Articulation points in complex networks

Abstract

An articulation point in a network is a node whose removal disconnects the network. Those nodes play key roles in ensuring connectivity of many real-world networks, from infrastructure networks to protein interaction networks and terrorist communication networks. Despite their fundamental importance, a general framework of studying articulation points in complex networks is lacking. Here we develop analytical tools to study key issues pertinent to articulation points, such as the expected number of them and the network vulnerability against their removal, in an arbitrary complex network. We find that a greedy articulation point removal process provides us a different perspective on the organizational principles of complex networks. Moreover, this process results in a rich phase diagram with two fundamentally different types of percolation transitions. Our results shed light on the design of more resilient infrastructure networks and the effective destruction of terrorist communication networks.

Figure 1. Articulation points and the greedy articulation points removal process.

(a) Articulation points in the terrorist communication network from the attacks on the United States on September 11, 2001 are highlighted in red. This network contains in total 62 nodes and 153 links. (b,c) At each time step, all the articulation points and the links attached to them are removed from the network. This greedy articulation points removal procedure can be considered as a network decomposition process: at each step, all the removed nodes (because of the removal of articulation points in the current network) form a layer in the network. We peel the network off one layer after another, until there is no articulation point left. We find that this terrorist communication network consists of 3 layers, shown in light yellow, blue, and green, respectively. (d) After 3 steps, a well-defined residual giant bicomponent is left, which contains 26 of the 62 nodes. Interestingly, 16 of the 19 hijackers (highlighted with squares) are in the residual giant bicomponent, which is statistically significant (Fisher's exact test yields a two-tailed test P value 1.13 × 10⁻⁵).

Figure 2. Articulation points and the residual giant bicomponent in real networks.

(a) Fraction of articulation points versus relative size of the residual giant bicomponent is plotted for a wide range of real networks, from infrastructure networks to technological, biological, and social networks. Most of the real networks analysed here have either a very small residual giant bicomponent or a rather big one (highlighted in light magenta and turquoise, separately). (b,c) Fraction of articulation points and relative size of the residual giant bicomponent , obtained from the fully randomized counterparts of the real networks, compared with the exact values ( and ). (d,e) Fraction of articulation points and relative size of the residual giant bicomponent , calculated from the degree-preserving randomized counterparts of the real networks, compared with the exact values ( and ). In b–e, all data points and error bars (standard error of the mean or s.e.m.) are determined from 100 realizations of the randomized networks, and the dashed lines (y=x) are guide for eyes. For detailed description of these real networks and their references, see Supplementary Note 7; Supplementary Tables 1–14.

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

(a) Erdős-Rényi random networks; (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 model 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.

Figure 4. Percolation transitions associated with greedy articulation points removal.

Two types of percolation transitions of different nature are shown for Erdős-Rényi random networks. (a) Relative size of the giant connected component (GCC) after t steps of greedy articulation points removal (GAPR), n_GCC(t, c), as a function of the mean degree c. Note that n_GCC(0, c) corresponds to the ordinary percolation (orange line); n_GCC(t, c) with finite t (only t=1, 2,..., 10 are shown here) corresponds to the GCC percolation (grey lines); and n_GCC(∞, c)=n_RGB(c) corresponds to the residual giant bicomponent (RGB) percolation (thick black line). (b) Total number of the GAPR steps T(c) for c<c* (magenta line) and the characteristic number of the GAPR steps for c>c* (turquoise line) as functions of the mean degree c. (c) The critical scaling behaviour of n_GCC(t, c) and n_RGB(c) for the GCC and RGB percolation transitions, respectively. (d) The divergence of T(c) and associated with the RGB percolation transition. (e–g) Temporal behaviours of fraction of the GCC (n_GCC(t, c)), fraction of APs (n_AP(t, c)), and average number of newly induced APs per single AP removal η(t, c) at critical (black lines), subcritical (magenta lines, c−c*=−2⁴ × 10⁻⁵, −2⁶ × 10⁻⁵, −2⁸ × 10⁻⁵, −2¹⁰ × 10⁻⁵, −2¹² × 10⁻⁵, respectively) and supercritical (turquoise lines, c − c*=2⁴ × 10⁻⁵, 2⁶ × 10⁻⁵, 2⁸ × 10⁻⁵, 2¹⁰ × 10⁻⁵, 2¹² × 10⁻⁵, respectively) regions of the RGB percolation transition. At criticality, n_AP(t, c*) decays in a power-law manner for large t (inset of f).

Figure 5. Residual giant bicomponent percolation transition and the phase diagram.

(a) Relative size of the residual giant bicomponent, n_RGB, as a function of the mean degree c in the Erdős-Rényi network (red), and scale-free networks with different degree exponents, λ=4.0 (green), 3.0 (blue) and 2.5 (yellow), constructed from the static model. Lines are our theoretical predictions. Simulations are performed with network size N=10⁶. Results (symbols) are averaged over 128 realizations and the error bars (s.e.m.) are generally smaller than the symbols, except at criticality. The deviation of simulation results from our theoretical prediction for λ=2.5 owes to degree correlations present in the constructed networks, which become prominent as λ→2. Inset displays the distribution of n_RGB at criticality generated from 51,200 Erdős-Rényi networks of size N=10⁶. The bimodal distribution of n_RGB indicates that it undergoes a discontinuous jump from nearly zero to a large finite value at the critical point. (b) Phase diagram associated with the greedy articulations point removal process in scale-free networks. The residual giant bicomponent percolation transition, the giant connected component percolation transitions (only t=1, 2,..., 10 are shown here), and the ordinary percolation transition are shown in thick solid line, thin dot-dashed lines and thick dashed line, respectively. In the limit of large λ, the phase boundaries, c*(t), for Erdős-Rényi networks are recovered (indicated by arrows). Here we only show c*(t=∞) (thick solid arrow), c*(t=1) and c*(t=2) (thin dot-dashed arrows), and c*(t=0) (thick dashed arrow).