Fig 1.
A schematic of our model of overload-based cascading failures on multiplex networks.
The system of multiplex networks is composed of two layers A and B (red and blue links respectively), which share the same set of nodes. At each time step t ≥ 0, each node has two load values and
, defined as the its betweenness centrality in each layer. Each node also has two fixed capacities, defined as
and
. At the beginning, the node with the largest total load
is attacked. This leads to the redistribution of loads among the remaining nodes. At the next time step, according to some pre-defined overload rule, some other nodes will also fail due to overload. This causes a cascading process of overload failures, until no more overload happens.
Fig 2.
Best choice among random/assortative/disassortative coupling schemes for two ER networks.
(a) OR rule. (b) SUM rule. (c) AND rule. N = 500, kA = kB = 6, averaged over M = 50 realizations. The color for certain αA and αB values indicates the coupling scheme with the largest mean G value. Since the largest mean G value (among the three coupling schemes) increases in general for larger αA and αB, a dashed line is added to each sub-figure to indicate the location where the best mean G reaches 0.5.
Fig 3.
Best choice among random/assortative/disassortative coupling schemes for two SF networks.
(a) OR rule. (b) SUM rule. (c) AND rule. N = 500, kA = kB = 6, averaged over M = 50 realizations. The color for certain αA and αB values indicates the coupling scheme with the largest mean G value. A dashed line is added to each sub-figure to indicate the location where the best mean G reaches 0.5, or 0.85 for (c).
Fig 4.
Best choice of overlap ratio r for two ER networks.
(a) OR rule. (b) SUM rule. (c) AND rule. N = 500, kA = kB = 6, averaged over M = 90 realizations. The color for certain αA and αB values indicates the optimal r that leads to the largest mean G value. A dashed line is added to each sub-figure to indicate the location where the best mean G reaches 0.5.
Fig 5.
Best choice of overlap ratio r for two SF networks.
(a) OR rule. (b) SUM rule. (c) AND rule. N = 500, kA = kB = 6, averaged over M = 90 realizations. The color for certain αA and αB values indicates the optimal r that leads to the largest mean G value. A dashed line is added to each sub-figure to indicate the location where the best mean G reaches 0.5.
Fig 6.
Best choice of coupling schemes for two ER networks.
(a) OR rule. (b) SUM rule. (c) AND rule. N = 500, kA = kB = 6, averaged over M = 50 realizations. The color for certain αA and αB values indicates the best coupling scheme having the largest mean G value. A dashed line is added to each sub-figure to indicate the location where the best mean G reaches 0.5.
Fig 7.
Best choice of coupling schemes for two SF networks.
(a) OR rule. (b) SUM rule. (c) AND rule. N = 500, kA = kB = 6, averaged over M = 50 realizations. The color for certain αA and αB values indicates the best coupling scheme having the largest mean G value. A dashed line is added to each sub-figure to indicate the location where the best mean G reaches 0.5, and 0.7 for (c).
Fig 8.
Best choice of network structures and coupling schemes.
(a) OR rule. (b) SUM rule. (c) AND rule. N = 500, kA = kB = 6, averaged over M = 50 realizations. The color for certain αA and αB values indicates the best type of networks and coupling scheme having the largest mean G value. A dashed line is added to each sub-figure to indicate the location where the best mean G reaches 0.5, and 0.7 for (c).
Fig 9.
Best choice of overlap ratio r for two SF networks, SUM rule.
(a) N = 1000, averaged over M = 30 realizations. (b) N = 2000, averaged over M = 30 realizations. (c) N = 3000, averaged over M = 5 realizations kA = kB = 6. The color for different αA and αB values indicates the optimal r that leads to the largest mean G value.
Fig 10.
Results on real-world coupled IPv4/IPv6 networks.
(a) Fraction of overlapping links, r1 and r2 and the system size N in the multiplex IPv4/IPv6 networks in January of different years. N = 1086, 2416, 4974, and 6550, respectively. (b) Mean G as a function of αA in the model with realistic IPv4/IPv6 networks and the SUM rule. January in different years, averaged over M = 30 realizations.