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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

The effect of a localized attack on a system with dependencies.(a), Propagation of local damage in a system of two interdependent diluted lattices with spatially constrained dependency links between the lattices (only one lattice shown here). The hole on the right is above the critical size  and spreads throughout the system while the hole on the left is below  and remains essentially the same size. (b), A localized circular failure of radius rh in a lattice with dependency links of length up to r. Outside the hole, the survival probability of a node increases with the distance ρ from the edge. The parameter ρc denotes the distance from the edge of the hole at which the occupation probability is equal to the percolation threshold of a lattice without dependencies pc ≈ 0.592736. (c), Phase diagram of a lattice with dependencies or two interdependent lattices. Depending on the average degree 〈k〉 and dependency length r, the system is either stable, unstable or metastable. The circles illustrate the increase (when 〈k〉 increases) of the critical attack size () that leads to system collapse in the metastable region. (d), As the system size grows, the minimal number of nodes which cause the system to collapse increases linearly for random attacks but stays constant (≈300) for localized attacks. This figure was obtained for a system of interdependent lattices diluted to 〈k〉 ≈ 2.9 and r = 15 (in the metastable phase-see c), with 1000 runs for each data point.
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f1: The effect of a localized attack on a system with dependencies.(a), Propagation of local damage in a system of two interdependent diluted lattices with spatially constrained dependency links between the lattices (only one lattice shown here). The hole on the right is above the critical size and spreads throughout the system while the hole on the left is below and remains essentially the same size. (b), A localized circular failure of radius rh in a lattice with dependency links of length up to r. Outside the hole, the survival probability of a node increases with the distance ρ from the edge. The parameter ρc denotes the distance from the edge of the hole at which the occupation probability is equal to the percolation threshold of a lattice without dependencies pc ≈ 0.592736. (c), Phase diagram of a lattice with dependencies or two interdependent lattices. Depending on the average degree 〈k〉 and dependency length r, the system is either stable, unstable or metastable. The circles illustrate the increase (when 〈k〉 increases) of the critical attack size () that leads to system collapse in the metastable region. (d), As the system size grows, the minimal number of nodes which cause the system to collapse increases linearly for random attacks but stays constant (≈300) for localized attacks. This figure was obtained for a system of interdependent lattices diluted to 〈k〉 ≈ 2.9 and r = 15 (in the metastable phase-see c), with 1000 runs for each data point.

Mentions: Even though different infrastructure systems have their own specific function and dynamics, they share a 2D spatial embedding that implies a fundamental restriction on their structure due the length limitation of connectivity and dependency links. Therefore we study here the general vulnerability of spatially embedded systems under localized attacks. We find here that localized attacks on spatially embedded systems with dependencies are significantly more damaging than random failures (see Fig. 1a), in marked contrast to single networks.


Localized attacks on spatially embedded networks with dependencies.

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

The effect of a localized attack on a system with dependencies.(a), Propagation of local damage in a system of two interdependent diluted lattices with spatially constrained dependency links between the lattices (only one lattice shown here). The hole on the right is above the critical size  and spreads throughout the system while the hole on the left is below  and remains essentially the same size. (b), A localized circular failure of radius rh in a lattice with dependency links of length up to r. Outside the hole, the survival probability of a node increases with the distance ρ from the edge. The parameter ρc denotes the distance from the edge of the hole at which the occupation probability is equal to the percolation threshold of a lattice without dependencies pc ≈ 0.592736. (c), Phase diagram of a lattice with dependencies or two interdependent lattices. Depending on the average degree 〈k〉 and dependency length r, the system is either stable, unstable or metastable. The circles illustrate the increase (when 〈k〉 increases) of the critical attack size () that leads to system collapse in the metastable region. (d), As the system size grows, the minimal number of nodes which cause the system to collapse increases linearly for random attacks but stays constant (≈300) for localized attacks. This figure was obtained for a system of interdependent lattices diluted to 〈k〉 ≈ 2.9 and r = 15 (in the metastable phase-see c), with 1000 runs for each data point.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f1: The effect of a localized attack on a system with dependencies.(a), Propagation of local damage in a system of two interdependent diluted lattices with spatially constrained dependency links between the lattices (only one lattice shown here). The hole on the right is above the critical size and spreads throughout the system while the hole on the left is below and remains essentially the same size. (b), A localized circular failure of radius rh in a lattice with dependency links of length up to r. Outside the hole, the survival probability of a node increases with the distance ρ from the edge. The parameter ρc denotes the distance from the edge of the hole at which the occupation probability is equal to the percolation threshold of a lattice without dependencies pc ≈ 0.592736. (c), Phase diagram of a lattice with dependencies or two interdependent lattices. Depending on the average degree 〈k〉 and dependency length r, the system is either stable, unstable or metastable. The circles illustrate the increase (when 〈k〉 increases) of the critical attack size () that leads to system collapse in the metastable region. (d), As the system size grows, the minimal number of nodes which cause the system to collapse increases linearly for random attacks but stays constant (≈300) for localized attacks. This figure was obtained for a system of interdependent lattices diluted to 〈k〉 ≈ 2.9 and r = 15 (in the metastable phase-see c), with 1000 runs for each data point.
Mentions: Even though different infrastructure systems have their own specific function and dynamics, they share a 2D spatial embedding that implies a fundamental restriction on their structure due the length limitation of connectivity and dependency links. Therefore we study here the general vulnerability of spatially embedded systems under localized attacks. We find here that localized attacks on spatially embedded systems with dependencies are significantly more damaging than random failures (see Fig. 1a), in marked contrast to single networks.

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