Considering the effect of the local topology structure of an edge on cascading failures, we investigate the cascading reaction behaviors on scale-free networks with respect to small edge-based initial attacks. Our aim is to explore the relationship between some parameters and universal robustness characteristics against cascading failures on scale-free networks. We find by the theoretical analysis that the Baraba'si-Albert (BA) scale-free networks can reach the strongest robustness level against cascading failures when α + β = 1, where the robustness is (...) quantified by a transition from normal state to collapse. And the network robustness has a positive correlation with the average degree. We furthermore confirm by the numerical simulations these results. (shrink)