  We consider communication over the binary erasure channel (BEC) using
low-density parity-check (LDPC) codes and belief propagation (BP) decoding. For
fixed numbers of BP iterations, the bit error probability approaches a limit as
blocklength tends to infinity, and the limit is obtained via density evolution.
On the other hand, the difference between the bit error probability of codes
with blocklength $n$ and that in the large blocklength limit is asymptotically
$\alpha(\epsilon,t)/n + \Theta(n^{-2})$ where $\alpha(\epsilon,t)$ denotes a
specific constant determined by the code ensemble considered, the number $t$ of
iterations, and the erasure probability $\epsilon$ of the BEC. In this paper,
we derive a set of recursive formulas which allows evaluation of the constant
$\alpha(\epsilon,t)$ for standard irregular ensembles. The dominant difference
$\alpha(\epsilon,t)/n$ can be considered as effects of cycle-free and
single-cycle structures of local graphs. Furthermore, it is confirmed via
numerical simulations that estimation of the bit error probability using
$\alpha(\epsilon,t)$ is accurate even for small blocklengths.
