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Localized attacks on spatially embedded networks with dependencies.

Berezin Y, Bashan A, Danziger MM, Li D, Havlin S - Sci Rep (2015)

Bottom Line: We develop a theoretical and numerical approach to describe and predict the effects of localized attacks on spatially embedded systems with dependencies.Though robust to random failures-even of finite fraction-if subjected to a localized attack larger than a critical size which is independent of the system size (i.e., a zero fraction), a cascading failure emerges which leads to complete system collapse.Our results demonstrate the potential high risk of localized attacks on spatially embedded network systems with dependencies and may be useful for designing more resilient systems.

View Article: PubMed Central - PubMed

Affiliation: Department of Physics, Bar Ilan University, Ramat Gan 52900, Israel.

ABSTRACT
Many real world complex systems such as critical infrastructure networks are embedded in space and their components may depend on one another to function. They are also susceptible to geographically localized damage caused by malicious attacks or natural disasters. Here, we study a general model of spatially embedded networks with dependencies under localized attacks. We develop a theoretical and numerical approach to describe and predict the effects of localized attacks on spatially embedded systems with dependencies. Surprisingly, we find that a localized attack can cause substantially more damage than an equivalent random attack. Furthermore, we find that for a broad range of parameters, systems which appear stable are in fact metastable. Though robust to random failures-even of finite fraction-if subjected to a localized attack larger than a critical size which is independent of the system size (i.e., a zero fraction), a cascading failure emerges which leads to complete system collapse. Our results demonstrate the potential high risk of localized attacks on spatially embedded network systems with dependencies and may be useful for designing more resilient systems.

No MeSH data available.


Related in: MedlinePlus

Dependence of the critical attack size  on the average degree 〈k〉 and the system dependency length r.(a, b), The value of  as a function of the dependency length r and average degree 〈k〉 represented as a log-scaled colormap. (a), Simulation results. We use a binary search algorithm to find the critical radius size, ie, the minimal rh for which the local attack spreads through the entire system. (b), Analytical results. The critical size is calculated as the smallest value of rh for which Eq. (3) has a self-consistent solution. For a numerical comparison between the simulation and analytical results in the metastable phase, see Supplementary Fig. 5. (c1, c2, c3), Critical attack size  as a function of average degree 〈k〉 for three r values, determined by simulations. The curves represent moving along vertical lines from bottom to top in (a) (cf. the circles in Fig. 1c). The shaded region represents the metastable region of 〈k〉 for each r. The area to the left of the shaded region is unstable and to the right is stable. (d), Critical attack size  as a function of system dependency length r for several 〈k〉 values, determined by simulations. The minimum of each curve represents the dependency length for which the system is most vulnerable to localized attacks. The numerical results in this figure were generated using a system of two interdependent diluted lattices. For comparison with a single diluted lattice composed of both, connectivity and dependency links, see Supplementary Information Fig. 1.
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f2: Dependence of the critical attack size on the average degree 〈k〉 and the system dependency length r.(a, b), The value of as a function of the dependency length r and average degree 〈k〉 represented as a log-scaled colormap. (a), Simulation results. We use a binary search algorithm to find the critical radius size, ie, the minimal rh for which the local attack spreads through the entire system. (b), Analytical results. The critical size is calculated as the smallest value of rh for which Eq. (3) has a self-consistent solution. For a numerical comparison between the simulation and analytical results in the metastable phase, see Supplementary Fig. 5. (c1, c2, c3), Critical attack size as a function of average degree 〈k〉 for three r values, determined by simulations. The curves represent moving along vertical lines from bottom to top in (a) (cf. the circles in Fig. 1c). The shaded region represents the metastable region of 〈k〉 for each r. The area to the left of the shaded region is unstable and to the right is stable. (d), Critical attack size as a function of system dependency length r for several 〈k〉 values, determined by simulations. The minimum of each curve represents the dependency length for which the system is most vulnerable to localized attacks. The numerical results in this figure were generated using a system of two interdependent diluted lattices. For comparison with a single diluted lattice composed of both, connectivity and dependency links, see Supplementary Information Fig. 1.

Mentions: We find that is entirely determined by the average degree 〈k〉 and the maximal dependency link length r. These are intensive system quantities and therefore does not grow with system size (Fig. 1d). Figs. 2c and 2d show how the critical damage size changes with respect to 〈k〉 and r for a system of size L × L = 1000 × 1000. In Fig. 2c we can see that the metastable region covers a wider range of 〈k〉 values when r increases. In the metastable phase, for every r, increases with 〈k〉 and jumps up dramatically at a certain 〈k〉 value which represents the end of the metastable phase and the beginning of the stable phase. Furthermore, we see that this jump occurs at larger 〈k〉 values for larger r values (Fig. 2c). In Fig. 2d, we see that above a certain minimum value, has an approximately linear dependence on r in the metastable region. This is due to the fact that a larger r means that a given node's dependency link can be located farther away. Thus the secondary damage from the localized attack is more dispersed and a larger attack size is required to initiate a cascade. Furthermore, we find that the critical damage size takes a minimal value and the system is most susceptible to small local attacks when r is near the stable phase. Extensive numerical simulations of over a high resolution grid of parameters 〈k〉 and r is shown in Fig. 2a and the theoretical prediction which is in good agreement is given in Fig. 2b. The theoretical description of the effect of 〈k〉 and r on is presented below.


