This paper investigates the consensus taking place on the edges of networks by defining dynamics to edges. Discrete-time and continuous-time edge consensus protocols are proposed in which each edge regulates itself according to the information of itself and its neighbouring edges. By mapping the original graph to its corresponding line graph, conditions guarantee the reaching of edge consensus are presented. Furthermore, the influence of network structure on the speed of reaching edge consensus is analysed by considering a scale-free network with a tunable exponent between 2 and . It is found that the convergence speed increases as the network becomes more heterogeneous. In addition, as original network becomes more homogeneous, i.e. as the exponent increases from 2, the corresponding line graph becomes more heterogeneous till reaches 2.5, after that, the line graph becomes more homogeneous.