  We define the graph product of unital completely positive maps on a universal
graph product of unital C*-algebras and show that it is unital completely
positive itself. To accomplish this, we introduce the notion of the
non-commutative length of a word, and we obtain a Stinespring construction for
concatenation. This result yields the following consequences. The graph product
of positive-definite functions is positive-definite. A graph product version of
von Neumann's Inequality holds. Graph independent contractions on a Hilbert
space simultaneously dilate to graph independent unitaries.
