  It is well-known that the McCormick relaxation for the bilinear constraint
$z=xy$ gives the convex hull over the box domains for $x$ and $y$. In network
applications where the domain of bilinear variables is described by a network
polytope, the McCormick relaxation, also referred to as linearization, fails to
provide the convex hull and often leads to poor dual bounds. We study the
convex hull of the set containing bilinear constraints $z_{i,j}=x_iy_j$ where
$x_i$ represents the arc-flow variable in a network polytope, and $y_j$ is in a
simplex. For the case where the simplex contains a single $y$ variable, we
introduce a systematic procedure to obtain the convex hull of the above set in
the original space of variables, and show that all facet-defining inequalities
of the convex hull can be obtained explicitly through identifying a special
tree structure in the underlying network. For the generalization where the
simplex contains multiple $y$ variables, we design a constructive procedure to
obtain an important class of facet-defining inequalities for the convex hull of
the underlying bilinear set that is characterized by a special forest structure
in the underlying network. Computational experiments are presented to evaluate
the effectiveness of the proposed methods.
