A quantitative method for determining the robustness of complex networks

•This paper proposes a quantitative method for determining the robustness.•This method can deal with both node and edge attacks and assure the similar results.•We provide a clear distinguishability on the robustness and fragility.•Some small-world networks are robust under selective node attacks and edge attacks.

Most current studies estimate the invulnerability of complex networks using a qualitative method that analyzes the decay rate of network performance. This method results in confusion over the invulnerability of various types of complex networks. By normalizing network performance and defining a baseline, this paper defines the invulnerability index as the integral of the normalized network performance curve minus the baseline. This quantitative method seeks to measure network invulnerability under both edge and node attacks and provides a definition on the distinguishment of the robustness and fragility of networks. To demonstrate the proposed method, three small-world networks were selected as test beds. The simulation results indicate that the proposed invulnerability index can effectively and accurately quantify network resilience and can deal with both the node and edge attacks. The index can provide a valuable reference for determining network invulnerability in future research.