Source-linked AI summary
Percolation in real interdependent networks
Filippo Radicchi
TL;DR
The appendix examines truncated multidimensional Taylor expansions around trivial solutions to approximate the interdependent-network equations. These expansions provide structural insight, but the resulting equation does not reduce the original problem to a simple eigenvalue/eigenvector equation.
Problem
The appendix addresses how to approximate the interdependent-network equations near their trivial solutions using Taylor expansions.
Method
It applies truncated multidimensional Taylor expansions with derivatives taken with respect to the relevant variables, including a second-order expansion for Eq. (9).
Results
The expansions yield approximations for the equations and provide insights into the solution structure.
Takeaways & Limitations
The Taylor-expansion approach helps analyze the equations, including contributions from the intersection graph.
Takeaways & Limitations
The resulting equation does not reduce the original problem to a simple eigenvalue/eigenvector equation.
Abstract
from arXiv · showhide
The function of a real network depends not only on the reliability of its own components, but is affected also by the simultaneous operation of other real networks coupled with it. Robustness of systems composed of interdependent network layers has been extensively studied in recent years. However, the theoretical frameworks developed so far apply only to special models in the limit of infinite sizes. These methods are therefore of little help in practical contexts, given that real interconnected networks have finite size and their structures are generally not compatible with those of graph toy models. Here, we introduce a theoretical method that takes as inputs the adjacency matrices of the layers to draw the entire phase diagram for the interconnected network, without the need of actually simulating any percolation process. We demonstrate that percolation transitions in arbitrary interdependent networks can be understood by decomposing these system into uncoupled graphs: the intersection among the layers, and the remainders of the layers. When the intersection dominates the remainders, an interconnected network undergoes a continuous percolation transition. Conversely, if the intersection is dominated by the contribution of the remainders, the transition becomes abrupt even in systems of finite size. We provide examples of real systems that have developed interdependent networks sharing a core of "high quality" edges to prevent catastrophic failures.
Taylor expansions
The appendix applies truncated Taylor expansions to the interdependent-network equations, including cases without and with an intersection graph. These expansions clarify solution structure but do not reduce the problem to a simple eigenvalue/eigenvector equation.
- Taylor expansions: Truncated multidimensional Taylor expansions reduce Eq. (5) to Eq. (8) by expanding around the trivial solution r = 0 with respect to ri→j.The expansion variables are the directed-edge components ri→j.
- Taylor expansions: When the intersection graph has no edges, Eq. (9) is expanded around s = 0, where the first derivatives vanish because SA−Bi and SB−Ai are zero.The resulting expansion therefore begins with higher-order terms.
- Taylor expansions: The zero values of SA−Bi and SB−Ai at s = 0 justify the second equality used in the expansion.These quantities are defined using the adjacency matrices of the network layers.
- Taylor expansions: The approximation without the intersection term is constructed from the adjacency matrices A and B and their non-intersection contributions.The intersection contribution can then be inserted into the approximation.
- Taylor expansions: The resulting equation provides structural insight but does not reduce the original interdependent-network problem to a simple eigenvalue/eigenvector equation.Similar considerations apply to the Taylor expansion of Eq. (11).