Figure 3. Fraction of articulation points in two canonical model networks.

(a) Erdős-Rényi random networks²⁰; (b) Scale-free networks with different degree exponents λ. In a, the fraction of articulation points (n_AP) is shown as red line. The probabilities of adding type-I (yellow dashed line) and type-II links (turquoise dashed line) are also shown. In b, we use the static model²⁶ to construct scale-free networks with asymptotically power-law degree distribution . Simulations are performed with network size N=10⁶ and the results (symbols) are averaged over 128 realizations with error bars (s.e.m.) smaller than the symbols. Lines are our theoretical predictions. (c–f) Illustrations of articulation points (red nodes), type-I links (yellow dashed lines) and type-II links (turquoise dashed lines) in Erdős-Rényi random networks of different mean degrees. Note that adding a single type-II link at most convert two normal nodes to articulation points (orange boxes), while adding a single type-I link could convert much more articulation points back to normal nodes (black boxes). This explains why the peak of n_AP emerges even though the probability of adding type-II links is still larger than that of adding type-I links. The largest connected component is highlighted in light blue in d–f.