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The effect of interdependence on the percolation of interdependent networks

J. Jiang, W. Li, X. Cai

arXiv:1402.6555v1physics.soc-phcs.SI

TL;DR

The paper asks how dependence relations and strength affect robustness in fully interdependent SF and ER networks under random failure. It develops two stochastic models with exponential or power-law dependence degrees and analyzes their percolation behavior theoretically and numerically. SF and ER networks respond oppositely to dependence strength, while power-law dependence is more robust than exponential dependence above a specified strength.

  • Problem

    The paper investigates how interdependence relations and strength influence the robustness and percolation of coupled networks under random attack.

    Source: §1 · p. 3

  • Method

    Two stochastic models generate fully interdependent SF or ER networks with random or preferential cross-network attachment, multi-support undirectional dependence links, and varied dependence strength.

    Source: §1 · p. 3, §2 · p. 4, §3 · p. 9

  • Results

    SF networks undergo second-order transitions and become less robust with stronger dependence, whereas ER networks show first-order transitions and increased robustness; power-law dependence is more robust than exponential dependence above a certain strength.

    Source: §4 · pp. 13–14

  • Takeaways & Limitations

    Dependence structure and strength produce network-class-specific robustness behavior, with power-law relations yielding greater robustness than exponential relations at sufficiently high dependence strength.

    Source: §4 · p. 14

    The paper notes that accurate theoretical analysis for the power-law case remains necessary, and future work could include inter-connectivity links between subnetworks.

    Source: §3 · p. 11, §4 · p. 14

Abstract

from arXiv · show

Two stochastic models are proposed to generate a system composed of two interdependent scale-free (SF) or Erd\H{o}s-R\'{e}nyi (ER) networks where interdependent nodes are connected with exponential or power-law relation, as well as different dependence strength, respectively. Each subnetwork grows through the addition of new nodes with constant accelerating random attachment in the first model but with preferential attachment in the second model. Two subnetworks interact with multi-support and undirectional dependence links. The effect of dependence relations and strength between subnetworks are analyzed in the percolation behavior of fully interdependent networks against random failure, both theoretically and numerically, and as a result, for both relations: interdependent SF networks show a second-order percolation phase transition and increased dependence strength decreases the robustness of the system, whereas, interdependent ER networks show the opposite results. In addition, power-law relation between networks yields greater robustness than exponential one at given dependence strength.

1 Introduction

Interdependence can trigger cascading failures and make network systems more vulnerable, but existing studies have largely assumed random cross-network connections. This work examines how dependence relations and strength affect robustness in interdependent SF and ER networks.

  • Interdependent-network failures can propagate recursively across subnetworks, producing cascades and increasing vulnerability under random attack.Source: §1 · p. 2
  • Most prior studies modeled randomly connected interdependent nodes, despite real systems often pairing highly connected nodes across subnetworks.Source: §1 · p. 2
  • Correlated interdependence has been reported to enhance robustness, while reduced dependence strength can change the percolation transition and improve resilience.Source: §1 · pp. 2–3
  • The paper generates exponential and power-law dependence relations, constructs interdependent SF and ER networks, and analyzes dependence strength under random attack.Source: §1 · p. 3

2 The first model

The first model constructs interdependent scale-free networks with exponential or power-law dependence relations and studies cascading failures under random attack. Dependence strength produces continuous percolation transitions, while sufficiently strong power-law dependence is more robust than exponential dependence.

Network construction

  • The model grows paired scale-free subnetworks with connectivity links and randomly or preferentially assigned dependence links, where q controls dependence strength.Each new node divides its links between intra-network connectivity and cross-network dependence, producing exponential or power-law dependence-degree distributions.Source: §2 · pp. 3–4

Cascade dynamics

  • Cascading failures begin with random removal in subnetwork A and iteratively remove nodes lacking giant-component membership or dependence links until the system stabilizes.The process retains only nodes in the giant component with at least one dependence link, propagating failures between subnetworks.Source: §2 · p. 4, Figure 1 · p. 5

Percolation behavior

  • For both dependence relations, ψ∞ decreases continuously to zero at pc > 0, indicating a second-order percolation transition.The simulations use N = 10^4 and m = 5, while the theoretical analysis uses generating functions for degree distributions and branching processes.Source: §2 · pp. 6–7, Figure 2 · p. 7

Dependence strength

  • Increasing dependence strength q raises pc for both relations, implying lower network resilience under random failure.The proposed explanation is that stronger dependence leaves fewer connectivity links, lowers mean connectivity degree, and accelerates inter-network failure cascades.Source: §2 · p. 7

Dependence relation

  • When q > 0.2, power-law dependence has a smaller pc than exponential dependence, indicating greater robustness at the same dependence strength.For q near zero, the scale-free network approaches the single-network result pc = 0; around q = 0.2, the two relations have similar pc.Source: §2 · p. 7, Figure 3 · p. 8
  • The greater robustness of power-law dependence is attributed to reducing the vulnerability created when hubs depend on weak nodes in the other network.This result is consistent with findings that positively correlated degrees among dependent nodes improve robustness relative to random coupling.Source: §2 · pp. 7–8

3 The second model

The second model generates two interdependent ER networks with exponential or power-law dependence relations and analyzes their percolation theoretically and numerically. Increasing dependence strength improves ER-network robustness, while power-law dependence is generally more robust than exponential dependence in simulations.

3 The second model

  • The model constructs two equivalent ER subnetworks by simultaneously adding nodes and randomly or preferentially attaching dependence links with strength DL.Random dependence produces exponential dependence degrees, whereas preferential dependence produces power-law dependence degrees.Source: §3 · pp. 8–9

3.1 Numerical simulations

  • Increasing dependence strength changes ER networks from first-order to second-order percolation behavior and improves robustness, approaching the single-network threshold pc = 0.25.For finite DL, giant components change discontinuously; as DL approaches infinity, the threshold approaches 0.25 and the transition becomes second-order.Source: §3 · p. 10, §3.1 · p. 11
  • For power-law dependence, simulations show slight theory deviations at DL = 1 because fat-tailed dependence degrees leave highly dependent nodes functional under larger attacks.The simulation threshold can therefore be smaller than the theoretical prediction.Source: §3 · p. 9, §3.1 · p. 11
  • The critical threshold pc decreases as DL increases for both dependence relations and approaches 0.25, the single-network random-percolation threshold.This limit is also obtained analytically as pc = 1/a for fixed a when DL approaches infinity.Source: §3 · p. 10, §3.1 · p. 12, Figure 5 · p. 13
  • In simulations, power-law dependence has a smaller pc than exponential dependence at each tested DL, indicating greater robustness except for weak theoretical dependence at DL = 1.The reported exception is attributed to weak dependence strength.Source: §3.1 · pp. 12–13, Figure 5 · p. 13

4 Conclusions and discussion

The paper develops stochastic models for fully interdependent SF and ER networks with distinct dependence relations and strengths under random attack. It finds that dependence strength has opposite robustness effects in SF and ER systems, while more complex interconnectivity remains a future extension.

  • The framework models exponential and power-law dependence through random and preferential attachment in fully interdependent SF and ER networks under random attack.Source: §4 · p. 13
  • Interdependent SF networks become more vulnerable as dependence strength increases and exhibit only second-order phase transitions.Source: §4 · p. 14
  • Interdependent ER networks show the opposite pattern: increased dependence strength enhances robustness and produces first-order phase transitions.Source: §4 · p. 14
  • The models can be extended by incorporating inter-connectivity links between subnetworks to study richer percolation behavior.Source: §4 · p. 14
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