  We investigate dynamics of an inference algorithm termed the belief
propagation (BP) when employed in spin glass (SG) models and show that its
macroscopic behaviors can be traced by recursive updates of certain auxiliary
field distributions whose stationary state reproduces the replica symmetric
solution offered by the equilibrium analysis. We further provide a compact
expression for the instability condition of the BP's fixed point which turns
out to be identical to that of instability for breaking the replica symmetry in
equilibrium when the number of couplings per spin is infinite. This
correspondence is extended to a SG model of finite connectivity to determine
the phase diagram, which is numerically supported.
