In this paper, we investigate the BV exponential stability of general networks of scalar conservation laws with positive velocities 4 and under dissipative boundary conditions. The paper is divided in two parts, the first one focusing on linear controls while the last one deals 5 with saturated laws. For the linear case, the global exponential BV stability is proven. For the saturated case, we argue that we cannot expect 6 to have a basin of attraction larger than the region of linearity in a BV context. We rather prove an L ∞ local stability result. An explicit 7 estimate of the basin of attraction is given. The Lyapunov functional is inspired from Glimm's seminal work [13] reconsidered in [7]. 8