  The data-processing inequality, that is, $I(U;Y) \le I(U;X)$ for a Markov
chain $U \to X \to Y$, has been the method of choice for proving impossibility
(converse) results in information theory and many other disciplines. Various
channel-dependent improvements (called strong data-processing inequalities, or
SDPIs) of this inequality have been proposed both classically and more
recently. In this note we first survey known results relating various notions
of contraction for a single channel. Then we consider the basic extension:
given SDPI for each constituent channel in a Bayesian network, how to produce
an end-to-end SDPI?
  Our approach is based on the (extract of the) Evans-Schulman method, which is
demonstrated for three different kinds of SDPIs, namely, the usual
Ahslwede-G\'acs type contraction coefficients (mutual information), Dobrushin's
contraction coefficients (total variation), and finally the $F_I$-curve (the
best possible non-linear SDPI for a given channel). Resulting bounds on the
contraction coefficients are interpreted as probability of site percolation. As
an example, we demonstrate how to obtain SDPI for an $n$-letter memoryless
channel with feedback given an SDPI for $n=1$.
  Finally, we discuss a simple observation on the equivalence of a linear SDPI
and comparison to an erasure channel (in the sense of "less noisy" order). This
leads to a simple proof of a curious inequality of Samorodnitsky (2015), and
sheds light on how information spreads in the subsets of inputs of a memoryless
channel.
