Abstract
In the analysis of the robustness of multiplex networks, it is commonly assumed that a node is functioning only if its interdependent nodes are simultaneously functioning. According to this model, a multiplex network becomes more and more fragile as the number of layers increases. In this respect, the addition of a new layer of interdependent nodes to a preexisting multiplex network will never improve its robustness. Whereas such a model seems appropriate to understand the effect of interdependencies in the simplest scenario of a network composed of only two layers, it may seem unsuitable to characterize the robustness of real systems formed by multiple network layers. In fact, it seems unrealistic that a real system evolved, through the development of multiple layers of interactions, towards a fragile structure. In this paper, we introduce a model of percolation where the condition that makes a node functional is that the node is functioning in at least two of the layers of the network. The model reduces to the commonly adopted percolation model for multiplex networks when the number of layers equals two. For larger numbers of layers, however, the model describes a scenario where the addition of new layers boosts the robustness of the system by creating redundant interdependencies among layers. We prove this fact thanks to the development of a message-passing theory that is able to characterize the model in both synthetic and real-world multiplex graphs.
2 More- Received 17 October 2016
- Publisher error corrected 8 February 2017
DOI:https://doi.org/10.1103/PhysRevX.7.011013
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Corrections
8 February 2017
Erratum
Publisher’s Note: Redundant Interdependencies Boost the Robustness of Multiplex Networks [Phys. Rev. X 7, 011013 (2017)]
Filippo Radicchi and Ginestra Bianconi
Phys. Rev. X 7, 019901 (2017)
Popular Summary
Global infrastructures are often dependent on one another. A power grid relies on a robust water supply, for example, and air traffic can affect usage of a rail system. Similar scenarios show up in financial networks as well as metabolic pathways within a cell. In all of these situations, a node in one layer of a network can control or regulate nodes in another layer. Typically in such multiplex networks with more than two layers, it is assumed that every node must be interdependent on each of its linked nodes in other layers. This assumption predicts that a multiplex network becomes increasingly susceptible to failure as the number of layers grows. We have come to the opposite conclusion: Increasing the number of layers can actually boost the robustness of a network if there are redundant interdependencies.
Network robustness is often characterized using what is known as a percolation model, a mathematical framework that monitors how global connectivity changes as individual components are damaged—for example, how travel across a subway system is impacted by closures of one or several stations. We have developed and analyzed a new percolation model where a node is still considered functional if it is operating in at least two network layers. In the case of the subway system, for instance, perhaps a bus line and an intercity rail system can still use the closed station.
Our model allows for a more comprehensive understanding of how complex systems respond to damage. This framework could be generalized, for example, to handle a variable number of minimal functioning layers or other realistic scenarios.