Source-linked AI summary
Assortativity Decreases the Robustness of Interdependent Networks
Di Zhou, Gregorio D'Agostino, Antonio Scala, H. Eugene Stanley
TL;DR
Interdependent networks can suffer cascading failures, motivating a study of how internal topology affects their robustness. Using a two-network interacting failure model with ER and scale-free networks, the paper varies assortativity and finds that higher assortativity lowers robustness, especially for scale-free systems.
Problem
The paper asks how internal topology affects cascading failures in interdependent networks, where failures can disrupt critical infrastructure.
Method
The study simulates two interdependent networks using the interacting failure model while varying assortativity in ER and scale-free network configurations.
Results
Higher assortativity increases the percolation threshold and therefore decreases robustness, with a much stronger threshold variation in scale-free than ER networks.
Takeaways & Limitations
The findings identify internal degree correlations as a topology feature relevant to assessing and improving the robustness of interdependent network architectures.
Abstract
from arXiv · showhide
It was recently recognized that interdependencies among different networks can play a crucial role in triggering cascading failures and hence system-wide disasters. A recent model shows how pairs of interdependent networks can exhibit an abrupt percolation transition as failures accumulate. We report on the effects of topology on failure propagation for a model system consisting of two interdependent networks. We find that the internal node correlations in each of the two interdependent networks significantly changes the critical density of failures that triggers the total disruption of the two-network system. Specifically, we find that the assortativity (i.e. the likelihood of nodes with similar degree to be connected) within a single network decreases the robustness of the entire system. The results of this study on the influence of assortativity may provide insights into ways of improving the robustness of network architecture, and thus enhances the level of protection of critical infrastructures.
I. INTRODUCTION
Interdependent critical infrastructures are vulnerable to cascading failures, while prior work has not established how the topology of each component network affects that vulnerability. Using complex-network models, the paper studies assortativity as a topology parameter and finds that it reduces robustness in coupled systems.
- Motivation: Critical infrastructures such as water, electricity, and communications are essential services whose interdependence can produce cascading failures and widespread disruption.The paper cites the September 2003 blackout across Italy as an example of catastrophic failure propagation.
- Motivation: Whole-system simulations can capture regional details but often require extensive data and effort, while privacy and access constraints limit their use for understanding general cascade mechanisms.The paper therefore motivates abstract approaches that isolate fundamental issues underlying cascading effects.
- Approach: The complex-network paradigm provides a way to investigate vulnerabilities and interdependencies among critical infrastructures without modeling every system detail.It is presented as a tool for focusing more detailed analyses.
- Scope: Interdependent-network research applies beyond critical infrastructures, including public cooperation, epidemic spreading, and human physiology.
- Research gap: The topology of each component network remains understudied, and interdependency levels may be fixed by economic or technological constraints.This motivates examining internal topology when interdependencies cannot be tuned.
- Contribution: Using the interacting failure model, the paper finds that assortativity decreases robustness to random failure in coupled ER and scale-free networks.For scale-free networks with power-law degree distributions, assortativity produces a sharp increase in fragility.
II. METHODS
The methods characterize network structure through degree distributions and assortativity, which measures degree similarity between neighboring nodes. The paper situates assortativity among competing structural effects and defines it through edge-based degree correlations.
- Network characterization: The node-degree distribution P(k) is a fundamental descriptor of network structure and can have heavy tails associated with effects such as disappearing epidemic thresholds.Other structural descriptors include triangle density, modular structure, communities, and motifs.
- Assortativity: Assortativity measures the tendency of neighboring nodes to have similar degrees through a link-averaged, two-point degree correlation.It extends a one-point average-degree characterization of network topology.
- Assortativity: The assortativity coefficient r is defined from correlations between the degrees of adjacent vertices, with edge averages taken over neighboring degree pairs.The supplied passages identify the coefficient's edge-based interpretation but do not provide a complete readable equation.
- Assortativity: Values near r ∼1 indicate similar-degree neighbors, values near r ∼−1 indicate high-degree-to-low-degree connections, and r ∼0 corresponds to random pairing.
- Related effects: For isolated networks, increasing assortativity can improve robustness to node removal and diffusion-driven dynamics but can also increase instability and reduce controllability.
A. Varying the assortativity
The paper varies assortativity while preserving degree information by sampling networks through link-swapping dynamics and a Hamiltonian-based statistical ensemble. ER and scale-free starting networks require different sampling considerations.
- Network sampling: The study tunes assortativity r while keeping the degree distribution fixed to test its effect on cascading fault propagation.
- Network sampling: A Hamiltonian-based ensemble assigns graph probabilities through µ(G) ∝ exp[−J H(G)], providing a flexible way to sample networks with different assortativity.The adjacency matrix represents links between nodes in the sampled graph.
- Network sampling: For a fixed degree distribution, the Hamiltonian's degree-product term determines how node linking changes assortativity, so larger J favors graphs with larger r.
- Link swapping: Link swapping preserves the initial degree distribution and each node degree while preferentially sampling assortative or disassortative configurations according to the swapping rule.
