Are Percolation Transitions always Sharpened by Making Networks Interdependent?

Abstract

We study a model for coupled networks introduced recently by Buldyrev et al., Nature 464, 1025 (2010), where each node has to be connected to others via two types of links to be viable. Removing a critical fraction of nodes leads to a percolation transition that has been claimed to be more abrupt than that for uncoupled networks. Indeed, it was found to be discontinuous in all cases studied. Using an efficient new algorithm we verify that the transition is discontinuous for coupled Erdos-Renyi networks, but find it to be continuous for fully interdependent diluted lattices. In 2 and 3 dimension, the order parameter exponent $\beta$ is larger than in ordinary percolation, showing that the transition is less sharp, i.e. further from discontinuity, than for isolated networks. Possible consequences for spatially embedded networks are discussed.

4 pages, including 6 figures; version 3 includes supplementary material (including 9 figures)

References in corpus (8)

• Cascade control and defense in complex networks
• Robustness of a Network of Networks
• Threshold effects for two pathogens spreading on a network
• "Explosive Percolation" Transition is Actually Continuous
• Explosive Percolation is Continuous, but with Unusual Finite Size Behavior
• Explosive percolation via control of the largest cluster
• Strongly discontinuous explosive percolation with multiple giant components
• Continuity of the Explosive Percolation Transition

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• Towards real-world complexity: an introduction to multiplex networks
• Structure of Triadic Relations in Multiplex Networks
• Cascading Failures in Complex Networks
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• Recent advances of percolation theory in complex networks
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• Cavity-based robustness analysis of interdependent networks: Influences of intranetwork and internetwork degree-degree correlations
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• Universal mechanism for hybrid percolation transitions
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• Efficient algorithm to study interconnected networks
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• Cascading failures in coupled networks with both inner-dependency and inter-dependency links
• Cascading Failures in Networks with Proximate Dependent Nodes
• The Unfolding and Control of Network Cascades
• Percolation transitions in the survival of interdependent agents on multiplex networks, catastrophic cascades, and SOS
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