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Percolation in Interdependent and Interconnected Networks: Abrupt Change from Second to First Order Transition

Yanqing Hu, Baruch Ksherim, Reuven Cohen, Shlomo Havlin

arXiv:1106.4128v1physics.soc-phcs.SI

TL;DR

Prior work examined coupled-network robustness with dependency links alone or connectivity links alone, while this paper studies systems with both. Using a percolation approach, it reports unusual transitions, including a jump followed by continuous decline and discontinuous changes at the boundary between transition orders.

  • Problem

    Prior studies considered coupled-network robustness with dependency links alone or connectivity links alone, leaving systems with both link types to be examined.

    Source: p. 1

  • Method

    The paper uses a percolation approach to study coupled networks with both interdependent and interconnected links.

    Source: pp. 1, 4

  • Results

    For qA = 0.35 and qB = 1, the giant component drops at ph_c = 0.66, then continuously decreases to zero as p falls to 0.556.

    Source: p. 4

  • Takeaways & Limitations

    The analysis finds that interconnected links increase robustness while interdependent links decrease it, and that their competition produces unusual phase transitions.

    Source: p. 1

    The detailed analysis is for ER networks, and the analysis assumes both dependency strengths qA and qB are greater than zero.

    Source: pp. 2, 4

Abstract

from arXiv · show

Robustness of two coupled networks system has been studied only for dependency coupling (S. Buldyrev et. al., Nature, 2010) and only for connectivity coupling (E. A. Leicht and R. M. D'Souza, arxiv:09070894). Here we study, using a percolation approach, a more realistic coupled networks system where both interdependent and interconnected links exist. We find a rich and unusual phase transition phenomena including hybrid transition of mixed first and second order i.e., discontinuities like a first order transition of the giant component followed by a continuous decrease to zero like a second order transition. Moreover, we find unusual discontinuous changes from second order to first order transition as a function of the dependency coupling between the two networks.

I. HOW TO GET gA(ϕ, φ) AND gB(ϕ, φ)

The section derives the giant-component fractions for both networks using a branching-process percolation approach. It builds generating functions for link types and adjusts them for node removal.

I. HOW TO GET gA(ϕ, φ) AND gB(ϕ, φ)

  • The branching-process approach models percolation using the networks’ degree-distribution generating functions.It constructs excess-degree generating functions for the four intra- and inter-network link types.Source: §I · p. 5
  • Removing fractions of nodes in networks A and B is represented by replacing the generating-function arguments with 1−ϕ(1−xA) and 1−φ(1−xB).The resulting functions give the fractions of each network in its giant component.Source: §I · p. 5
  • For Poisson degree distributions, the model uses average intra-network degrees kA and kB and inter-network degrees kAB and kBA.The inter-network averages may differ because the network sizes may differ.Source: §I · p. 5

II. ABRUPT JUMP ON THE BOUNDARY

At the boundary between first- and second-order transitions, the system can have two intersections satisfying the tangency condition. The analysis selects threshold solutions from the system’s nonnegative solutions.

II. ABRUPT JUMP ON THE BOUNDARY

  • The boundary is found by setting the first- and second-order critical thresholds equal and evaluating uA and uB in system (19).When reducing the equations, the selected uA and uB are maximum nonnegative solutions in [0,1].Source: §II · p. 5
  • When system (19) has multiple solutions, the threshold is the minimum nonnegative p solution, paired with the corresponding maximum u solution.The excerpt identifies this selection rule but does not state the complete resulting order-parameter jump.Source: §II · p. 5
  • At the transition boundary, two intersections can satisfy the tangency condition, one at the origin.This occurs when the system has more than one solution; the analysis selects the minimum threshold and corresponding maximum nonnegative solution.Source: §II · p. 5

III. HYBRID PHASE TRANSITION

The paper identifies hybrid transitions in which the solution jumps to a smaller positive value and then decreases continuously to zero. It also describes a first-order threshold example in which multiple tangential intersections occur.

III. HYBRID PHASE TRANSITION

  • When system (19) has another solution at ph_c in (0, 1), the transition can be hybrid: just below ph_c, uA and uB jump to smaller values, then approach zero continuously at pc.The hybrid transition is illustrated in Fig. 5.Source: §III · p. 5
  • For the example with pc = 0.5464, the maximum intersection is the physical solution and the transition is first order, although both intersections satisfy tangency.One of the tangential intersections is at the origin.Source: §III · p. 6
  • For K = 0.1, pc ≈0.556 and ph ≈0.66; as p decreases continuously, the solution jumps from the maximum intersection S to the minimum intersection Q, then decreases continuously to zero.Source: §III · p. 6
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