  The coding problem over the Gray-Wyner network is studied from the
second-order coding rates perspective. A tilted information density for this
network is introduced in the spirit of Kostina-Verd\'u, and, under a certain
regularity condition, the second-order region is characterized in terms of the
variance of this tilted information density and the tangent vector of the
first-order region. The second-order region is proved by the type method: the
achievability part is proved by the type-covering argument, and the converse
part is proved by a refinement of the perturbation approach that was used by
Gu-Effros to show the strong converse of the Gray-Wyner network. This is the
first instance that the second-order region is characterized for a
multi-terminal problem where the characterization of the first-order region
involves an auxiliary random variable.
