Percolation in interdependent and interconnected networks: Abrupt change from second- to first-order transitions

Robustness of two coupled networks systems has been studied separately only for dependency coupling [Buldyrev et al., Nature (London) 464, 1025 (2010)] and only for connectivity coupling [Leicht and D’Souza, e-print arXiv:0907.0894]. Here we study, using a percolation approach, a more realistic coupled networks system where both interdependent and interconnected links exist. We find rich and unusual phase-transition phenomena including hybrid transition of mixed first and second order, i.e., discontinuities like in a first-order transition of the giant component followed by a continuous decrease to zero like in a second-order transition. Moreover, we find unusual discontinuous changes from second-order to first-order transition as a function of the dependency coupling between the two networks.

Figure 1

Two types of interlinks where the dependency links (dashed arrows) are not necessarily bidirectional. The nodes of A and B are randomly connected with connectivity links (full line). The functionality of some of the A nodes (red open circles) depends on B nodes (purple solid circles) and vice versa.

Figure 2

(a) Giant components $P_{\infty }^A$ and $P_{\infty }^B$ vs a fraction of the remaining nodes, $p$, for $N=10000$, $\overline{k}=2$, and $\overline{K}=1$. Networks $A$ (open symbols) and $B$ (full symbols) are shown for different $(q_A,q_B)$ pairs: $(0.8,0.1)$ ($\circ$), $(0.1,0.8)$ ($\diamond$), and $(0.8,0.8)$ ($\square$). The symbols represent simulations and the lines represent the theory. We see three different types of behaviors: no phase transition ($\circ$), second-order phase transition ($\diamond$), and first-order phase transition ($\square$). (b) Phase diagram showing the first-order, second-order and hybrid phase transition regimes and the boundaries, for $q_B=1$ and $\overline{k}=3$. In the second-order transition regime, between the two dashed curves (red and blue), there exists a hybrid phase-transition regime [see details in Fig. 3c]. Since the hybrid transition is continuous in the neighborhood of $p_c$ and the jump occurs well above $p_c$, we classify this hybrid phase transition as a second-order phase transition.

Figure 3

(a) Size of giant components vs dependency and connectivity links strength, for $q_B=1$ and $\overline{k}=3$. The giant components size at $p_c$ changes from zero to a finite value while changing $q_A$ and $\overline{K}$. When $q_A$ and $\overline{K}$ are at the boundary of different phase transitions, the jump occurs [see Fig. 2b]. (b) The values of $P_{\infty }^A\left(p_c\right)$ ($\circ$) and $P_{\infty }^B\left(p_c\right)$ ($\square$) along the boundary for $q_B=1$ and $\overline{k}=3$. (c) A hybrid phase transition, for $q_B=1,q_A=0.35$, $\overline{k}=3$, and $\overline{K}=0.1$. According to Eqs. (5), $P_{\infty }^A$ and $P_{\infty }^B$ have the same properties as $u_A$ and $u_B$, respectively. At $p=p_c^h\approx 0.66$ the values of $u_A$ and $u_B$ jump, and then for lower $p$ values they continuously approach zero. In the inset, simulation and theoretical results are shown as symbols and lines, respectively.

Figure 4

Tangential conditions. (a) An abrupt jump on the boundary for $q_A=0.394,q_B=0.8$ and $\overline{k}=3,\overline{K}=0.2$. Here $p_c^{\mathrm{I}}=p_c^{\mathrm{II}}=0.5464$, which is the threshold of the system. Although both intersections (one of which is at the origin) satisfy the tangential condition, the $u_A^{\max },u_B^{\max }$ values are the physical solution and the transition is of the first order. (b) Hybrid transition analysis, for $q_B=1,q_A=0.35,\overline{k}=3$, and $\overline{K}=0.1$. Here $p_c\approx 0.556$ and $p_c^h\approx 0.66$. The maximal intersection S satisfies the tangential condition. When continuously decreasing $p$, the solution of the system jumps from the maximal intersection S to the minimal intersection Q and then continuously decreases to zero.