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We revisit the Wyner-Ahlswede-K&#x00F6;rner network, focusing especially on the converse part of the dispersion analysis, which is known to be challenging. Using the functional-entropic duality and the reverse hypercontractivity of the transposition semigroup, we lower bound the error probability for each joint type. Then by averaging the error probability over types, we lower bound the c-dispersion (which characterizes the second-order behavior of the weighted sum of the rates of the two compressors when a nonvanishing error probability is small) as the variance of the gradient of <inline-formula> <tex-math notation="LaTeX">$\inf _{ {P}_{ {U}| {X}}}\{{ { cH}}( {Y}| {U})+ {I}( {U}; {X})\}$ </tex-math></inline-formula> with respect to <inline-formula> <tex-math notation="LaTeX">$ {Q}_{{{ XY}}}$ </tex-math></inline-formula>, the per-letter side information and source distribution. In comparison, using standard achievability arguments based on the method of types, we upper-bound the c-dispersion as the variance of <inline-formula> <tex-math notation="LaTeX">$ {c}\imath _{ {Y}| {U}}( {Y}| {U})+\imath _{ {U}; {X}}( {U}; {X})$ </tex-math></inline-formula>, which improves the existing upper bounds but has a gap to the aforementioned lower bound. Our converse analysis should be immediately extendable to other distributed source-type problems, such as distributed source coding, common randomness generation, and hypothesis testing with communication constraints. We further present improved bounds for the general image-size problem via our semigroup technique.

Dispersion Bound for the Wyner-Ahlswede-Körner Network via a Semigroup Method on Types