A quantitative method for determining the robustness of complex networks

Most current studies estimate the invulnerability of complex networks using a qualitative method that analyzes the inaccurate decay rate of network efficiency. This method results in confusion over the invulnerability of various types of complex networks. By normalizing network efficiency and defining a baseline, this paper defines the invulnerability index as the integral of the difference between the normalized network efficiency curve and the baseline. This quantitative method seeks to establish a benchmark for the robustness and fragility of networks and to measure network invulnerability under both edge and node attacks. To validate the reliability of 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. The index should provide a valuable reference for determining network invulnerability in future research.

💡 Research Summary

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The paper introduces a quantitative metric, the “invulnerability index” (I), to assess the robustness of complex networks in a clear, mathematically grounded manner. Existing studies typically rely on qualitative judgments of network efficiency decay, which leads to ambiguous conclusions, especially when comparing node‑based and edge‑based attacks. The authors first define network performance as the relative size of the giant component, s(r)=˜s(T)/N, where ˜s(T) is the absolute size of the giant component after a set T of edges has been removed, N is the original node count, and r=|T|/E denotes the fraction of removed edges (E is the total number of edges).

A baseline function f(r)=1−r is then introduced. This linear baseline represents the hypothetical case where the loss of nodes is exactly proportional to the fraction of removed edges. If the empirical performance curve s(r) lies above the baseline, the network is considered robust for that attack; if it lies below, the network is deemed fragile.

The invulnerability index for a given removal fraction α is defined as the signed area between s(r) and the baseline from r=0 to r=α:
Iα = ∫₀^α