Geometric Correlations Mitigate the Extreme Vulnerability of Multiplex Networks against Targeted Attacks

We show that real multiplex networks are unexpectedly robust against targeted attacks on high-degree nodes and that hidden interlayer geometric correlations predict this robustness. Without geometric correlations, multiplexes exhibit an abrupt breakdown of mutual connectivity, even with interlayer degree correlations. With geometric correlations, we instead observe a multistep cascading process leading into a continuous transition, which apparently becomes fully continuous in the thermodynamic limit. Our results are important for the design of efficient protection strategies and of robust interacting networks in many domains.

Figure 1

(a) Relative size of the MCC against the fraction $p$ of nodes remaining in the system for the arXiv (layers 1 and 2) collaboration multiplex (green lines) and for its reshuffled counterpart (red dashed lines). Different lines correspond to different realizations of the targeted attacks process. (b) shows the same for the C. Elegans multiplex (layers 2 and 3), (c) for Drosophila (layers 1 and 2), and (d) for Sacc Pomb (layers 3 and 4).

Figure 2

Targeted attacks on synthetic multiplex networks generated by the GMM model (see the text). Each layer has a power law degree distribution with exponent $\gamma =2.6$, average node degree $⟨k⟩\approx 6$, and clustering $\overline{c}=0.35$. (a)–(d) Here, each layer has $N=500$ nodes, and we have set $g=1$ and $\nu =0$. (a) The relative size of the MCC as a function of the fraction of nodes remaining in the system $p$. (b) The MCC after the removal of four nodes [corresponding to the dashed blue line in (a)]. (c) The same as in (b) after the removal of 23 nodes [dashed red line in (a)]. (d) The MCC after the removal of 42 nodes [dashed yellow line in (a)]. (e) Evolution of the MCC in a two-layer synthetic multiplex with layers of size $N=2\times 10^3$ nodes. The inset shows the same for $10^6$ nodes. (f) The critical number of nodes, $\mathrm{\Delta }N$, as a function of the system size $N$ when there are no angular correlations, $g=0$, and for different radial correlation strengths. The results are averages over 60 realizations (for $N<10000$, we performed 1000 realizations). (g) The same as in (f) but for different values of the angular correlation strength $g$ and for fixed $\nu =1$. (h) shows the largest and second-largest cascade sizes (relative to the system size).

Figure 3

(a)–(c) show the distribution of component sizes (probability density function) during the evolution of the MCC for two-layer synthetic multiplexes constructed with the GMM model. Each layer has a power law degree distribution with exponent $\gamma =2.6$, average node degree $⟨k⟩=6$, and clustering $\overline{c}=0.35$. In (a)–(c), each layer has $N=5\times 10^4$ nodes. The distribution of component sizes (a) directly before the transition ($p=0.94539$) and (b) directly after ($p=0.94540$), when there are no radial or angular correlations, $\nu =0$, and $g=0$. (c) The distribution of component sizes at $p=0.9078$ when there are maximal angular correlations, $g=1$, and no radial correlations, $\nu =0$. (d) The absolute size of the second-largest MCC as a function of $p$ for different layer sizes $N$ as indicated in the legend ($\times 10^3$); for each size, the results are averages over 60 realizations of the multiplex [as in (e)]. (e) The scaling of the maximum of the second-largest MCC. The black dashed line shows a fit $\propto N^{0.84}$, while the inset shows the value of $p=p_c$ where the maximum is realized.

Figure 4

Relative mitigation of vulnerability $\mathrm{\Omega }$ [Eq. (1)] as a function of the NMI, which is a measure of the strength of angular correlations between the layers of the considered real systems (see Supplemental Material [19], Sec. IX, for details).