  We consider a dynamical system for computing Nash bargaining solutions on
graphs and focus on its rate of convergence. More precisely, we analyze the
edge-balanced dynamical system by Azar et al and fully specify its convergence
for an important class of elementary graph structures that arise in Kleinberg
and Tardos' procedure for computing a Nash bargaining solution on general
graphs. We show that all these dynamical systems are either linear or
eventually become linear and that their convergence times are quadratic in the
number of matched edges.
