This paper considers the robustness of weighted complex networks against cascading failures. We propose a betweenness-dependent node weighting method, and construct a cascading model by applying the local weighted flow redistribution rule. The cascading failures triggered by small node attacks are studied numerically and theoretically for four typical complex networks. The results show that all studied networks in which the weight of each node is its betweenness reach the strongest robustness level. Furthermore, the strategy in which the weight...