Multiple phase transitions in networks of directed networks

The robustness in real-world complex systems with dependency connectivities differs from that in isolated networks. Although most complex network research has focused on interdependent undirected systems, many real-world networks—such as gene regulatory networks and traffic networks—are directed. We thus develop an analytical framework for examining the robustness of networks made up of directed networks of differing topologies. We use it to predict the phase transitions that occur during node failures and to generate the phase diagrams of a number of different systems, including treelike and random regular (RR) networks of directed Erdős-Rényi (ER) networks and scale-free networks. We find that the the phase transition and phase diagram of networks of directed networks differ from those of networks of undirected networks. For example, the RR networks of directed ER networks show a hybrid phase transition that does not occur in networks of undirected ER networks. In addition, system robustness is affected by network topology in networks of directed networks. As coupling strength $q$ increases, treelike networks of directed ER networks change from a second-order phase transition to a first-order phase transition, and RR networks of directed ER networks change from a second-order phase transition to a hybrid phase transition, then to a first-order phase transition, and finally to a region of collapse. We also find that heterogenous network systems are more robust than homogeneous network systems. We note that there are multiple phase transitions and triple points in the phase diagram of RR networks of directed networks and this helps us understand how to increase network robustness when designing interdependent infrastructure systems.

Figure 1

Schematic of NODNs. (a)–(c) are treelike NODNs without loops. (d) and (e) are random regular NODNs with loops. (f) A partially dependent pair of two directed networks: network 3 and network 5 are coupled by directed dependency links (dotted green lines) with a no-feedback condition [19]. A dotted directed line from node $i$ in one network to node $j$ in the other network indicates that a failure of node $i$ will cause node $j$ to fail. There is a fraction $q_{35}=5/11$ of nodes in network 3 that depend on the nodes of network 5, and the $q_{53}=6/12$ fraction of nodes in network 5 depends on the nodes of network 3.

Figure 2

The percolation behaviors of treelike networks of (a) ER networks ($\langle k\rangle =10$), (b) RR networks ($k=10$), and (c) SF networks ($\lambda =2.5$). For a single network ($n=1$), the size of the final GSCC continuously decreases to zero when the fraction of remaining nodes $p$ decreases. When the system composes more than two networks ($n\ge 2$), the size of the final GSCC discontinuously jumps to zero at a critical point $p_c^{\mathrm{I}}$. The symbols represent simulation results (each network layer contains $N={10}^6$ nodes), the solid lines are theoretic results, and they agree well with each other.

Figure 3

The final GSCC sizes of the treelike networks of (a) the ER networks with different average degrees, (b) the RR networks with different degrees, and (c) the SF networks ($\lambda =2.5$) with different minimum in degrees and out degrees. The system contains $n=4$ networks. The theoretic results (solid lines) agree well with the simulation results (symbols). When the fraction of remaining nodes $p$ changes, these systems all show first order phase transitions.

Figure 4

The critical point $p_c^{\mathrm{I}}$ of treelike networks of $n$ (a) ER networks with different average degrees $\langle k\rangle$, (b) RR networks with different degrees $k$, and (c) SF networks with different degree distribution exponents $\lambda$. The results of the ER networks and the RR network are computed by Eqs. (28) and (34), respectively. The results of the SF networks are computed by substituting Eq. (37) with Eq. (17). They are in good agreement with simulations. As more networks are added in, the critical point $p_c^{\mathrm{I}}$ increase, indicating the more vulnerable the system is.

Figure 5

The minimum average degree ${\langle k\rangle }_{\min }$ in treelike networks of $n$ ER networks. The treelike networks of the ER networks with $\langle k\rangle <{\langle k\rangle }_{\min }$ completely collapse even when no node is removed ($p=1$).

Figure 6

Percolation behavior of the RR networks of (a) the ER networks ($\langle k\rangle =15$) and (b) the SF network ($\lambda =2.5$) with different coupling strengths $q$ and $l=3$. For both RR networks of the ER networks and RR networks of the SF networks, systems show second order, hybrid, and first order phase transitions with different coupling strengths $q$. The solid lines represent theoretic results, and they are in perfect agreement with the simulation results (symbols). In simulations, each layer network contains ${10}^6$ nodes, and each data point is averaged over 30 realizations.

Figure 7

The final GSCC size of the RR networks of (a) ER networks with $\langle k\rangle =15$ and $q=0.45$, and (b) SF networks with $\lambda =2.5$ and $q=0.32$. When the degree of every network of the RR network equals one, the system contains a single network and shows a second order phase transition. As $l$ increases, the system becomes more vulnerable. The solid lines represent theoretic results, and they are in perfect agreement with the simulation results (symbols, $N={10}^6$ and averaged over 30 realizations).

Figure 8

The solution of $R(z,q)$ as a function of $z$ in the RR networks of the ER networks with different coupling strengths $q$, where $\langle k\rangle =15$ and each network node has three neighbors. When $q<q_{c2}=0.4348,R(z,q)$ is a nondecreasing function of $z$, and when $z\to 1,R(z,q)$ reaches its maximum value as the cyan solid line shows. When $q_{c2}<q<q_{c1}=0.4826,R(z,q)$ shows a peak in the region $z\in (0,1)$ (red triangle), but the maximum value continues to be ${\lim }_{z\to 1}R(z,q)$ as the red dashed line shows. When $q_{c1}<q<q_{\max }=0.5136,R(z,q)$ shows a peak (blue triangle), and the peak value is also its maximum value as the blue dashed-dotted line shows. When $q\ge q_{\max },R(z,q)\le 1$ for all the $z\in [0,1]$ as the black dot line shows. The magenta solid line shows the reference line of $R(z,q)=1$. The cyan circle and the red circle represent the percolation threshold of $p_c^{\mathrm{II}}=1/{\lim }_{z\to 1}R(z,q)$, and the red triangle and blue triangle represent another percolation threshold $p_c^{\mathrm{I}}=1/R\left(z_c\right)$, where $z_c$ is the critical point where the peak appears.

Figure 9

The percolation thresholds of the RR networks of (a) ER networks with different $\langle k\rangle$'s and (b) SF networks with different $\lambda$'s. As the coupling strength $q$ increases, the system where each network node has three neighbors shows different phase transitions: (a) The RR networks of the ER networks with small average degrees, such as $\langle k\rangle =6$, show second order phase transitions until $q_{\max }$ is reached where the system suffers from collapse; when the average degree is larger, such as $\langle k\rangle =8$, the system shows a second order phase transition then changes into a hybrid phase transition and collapse when $q_{\max }$ is reached; for even greater average degrees, such as $\langle k\rangle =12$, the system displays a second order phase transition through a hybrid and then changes into a first order phase transition. (b) For the RR networks of the SF networks with small degree exponents, such as $\lambda =2.2$, the system shows a second order then a hybrid phase transition and collapse at last. For larger degree exponents, systems undergo a second order through a hybrid to a first order phase transition. The dashed-dotted lines represent the percolation threshold $p_c^{\mathrm{II}}$, and the solid lines are another percolation threshold of $p_c^{\mathrm{I}}$. These two thresholds both appear under the same $q$, meaning the system shows a hybrid phase transition.

Figure 10

Phase diagrams of the random regular networks of (a) the ER networks and (b) the SF networks where each network node has three neighbors. The green region labeled Phase II is the region of the second order phase transition. The purple region labeled Hybrid is the region of the hybrid phase transition. The blue region labeled Phase I is the region of the first order phase transition. The orange region labeled Collapse is the region where the system collapse even without removing nodes. Triple points (red dots) appear in the phase diagram.