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Network robustness of multiplex networks with interlayer degree correlations
Byungjoon Min, Su Do Yi, Kyu-Min Lee, K. -I. Goh
TL;DR
The paper asks how interlayer degree correlations affect the robustness of multiplex networks across ordinary connectivity, mutual connectivity, and biconnectivity. It uses generating-function formalism to analyze these properties under node removals and derives contrasting robustness behavior for maximally correlated and anti-correlated duplex networks. Correlation can lower percolation thresholds, while positive and negative coupling produce distinct biconnectivity and failure responses.
Problem
The paper addresses the lack of unified understanding of how interlayer degree correlations affect diverse robustness properties of multiplex networks.
Method
The paper uses generating-function formalism based on joint degree distributions and coupled self-consistency equations to analyze multiplex connectivity and biconnectivity.
Results
More correlated coupling lowers the percolation threshold; MP duplexes have equal giant bicomponent and unicomponent sizes, whereas MN bicomponents emerge later and grow more slowly near threshold.
Takeaways & Limitations
MP coupling provides a well-connected structure even at sparse link density, while MN coupling hinders low-density bicomponent emergence but can establish biconnected structure at higher density.
Abstract
from arXiv · showhide
We study the robustness properties of multiplex networks consisting of multiple layers of distinct types of links, focusing on the role of correlations between degrees of a node in different layers. We use generating function formalism to address various notions of the network robustness relevant to multiplex networks such as the resilience of ordinary- and mutual connectivity under random or targeted node removals as well as the biconnectivity. We found that correlated coupling can affect the structural robustness of multiplex networks in diverse fashion. For example, for maximally-correlated duplex networks, all pairs of nodes in the giant component are connected via at least two independent paths and network structure is highly resilient to random failure. In contrast, anti-correlated duplex networks are on one hand robust against targeted attack on high-degree nodes, but on the other hand they can be vulnerable to random failure.
I. INTRODUCTION
Multiplex networks contain interacting layers whose interlayer degree correlations shape robustness, but their effects across connectivity notions remain insufficiently unified. The paper studies these effects using representative positive, uncorrelated, and negative coupling structures.
- I. INTRODUCTION: Interlayer degree correlations are common in real-world multiplex networks and can align or oppose node degrees across layers.Positive correlation makes hubs in one layer hubs in another, whereas negative correlation pairs hubs with low-degree nodes across layers.
- I. INTRODUCTION: Network robustness concerns structural resilience to external perturbations and has practical relevance for designing structures resistant to random breakdowns or intentional attacks.Biconnectivity captures backup pathways that preserve communication after removal of one route.
- I. INTRODUCTION: The paper examines robustness through ordinary connectivity, mutual connectivity, and biconnectivity under random or degree-targeted node or link removal.Mutual connectivity requires simultaneous connectivity through every coupled layer, while biconnectivity requires at least two independent paths between nodes.
- I. INTRODUCTION: The analysis compares maximally-positive, uncorrelated, and maximally-negative duplex networks as theoretically simple limiting structures.In maximally-positive coupling, degree order matches across layers; in maximally-negative coupling, it is reversed, although real networks generally lie between these limits.
- I. INTRODUCTION: A giant bicomponent spanning a finite fraction of the network is important for stable connectivity because each node has an alternative route preserving its connection.This redundancy can allow nodes to endure failures when restoration is faster than successive failures.
A. Generating function method
The paper generalizes generating-function methods to multiplex networks by modeling joint degree distributions and coupled edge-following probabilities. It derives giant bicomponent sizes and their existence condition from self-consistency equations and a Jacobian eigenvalue.
- A. Generating function method: The method defines a generating function for the joint degree distribution P(k⃗) across n layers with distinct link types.The degree vector contains each node’s degree in every layer, and auxiliary variables couple to these degrees.
- A. Generating function method: Following a randomly chosen i-type link defines the remaining-degree generating function used to formulate the multiplex connectivity equations.The approach uses auxiliary variables x⃗=(x_1,x_2,· · ·,x_n) coupled to the layer-specific degree vector.
