Source-linked AI summary
The extreme vulnerability of interdependent spatially embedded networks
Amir Bashan, Yehiel Berezin, Sergey V. Buldyrev, Shlomo Havlin
TL;DR
The paper analyzes interdependent lattice networks and their percolation transitions. It finds that any finite coupling produces an abrupt collapse, with zero critical dependency.
Problem
The paper examines the stability of spatially embedded interdependent networks, extending analysis beyond non-embedded networks.
Method
The paper analytically studies coupled lattice networks and their percolation transitions, including symmetric and non-symmetric interdependence.
Results
Any finite coupling between lattices produces a discontinuous, first-order percolation transition and zero critical dependency.
Takeaways & Limitations
Very weakly coupled lattice networks remain extremely vulnerable because they abruptly collapse for any finite interdependence.
Abstract
from arXiv · showhide
Recent studies show that in interdependent networks a very small failure in one network may lead to catastrophic consequences. Above a critical fraction of interdependent nodes, even a single node failure can invoke cascading failures that may abruptly fragment the system, while below this "critical dependency" (CD) a failure of few nodes leads only to small damage to the system. So far, the research has been focused on interdependent random networks without space limitations. However, many real systems, such as power grids and the Internet, are not random but are spatially embedded. Here we analytically and numerically analyze the stability of systems consisting of interdependent spatially embedded networks modeled as lattice networks. Surprisingly, we find that in lattice systems, in contrast to non-embedded systems, there is no CD and \textit{any} small fraction of interdependent nodes leads to an abrupt collapse. We show that this extreme vulnerability of very weakly coupled lattices is a consequence of the critical exponent describing the percolation transition of a single lattice. Our results are important for understanding the vulnerabilities and for designing robust interdependent spatial embedded networks.
I. SYMMETRIC INTERDEPENDENT NETWORKS
For two symmetric interdependent networks, lattice structure makes every nonzero coupling produce an abrupt first-order collapse. The transition threshold depends on single-network critical behavior, with lattice networks yielding zero critical dependency because their derivative diverges at percolation criticality.
- Model and assumptions: A fraction q of nodes in each lattice randomly depends on nodes in the other network, under a symmetric and no-feedback setup.The two networks share the same degree distribution, and dependency links are selected randomly.
- Percolation behavior: For q = 0, lattices show a continuous second-order percolation transition, whereas any q > 0 changes it to a discontinuous first-order transition.Theory and simulations are reported to be in excellent agreement.
- Critical dependency: The critical dependency qc depends on single-network behavior near criticality rather than the entire percolation curve.The transition condition is obtained by solving the self-consistency equation together with the tangency condition.
- Critical dependency: For two-dimensional lattices, β = 5/36 < 1 makes g′(pc) diverge, yielding qc = 0 and an abrupt transition for every nonzero coupling.By contrast, Erdős–Rényi networks have β = 1 and finite g′(pc), producing a finite critical dependency.
- Collapse size: At the collapse point, the lattice giant component remains finite before abruptly jumping to zero, unlike the coupled random-network case.For coupled lattices, collapse occurs when the giant component is about 1/2 of the original network, while coupled random networks at q = qc = 0.5 have zero size at criticality.
- Starlike network-of-lattices: In a starlike network-of-lattices, the root lattice collapses first for n > 2, while peripheral lattices either collapse with it or later undergo a second-order transition.For large n, peripheral lattices can remain functional after the root collapses and continuously disintegrate at pµ.
II. NON-SYMMETRIC INTERDEPENDENT NETWORKS
The paper formulates mutual percolation for two non-symmetric interdependent networks and derives conditions for their steady state and transition behavior. Unequal dependency fractions or network structures generally cause the networks to disintegrate at different points, with discontinuous behavior analyzed through self-consistency and tangential conditions.
- Steady-state formulation: The steady state of two non-symmetric interdependent networks is determined by a pair of coupled equations for the surviving variables x1 and x2.The giant component of network i is P∞,i = x_i g_i(x_i), while p_i denotes initial occupation and q_ij the relevant dependency fraction.
- Steady-state formulation: The analysis assumes equal initial occupation, p1 = p2 = p, while allowing unequal dependency fractions and network response functions.Specifically, q12 may differ from q21 and g1 may differ from g2.
- Transition behavior: Unequal dependency fractions or network structures generally prevent the two networks from disintegrating together.Network 1 is defined as the network that first collapses when p reaches its critical value.
- Transition behavior: At network 1’s critical point, x1 discontinuously jumps from above its single-network threshold to below it as p crosses pµ c.The critical solution is identified using expansions around the limiting value of x2 and a tangential condition derived from the self-consistency equation.
- Transition behavior: Decreasing q12 and q21 lowers the critical giant-component size xµ 1g1(xµ 1), and the equations identify conditions for discontinuous behavior.The solution of the self-consistency and tangential equations provides the critical point for the first-order transition.
III. INTERDEPENDENT NETWORKS WITH FEEDBACK-DEPENDENCY-LINKS
The paper extends interdependent-lattice analysis to feedback dependency links, where dependency chains can span multiple nodes and networks. Feedback makes interdependent lattices more vulnerable, increasing both the critical threshold and the giant-component size at criticality, while chain lengths diverge as dependency approaches one.
- Feedback dependency links: Removing the no-feedback constraint allows randomly chosen dependency links to form chains across the interdependent networks.A node in one network can depend on a node in the other, which may depend on another node in the first network.
- Feedback dependency links: For feedback dependency links, the governing equations are modified from the no-feedback formulation to account for these chains.The paper states the corresponding equation change and compares analytical solutions with simulations.
- Feedback dependency links: Feedback-linked interdependent lattices are more vulnerable than lattices without feedback links.Figure 5 compares critical thresholds and collapse sizes for the two dependency rules using simulations and analytical curves.
- Dependency-chain mechanism: For small dependency probability q, chain-length probabilities decay exponentially as p(l) ∼ q^l, giving chains a characteristic finite length.Under these conditions, the results resemble those for the no-feedback case.
- Dependency-chain mechanism: As q approaches 1, dependency-chain lengths diverge, system-sized chains appear, and the system becomes extremely unstable.The feedback-dependent lattice formulation also extends to a starlike network of networks with one root lattice and n − 1 peripheral lattices.