  In this paper a double quasi Poisson bracket in the sense of Van den Bergh is
constructed on the space of noncommutative weights of arcs of a directed graph
embedded in a disk or cylinder $\Sigma$, which gives rise to the quasi Poisson
bracket of G.Massuyeau and V.Turaev on the group algebra $\mathbf
k\pi_1(\Sigma,p)$ of the fundamental group of a surface based at
$p\in\partial\Sigma$. This bracket also induces a noncommutative Goldman
Poisson bracket on the cyclic space $\mathcal C_\natural$, which is a $\mathbf
k$-linear space of unbased loops. We show that the induced double quasi Poisson
bracket between boundary measurements can be described via noncommutative
$r$-matrix formalism. This gives a more conceptual proof of the result of N.
Ovenhouse that traces of powers of Lax matrix form an infinite collection of
noncommutative Hamiltonians in involution with respect to noncommutative
Goldman bracket on $\mathcal C_\natural$.