- A. Generating function method: On locally tree-like networks, u_i is the probability that a node reached through an i-type edge does not belong to the giant component.The probabilities u_i satisfy coupled self-consistency equations for i=1,2,· · ·,n.
- A. Generating function method: The giant bicomponent size B is computed as the complementary probability that a randomly chosen node has at most one link leading to the giant component.The resulting expression separates the giant unicomponent size S from the difference between S and B.
- A. Generating function method: A giant bicomponent exists when the largest eigenvalue of the Jacobian at (1,···,1) exceeds unity.For duplex networks, this eigenvalue is expressed using layer-specific κ_i and the cross-layer degree moment K_i=⟨k_1k_2⟩.
B. Results
Correlations strongly reshape bicomponent robustness in duplex ER networks: positive coupling supports bic connectivity at arbitrarily low nonzero density, whereas negative coupling delays its emergence but eventually spans the network.
- More correlated coupling lowers the percolation threshold for duplex ER networks.
- For MP coupling, the giant bicomponent equals the giant unicomponent, so every pair in the giant unicomponent has at least two independent paths.
- MP coupling supports a giant bicomponent at any nonzero link density, providing well-connected structure even in sparse networks.
- For MN coupling, the giant bicomponent emerges only after zMN_c = 0.838..., then grows more slowly than the giant unicomponent.
- Near zMN_c, BMN scales as (z − zc)^βB with βB = 2, while the unicomponent has βS = 1 for all coupling types.
- When z > z* = 1.146..., MN coupling connects the entire network into one bicomponent, eliminating the gap between BMN and SMN.
III. ERROR AND ATTACK TOLERANCE
The paper extends generating-function analysis to robustness after random or degree-targeted node removal in multiplex networks, combining analytic calculations with simulations on duplex ER networks.
- The section examines error and attack tolerance in multiplex networks with interlayer degree correlations.
- A node-removal probability φ(k⃗) encodes the removal strategy, including uniform random failure and total-degree-based intentional attack.
- The method defines generating functions for remaining joint and excess degrees after node removal.
- Coupled self-consistency equations give the probability vi that a node reached through an i-type link is outside the giant component.
- The remaining giant-component size S is obtained using the selected removal function φ(k⃗), with analytic predictions compared against numerical simulations.
- The main analyses use duplex Erdős-Rényi networks with equal layer density z before outlining other graph ensembles and coupling types.
B. Error tolerance: Random node removals
Under random node removals, MP coupling is consistently the most resilient and MN coupling the most vulnerable among the compared duplex ER structures, with correlation effects weakening in dense networks.
- MP coupling is more resilient and MN coupling more vulnerable than the other couplings under random node removal.
- The MP critical failure fraction is always larger than those of UC and MN, so more nodes must be removed to destroy connectivity at fixed z.
- The rescaled giant-component size S/S(0) is larger for MP coupling than for the other cases at every removal fraction f.
- The authors associate MP’s high robustness with the skewness of its total-degree distribution, while MN is more vulnerable for the opposite reason.
- Correlation effects become less significant as network density increases.
C. Attack vulnerability: Targeted node removals
Under attacks targeting nodes by total degree, robustness depends jointly on coupling type and density: MP and MN reverse their relative advantages across sparse and dense regimes.
- For total-degree attacks, structural robustness depends on both multiplex coupling type and link density.
- When z < zα = 1.460..., MP coupling is more robust than UC, but when z > zα it is more vulnerable.
- MN coupling shows the opposite density dependence: it is more robust than UC in dense networks but more vulnerable in sparse networks.
- The critical attack fraction versus mean degree has a more complicated pattern than random-failure results, including anomalous decreases in narrow windows.
- Despite degree skewness, MP coupling can remain more attack-robust than UC at sufficiently sparse link density.
D. Other multiplex coupling factors
The paper examines unequal layer densities, partial interlayer degree correlations, and triplex coupling combinations to determine how these factors alter robustness under failure and attack.
- Unequal mean degrees: For layers with z1 = 3z2, random-failure robustness matches equal-total-degree results, with MP most robust and MN least robust.