Localized attacks on spatially embedded networks with dependencies.

Berezin Y, Bashan A, Danziger MM, Li D, Havlin S - Sci Rep (2015)

Dependence of the critical attack size  on the average degree 〈k〉 and the system dependency length r.(a, b), The value of  as a function of the dependency length r and average degree 〈k〉 represented as a log-scaled colormap. (a), Simulation results. We use a binary search algorithm to find the critical radius size, ie, the minimal rh for which the local attack spreads through the entire system. (b), Analytical results. The critical size is calculated as the smallest value of rh for which Eq. (3) has a self-consistent solution. For a numerical comparison between the simulation and analytical results in the metastable phase, see Supplementary Fig. 5. (c1, c2, c3), Critical attack size  as a function of average degree 〈k〉 for three r values, determined by simulations. The curves represent moving along vertical lines from bottom to top in (a) (cf. the circles in Fig. 1c). The shaded region represents the metastable region of 〈k〉 for each r. The area to the left of the shaded region is unstable and to the right is stable. (d), Critical attack size  as a function of system dependency length r for several 〈k〉 values, determined by simulations. The minimum of each curve represents the dependency length for which the system is most vulnerable to localized attacks. The numerical results in this figure were generated using a system of two interdependent diluted lattices. For comparison with a single diluted lattice composed of both, connectivity and dependency links, see Supplementary Information Fig. 1.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4355725&req=5

f2: Dependence of the critical attack size on the average degree 〈k〉 and the system dependency length r.(a, b), The value of as a function of the dependency length r and average degree 〈k〉 represented as a log-scaled colormap. (a), Simulation results. We use a binary search algorithm to find the critical radius size, ie, the minimal rh for which the local attack spreads through the entire system. (b), Analytical results. The critical size is calculated as the smallest value of rh for which Eq. (3) has a self-consistent solution. For a numerical comparison between the simulation and analytical results in the metastable phase, see Supplementary Fig. 5. (c1, c2, c3), Critical attack size as a function of average degree 〈k〉 for three r values, determined by simulations. The curves represent moving along vertical lines from bottom to top in (a) (cf. the circles in Fig. 1c). The shaded region represents the metastable region of 〈k〉 for each r. The area to the left of the shaded region is unstable and to the right is stable. (d), Critical attack size as a function of system dependency length r for several 〈k〉 values, determined by simulations. The minimum of each curve represents the dependency length for which the system is most vulnerable to localized attacks. The numerical results in this figure were generated using a system of two interdependent diluted lattices. For comparison with a single diluted lattice composed of both, connectivity and dependency links, see Supplementary Information Fig. 1.
Mentions: We find that is entirely determined by the average degree 〈k〉 and the maximal dependency link length r. These are intensive system quantities and therefore does not grow with system size (Fig. 1d). Figs. 2c and 2d show how the critical damage size changes with respect to 〈k〉 and r for a system of size L × L = 1000 × 1000. In Fig. 2c we can see that the metastable region covers a wider range of 〈k〉 values when r increases. In the metastable phase, for every r, increases with 〈k〉 and jumps up dramatically at a certain 〈k〉 value which represents the end of the metastable phase and the beginning of the stable phase. Furthermore, we see that this jump occurs at larger 〈k〉 values for larger r values (Fig. 2c). In Fig. 2d, we see that above a certain minimum value, has an approximately linear dependence on r in the metastable region. This is due to the fact that a larger r means that a given node's dependency link can be located farther away. Thus the secondary damage from the localized attack is more dispersed and a larger attack size is required to initiate a cascade. Furthermore, we find that the critical damage size takes a minimal value and the system is most susceptible to small local attacks when r is near the stable phase. Extensive numerical simulations of over a high resolution grid of parameters 〈k〉 and r is shown in Fig. 2a and the theoretical prediction which is in good agreement is given in Fig. 2b. The theoretical description of the effect of 〈k〉 and r on is presented below.

Bottom Line: We develop a theoretical and numerical approach to describe and predict the effects of localized attacks on spatially embedded systems with dependencies.Though robust to random failures-even of finite fraction-if subjected to a localized attack larger than a critical size which is independent of the system size (i.e., a zero fraction), a cascading failure emerges which leads to complete system collapse.Our results demonstrate the potential high risk of localized attacks on spatially embedded network systems with dependencies and may be useful for designing more resilient systems.

View Article: PubMed Central - PubMed

Affiliation: Department of Physics, Bar Ilan University, Ramat Gan 52900, Israel.

ABSTRACT
Many real world complex systems such as critical infrastructure networks are embedded in space and their components may depend on one another to function. They are also susceptible to geographically localized damage caused by malicious attacks or natural disasters. Here, we study a general model of spatially embedded networks with dependencies under localized attacks. We develop a theoretical and numerical approach to describe and predict the effects of localized attacks on spatially embedded systems with dependencies. Surprisingly, we find that a localized attack can cause substantially more damage than an equivalent random attack. Furthermore, we find that for a broad range of parameters, systems which appear stable are in fact metastable. Though robust to random failures-even of finite fraction-if subjected to a localized attack larger than a critical size which is independent of the system size (i.e., a zero fraction), a cascading failure emerges which leads to complete system collapse. Our results demonstrate the potential high risk of localized attacks on spatially embedded network systems with dependencies and may be useful for designing more resilient systems.

No MeSH data available.


Related in: MedlinePlus