Enhancing the robustness of a multiplex network leads to multiple discontinuous percolation transitions

Determining design principles that boost the robustness of interdependent networks is a fundamental question of engineering, economics, and biology. It is known that maximizing the degree correlation between replicas of the same node leads to optimal robustness. Here we show that increased robustness might also come at the expense of introducing multiple phase transitions. These results reveal yet another possible source of fragility of multiplex networks that has to be taken into the account during network optimization and design.

Figure 1

The phase diagram for interdependent percolation in multiplex networks with degree-activity distribution $P(k,B)={\delta }_{B,20}\widehat{P}\left(k\right)$, and $\widehat{P}\left(k\right)=c_1{\delta }_{k,3}+c_2{\delta }_{k,9}+c_3{\delta }_{k,30}$ is depicted with a barycentric plot featuring domains ${\mathrm{\Omega }}_{ij}$ with $i$ continuous and $j$ discontinuous phase transitions. Panels: The fraction $S$ of nodes in the Mutually connected giant component (MCGC) for the points $A,B,C,D$, of barycentric coordinates ($c_1,c_2,c_3$): $A=(0.095,0.095,0.810)$; $B=(0.87,0.04,0.09)$; $C=(0.75,0.2,0.05)$, and $D=(0.73,0.20,0.065)$. The dashed lines indicate the unstable branch and the vertical guidelines indicate the predicted positions of the discontinuous phase transitions.

Figure 2

Evolution of the size and structure of MCGC during interdependent percolation. (a)–(c) Theoretical predictions (solid lines) and simulation results (dots) for size $S$ of MCGC and the fraction ${\sigma }_k$ of nodes that have intralayer degree $k$ are plotted at different points of the phase diagram from Fig. $1$: $C$, panel (a); $B$, panel (b); and $A$, panel (c). (d)–(f) The fractions $S$ of nodes in the MCGC (solid lines) are compared against the randomized network in which replica degrees are independent random variables (RVs) (dashed lines) and shown at different points from the phase diagram: $C$, panel (d); $B$, panel (e); and $A$, panel (f). The simulations in panels (a)–(c) are averaged over 20 realizations of networks with ${10}^5$ nodes and $M=20$ layers.

Figure 3

The influence of multimodality of degree distributions on the number of phase transitions. Three examples of degree distributions given by $P(k,B)={\delta }_{B,20}{\widehat{P}}^{\prime }\left(k\right)$ are shown together with the analytically obtained fraction $S$ of nodes in the MCGC. Distributions ${\widehat{P}}^{\prime }\left(k\right)$ are obtained by applying Gaussian filters to $\widehat{P}\left(k\right)=c_1{\delta }_{k,3}+c_2{\delta }_{k,9}+c_3{\delta }_{k,30}$ with $(c_1,c_2,c_3)=(0.75,0.2,0.05)$. Case (a) depicts the unmodified $\widehat{P}\left(k\right)$ distribution; cases (b) and (c) are obtained by applying the Gaussian window filter with standard deviation $\sigma =0.55$ and $\sigma =1.1$, respectively.

Figure 4

The phase diagram for interdependent percolation in a regular multiplex network defined by degree distribution $P(k,B)={\delta }_{k,3}\tilde{P}\left(B\right), \tilde{P}\left(k\right)=c_1{\delta }_{B,1}+c_2{\delta }_{B,2}+c_3{\delta }_{B,30}$. The barycentric plot features domains ${\mathrm{\Omega }}_{ij}$ with $i$ continuous and $j$ discontinuous phase transitions. Panels: The fraction $S$ of nodes in the MCGC for points $A,B,C,D$, and $E$ having the barycentric coordinates $(c_1,c_2,c_3)$: $A=(0.85,0.06,0.09), B=(0.33,0.33,0.33), C=(0.68,0.04,0.28), D=(0.6,0.16,0.24)$, and the shared accumulation point $E=(0.70,0.11,0.18)$.

Figure 5

Barycentric plot: The phase diagram for interdependent percolation in the multiplex network with the degree-activity distribution given by $P(k,B)=c_1{\delta }_{k,5}{\delta }_{B,1}+c_2{\delta }_{k,4}{\delta }_{B,2}+c_3{\delta }_{k,3}{\delta }_{B,30}$. Panels: Comparison of the MCGC size $S$ versus the randomized null model at the points $A,B,C,D$, and $E$ having coordinates $(c_1,c_2,c_3)$: $A=(0.62,0.10,0.28), B=(0.36,0.08,0.56), C=(0.1,0.2,0.7), D=(0.32,0.48,0.2)$, and $E=(0.22,0.30,0.48).$

Figure 6

The effect of the number of layers on network robustness. The size of MCGC $S$ in a network with joint degree-activity distributions given by (a) $P(k,B)={\delta }_{B,M}\widehat{P}\left(k\right), \widehat{P}\left(k\right)=c_1{\delta }_{k,3}+c_2{\delta }_{k,9}+c_3{\delta }_{k,30}$ with $(c_1,c_2,c_3)=(0.75,0.2,0.05)$, and (b) $P(k,B)={\delta }_{B,M}(c_1{\delta }_{k,3^1}+c_2{\delta }_{k,3^2}+c_3{\delta }_{k,3^3}+c_4{\delta }_{k,3^4}+c_5{\delta }_{k,3^5}+c_6{\delta }_{k,3^6}$ where $(c_1,c_2,c_3,c_4,c_5,c_6)=(0.5718,0.3049,0.0953,0.0210,0.0057,0.0013)$. The numbers of layers $M$ are indicated in the legends.