  We study the stability of low-density parity-check (LDPC) codes under
blockwise or bitwise maximum $\textit{a posteriori}$ (MAP) decoding, where
transmission takes place over a binary-input memoryless output-symmetric
channel. Our study stems from the consideration of constructing universal
capacity-achieving codes under low-complexity decoding algorithms, where
universality refers to the fact that we are considering a family of channels
with equal capacity. Consider, e.g., the right-regular sequence by Shokrollahi
and the heavy-tail Poisson sequence by Luby $\textit{et al}$. Both sequences
are provably capacity-achieving under belief propagation (BP) decoding when
transmission takes place over the binary erasure channel (BEC).
  In this paper we show that many existing capacity-achieving sequences of LDPC
codes are not universal under BP decoding. We reveal that the key to showing
this non-universality result is determined by the stability of the underlying
codes. More concretely, for an ordered and complete channel family and a
sequence of LDPC code ensembles, we determine a stability threshold associated
with them, which further gives rise to a sufficient condition such that the
sequence is not universal under BP decoding. Furthermore, we show that the same
stability threshold applies to blockwise or bitwise MAP decoding as well. We
present how stability can determine an upper bound on the corresponding
blockwise or bitwise MAP threshold, revealing the operational significance of
the stability threshold.