- Sampling behavior: The experiments begin with Barabasi-Albert and ER networks, but scale-free networks reach a narrower assortativity range as system size increases for comparable simulation times.ER networks reach equilibrium after a number of steps per node that appears independent of system size, whereas scale-free sampling is more complicated.
B. Interdependent failures model
The interdependent failure model repeatedly alternates connectivity pruning and dependency-based node removal across two coupled networks. This iterative process can undergo an abrupt transition at a critical surviving-node fraction, while analytical treatment of assortativity remains limited.
- Model setup: The model uses two equally sized networks with matching topology and assortativity, pairing each node in one network with a unique dependent node in the other.
- Cascade dynamics: A random attack removes fraction 1−x of network A, after which dependency links and giant-component filtering propagate removals between networks.
- Cascade dynamics: The cascade alternates largest-component identification, removal of nodes outside that component, and removal of nodes lacking surviving dependent partners until no further A-nodes are eliminated.The listed steps are s1 through s7 of the iterative algorithm.
- Cascade dynamics: The final cascade state is represented as a fixpoint of operators that select each network's largest component and retain nodes linked to surviving nodes in the other network.
- Analytical scope: Generating-function methods have not yet been shown to incorporate assortativity generally, and the relevant functions rarely have analytical closed forms, making numerical simulation the main investigative tool.
- Transition behavior: For ER and scale-free networks, the model has an abrupt transition: attacks beyond the critical surviving fraction leave the system completely fragmented, whereas smaller attacks leave a finite surviving fraction.
C. Percolation
The paper analyzes the interdependent-network percolation transition using the giant-component fraction as its order parameter and estimates the threshold with fluctuation- and derivative-based methods.
- C. Percolation: The giant-component fraction s = S/N is the order parameter, with s = 0 below the critical threshold and s > 0 above it.Finite-size effects can blur the distinction between weak first-order and second-order transitions.
- C. Percolation: The simulations average s across network layers with the same assortativity and across independent IFM runs.
- C. Percolation: The percolation threshold xc is estimated from the maximum fluctuation of the giant component and from the peak of its numerical derivative.The derivative uses ϵ = 10^-3.
- C. Percolation: The two threshold estimates are expected to converge as system size increases, although finite-size effects can make them differ.
A. Generating the networks
Networks with controlled assortativity are generated by Monte Carlo rewiring from fixed-degree-distribution configurations, then paired into two interdependent layers.
- A. Generating the networks: Monte Carlo rewiring samples networks with a fixed degree distribution using probability exp [−J H(G)].
- A. Generating the networks: Positive J favors links between similar-degree nodes and produces assortative networks, whereas negative J favors links between different-degree nodes.
- A. Generating the networks: Each rewired configuration is duplicated into two identical monolayers, then one layer’s labels are randomly permuted before one-to-one dependency links are added.This procedure avoids degree correlations between the two layers.
- A. Generating the networks: ER and SF networks are generated with N = 10,000 nodes and average degree ⟨k⟩ = 6, spanning assortativity values from r = −0.8 to r = 0.8 for ER networks.
B. Breakdown of coupled SF networks
Assortativity reduces the robustness of coupled networks, with a much stronger effect on scale-free than Erdős–Rényi networks; the critical threshold is identified through component fluctuations and derivative peaks.
- B. Breakdown of coupled SF networks: The damage needed to destroy the giant component decreases with assortativity, while the transition becomes less sharp at fixed system size.
- B. Breakdown of coupled SF networks: Repeated single-network percolation steps amplify assortativity’s effect because each iteration removes a larger fraction of nodes.
- B. Breakdown of coupled SF networks: The numerical derivative of the giant-component fraction peaks in the critical region, and its inflection-point estimate agrees with the fluctuation maximum within error bars.
- B. Breakdown of coupled SF networks: The scale-free percolation threshold xc increases with assortative coefficient r, indicating decreasing robustness as assortativity rises.
- B. Breakdown of coupled SF networks: The number of IFM iterations peaks near the critical threshold and measures the stability of the largest component under further node removal.
- B. Breakdown of coupled SF networks: For ER networks, increasing assortativity also requires more damage for transition, but the critical threshold is almost flat compared with the dramatic SF response.
C. Order of the phase transition
The simulations support an abrupt, first-order transition in both network types, while the model lacks the equilibrium free-energy description normally associated with hysteresis analysis.
- C. Order of the phase transition: Second-order behavior would produce growing and narrowing fluctuation peaks with system size, but the simulations show no second-order divergence.
- C. Order of the phase transition: The fluctuation and second-cluster analyses support a first-order transition, consistent with mean-field network-of-networks models.
- C. Order of the phase transition: The second-largest cluster has no peak and remains of order ∼1/N across the simulations.
- C. Order of the phase transition: Hysteresis is not a direct equilibrium diagnostic here because IFM final states are characterized as operator fixpoints rather than by a free-energy description.
IV. CONCLUSION
The study examines how assortativity affects robustness in interdependent systems modeled by two interacting networks, including scale-free and Erdős-Rényi networks.
- The simulations examine cascading faults in interdependent systems comprising both scale-free and Erdős-Rényi network models.