Cascading failures in coupled networks with both inner-dependency and inter-dependency links

Abstract

We study the percolation in coupled networks with both inner-dependency and inter-dependency links, where the inner- and inter-dependency links represent the dependencies between nodes in the same or different networks, respectively. We find that when most of dependency links are inner- or inter-ones, the coupled networks system is fragile and makes a discontinuous percolation transition. However, when the numbers of two types of dependency links are close to each other, the system is robust and makes a continuous percolation transition. This indicates that the high density of dependency links could not always lead to a discontinuous percolation transition as the previous studies. More interestingly, although the robustness of the system can be optimized by adjusting the ratio of the two types of dependency links, there exists a critical average degree of the networks for coupled random networks, below which the crossover of the two types of percolation transitions disappears, and the system will always demonstrate a discontinuous percolation transition. We also develop an approach to analyze this model, which is agreement with the simulation results well.

Figure 1. The minimum values of S^B, labeled as , as a function of the parameter β for different average degrees.

The value of jumps from to zero abruptly at the critical point . The lines denote the numerical solutions and the symbols denote the simulation results from 20 time realizations on networks with 10⁵ nodes.

Figure 2. Graphical solutions for eqs (5) and (6) with 〈k〉 = 8.

(a–c), , p_c ≈ 0.2513 with nonzero and . (d–f), , p_c ≈ 0.3136 with and nonzero . (g–i), . p_c ≈ 0.4539 with nonzero and .

Figure 3. The sizes of the giant components S^A and S^B vs. p.

Panels (a,b) show the results for network A and network B in coupled random networks with 〈k〉 = 8, respectively. Panels (c,d) show the results for network A and network B in coupled scale-free networks with k_min = 4, k_max = 316 and λ = 2.7, respectively. The solid lines show the theoretical predictions, and the symbols represent simulation results from 20 time realizations on networks with 10⁵ nodes.

Figure 4. The critical point p_c for different values of β.

Panel (a) shows the results for coupled random networks with different average degree. For 〈k〉 = 8, the first tricritical point β_c = 0.3929 and the second tricritical point . For 〈k〉 = 6, β_c = 0.4511 and . For 〈k〉 = 4, the two tricritical points are merged together and the coupled networks always demonstrate discontinuous percolation transition. The theoretical prediction for the continuous percolation transition points are the results of eq. (13) and the discontinuous percolation transition points are obtained as the way shown in Fig. 2. Panel (b) shows the results for coupled scale-free networks with different lower bounds k_min and the same upper bound k_min = 316. For k_min = 2, β_c = 0.1068 and . For k_min = 3, β_c = 0.1266 and . For k_min = 4, β_c = 0.1355 and . In both panels, the symbols represent simulation results from 20 time realizations on networks with 10⁵ nodes, and the solid lines represent the theoretical predictions.