- Unequal mean degrees: Under targeted attack for z1 = 3z2 = 3, the ordering reverses relative to random failure.
- Unequal mean degrees: For z1 = 3z2, MP becomes more vulnerable than UC when total mean degree exceeds (z1 + z2)α = 2.522..., shrinking MP's attack-robust regime.
- Non-maximal correlations: Partial correlation mixes a fraction q of maximally correlated node pairs with a fraction 1 − q of randomly coupled pairs.
- Triplex coupling: In triplex networks, MP-MP is most robust to random failure but fragile to targeted attack, whereas MN-MP shows the opposite pattern.
- Triplex coupling: MP-UC and MN-UC produce intermediate robustness between their corresponding maximally correlated and UC-UC configurations.
E. Multiplex scale-free networks
The paper studies correlated multiplex scale-free layers using static-model networks with γ = 5/2 and compares their robustness under random failure and targeted attack.
- Figure 6 compares giant-component size for random failure at z = 1 and total-degree attacks at z = 1 and 4.
- Static-model layers generate scale-free degree distributions with exponent γ = (µ + 1)/µ and tunable mean degree z.
- For γ = 5/2 ≤ 3, all coupling types remain highly robust to random failure because each layer has strong degree heterogeneity.
- Under random failure, MP coupling is most robust and MN coupling least robust, with only small differences among coupling types.
- Under targeted attack, MN coupling is more resilient in dense networks but more vulnerable in sparse networks.
IV. MUTUAL CONNECTIVITY
This section develops generating-function analyses of mutual connectivity in duplex and triplex multiplex networks and compares analytical predictions with numerical results.
- Mutual connectivity requires simultaneous connectivity across every interdependent layer, and its giant component size is obtained using generating functions.
- The giant mutual component emerges discontinuously, unlike the continuous transition of ordinary percolation.
- For duplex ER networks, MP coupling has the lowest mutual-percolation threshold, whereas MN coupling requires denser networks for emergence.
B. Mutual connectivity under node removals
The paper combines generating-function theory with simulations to analyze giant mutual-component robustness under random node removal and degree-targeted attack.
- Coupled self-consistency equations determine the probability that a node reached through layer i does not belong to the giant mutual component after node deletion.
- The giant mutual-component size M after node removals is computed from the resulting probabilities.
- Under random removal in equal-density duplex ER networks, MP is more robust and MN more vulnerable than the other coupling types.
- Under targeted attack, robustness rankings vary with density: MP is favored near z ≈ 3.2, MN near z ≈ 4, and MN again at z ≈ 8 while MP is most vulnerable.
- These attack results show that correlated multiplexity affects mutual-connectivity robustness non-monotonically and depends strongly on interdependency details.
V. A REAL-WORLD EXAMPLE
The real-world Italian Internet–power transmission multiplex network is tested under random failure and degree-targeted attack, with rewired couplings used for comparison. It tolerates extensive random removals but rapidly disintegrates when highest-degree interdependent nodes are targeted, while its behavior resembles MN coupling.
- Real-world network: The study evaluates functional interdependent nodes in Italy’s Internet backbone and high-voltage electrical transmission multiplex network under node removals.The analysis uses an interdependent cascade model and examines both random failure and degree-based targeted attack.
- Random failure: Around 80% interdependent-node removals can be endured under random failure, based on the rescaled functional-node fraction Φ/Φ(0).Here Φ(0) is the fraction of functional nodes at zero removals.
- Targeted attack: Targeted removal of as few as 20% of the highest-degree interdependent nodes causes rapid disintegration.The targeted attack is degree-based and focuses on the highest-degree interdependent nodes.
- Correlated couplings: MN coupling is more robust to targeted attacks on high-degree nodes than MP coupling in rewired multiplex networks.The rewired-network comparison isolates the effect of correlated interdependency couplings.
- Correlated couplings: The real-world network’s behavior lies close to MN coupling despite significant differences in their actual interdependency patterns.This comparison concerns the targeted-attack vulnerability shown for the real-world and rewired networks.