorithms for
satisfiability as well as the peeling process for identifying the $k$-core of a
random graph. Here we analyse Warning Propagation in full generality on a very
general class of multi-type random graphs. We prove that under mild assumptions
on the random graph model and the stability of the the message limit, Warning
Propagation converges rapidly. In effect, the analysis of the fixed point of
the message passing process on a random graph reduces to analysing the process
on a multi-type Galton-Watson tree. This result corroborates and generalises a
heuristic first put forward by Pittel, Spencer and Wormald in their seminal
$k$-core paper (JCTB 1996).
