In dynamical networks, usually there are time delays among nodes during their communication. Different pairs of nodes generally have different time delays (i.e., having non-uniform communication delays). It has more practical significance to study the successive lag synchronization on dynamical networks with non-uniform communication delays. So, in this paper we construct a dynamical network model with non-uniform communication delay. Then, by designing linear feedback control and adaptive feedback control, and by using the Lyapunov function method, we obtain sufficient conditions for guaranteeing the stability of successive lag synchronization. Finally, in the numerical simulation, we choose the Chua's circuit as the local nonlinear dynamic and two kinds of topological structures for dynamical network to verify the effectiveness and correctness of obtained results.