Controlling the uncertain response of real multiplex networks to random damage

We reveal large fluctuations in the response of real multiplex networks to random damage of nodes. These results indicate that the average response to random damage, traditionally considered in mean-field approaches to percolation, is a poor metric of system robustness. We show instead that a large-deviation approach to percolation provides a more accurate characterization of system robustness. We identify an effective percolation threshold at which we observe a clear abrupt transition separating two distinct regimes in which the most likely response to damage is either a functional or a dismantled multiplex network. We leverage our findings to propose a metric, named safeguard centrality, able to single out the nodes that control the response of the entire multiplex network to random damage. We show that safeguarding the function of top-scoring nodes is sufficient to prevent system collapse.

Figure 1

The mean-field approach to percolation can be misleading for characterizing the robustness of finite multiplex networks. Single realizations of the percolation process described by the corresponding size $R$ of the MCGC as a function of $p$ (blue and orange curves) are plotted together with the average size $\overline{R}$ or the MCGC measured over ${10}^6$ realizations of the initial damage. The different panels correspond to four different datasets: the American Airlines–Delta Airlines multiplex network (a), Delta Airlines–United Airlines multiplex network (b), the Drosophila Melanogaster genetic network (c), and the C. Elegans connectome (d).

Figure 2

Typical vs average size of the MCGC. We report $\overline{R}$ and $\widehat{R}$ for four different data sets: the American Airlines–Delta Airlines multiplex network (a), Delta Airlines–United Airlines multiplex network (b), the Drosophila Melanogaster genetic network (c), the C. Elegans connectome (d). The curves $\overline{R}$ vs $p$ and $\widehat{R}$ vs $p$ have been numerically calculated from the distributions $\pi \left(R\right)$ of observing a MCGC with relative size $R$. The distribution $\pi \left(R\right)$ is constructed for each value of $p$ by performing $P$ initial realizations of the damage. We use $Q={10}^6$ for all the multiplex network datasets.

Figure 3

The large deviation study of percolation on the American Airlines–United Airlines duplex network. The probability distribution $\pi \left(R\right)$ of observing a MCGC of relative size $R$ in the American Airlines–United Airlines duplex network is shown in (a) for different values of $p$. The average $\overline{R}$ and the most likely $\widehat{R}$ size of the MCGC of the same dataset are plotted as a function of $p$ in (b). These results are obtained from numerical simulations of $Q={10}^6$ random initial realizations of the damage.

Figure 4

The characterization of the two modes of the distribution $\pi \left(R\right)$. The probabilities $P(R\ge R_{\text{min}})$, $P(R<R_{\text{min}})$ (a) and the average size $\langle R|R\ge R_{\text{min}}\rangle$ and $\langle R|R>R_{\text{min}}\rangle$ of the MCGC corresponding to the two modes (b) are plotted as a function of $p$ for the American Airlines–United Airlines duplex network. These results are obtained by performing $Q={10}^6$ realizations of the initial damage.

Figure 5

Correlations and specific heat for the American Airlines–United Airlines duplex network. The specific heat of percolation $C$, the correlation coefficient ${\chi }_{\text{NN}}$ among neighbor nodes, and the correlation coefficient $\chi$ among any pair of nodes calculated for the American Airlines–United Airlines dataset are plotted as a function of $p$. The solid dashed line indicates the effective critical point $p=p_c$. These results are obtained from numerical simulations of $Q={10}^6$ random realizations of the initial damage.

Figure 6

The mean value $\overline{q}$ (a) and the standard deviation ${\sigma }_{\overline{q}}$ (b) of the overlap distribution $\rho \left(q\right)$ are plotted vs $p$ for the American Airlines–United Airlines multiplex network. These statistical properties have been numerically evaluated starting from $\tilde{Q}={10}^6$ pairs of random realizations of the initial damage.

Figure 7

The overlap distributions $\rho \left(q\right)$ among the MCGCs of the American Airlines–United Airlines duplex network are plotted for values of $p$ given by $p=0.1,0.2,0.3,0.4,0.5,0.6$ (a) and $p=0.6,0.7,0.7,0.9,0.99$ (b). These results indicate that for $p<0.6$ the typical overlap among MCGCs decreases with increasing values of $p$ while for $p>0.6$ the typical overlap among MCGCs increases as $p$ increases. This distributions are calculated starting from $\tilde{Q}={10}^6$ pairs of random realizations of the initial damage.

Figure 8

Finite-size effects of percolation on a Poisson duplex network. The distribution $\pi \left(R\right)$ of the size of the MCGC, the average $\overline{R}$, and the typical $\widehat{R}$ size of the MCGC are plotted vs $p$ for Poisson duplex networks with average degree $z=5$ and network sizes $N={10}^2$ (a), $N={10}^3$ (b), $N={10}^4$ (c). These numerical results are obtained by running $Q={10}^6$ (a),(b) and $Q={10}^5$ (c) realizations of the initial damage configurations. The dashed solid line indicates the theoretically predicted percolation threshold in the limit $N\to \infty$.

Figure 9

Effect of the damage and the safeguard of the top-ranked nodes on the robustness of the American Airlines–United Airlines duplex network. The centrality measure $\mathrm{\Delta }s$ for each of the $N=73$ airports of the American Airlines–United Airlines dataset is shown in (a). The centrality measures are evaluated by considering $P={10}^6$ realization of the initial damage at $p=p_c=0.40$ taking $R^{★}=1/\sqrt{N}<R_c=0.27$. The distribution $\pi \left(R\right)$ of the size $R$ of the MCGC at $p=p_c$ is compared to the distribution $\pi \left(R\right)$ obtained when the top-ranked nodes according to $\mathrm{\Delta }s$ are damaged for sure (b) or safeguarded (c) while the other nodes are damaged with probability $p=p_c=0.40$. The distribution in (b) and (c) are obtained considering ${10}^6$ realizations of the initial damage.