[Paper Review] Robustness of Bidirectional Interdependent Networks: Analysis and Design

Marzieh Parandehgheibi, Eytan Modiano|arXiv (Cornell University)|May 3, 2016

Infrastructure Resilience and Vulnerability Analysis22 references3 citations

TL;DR

This paper proposes a deterministic model to analyze the robustness of bidirectional interdependent networks with known topologies, introducing the metric $\mathcal{MR}(D)$ to quantify the minimum node removals needed to cause $D$ failures in the dependent network. It proves $\mathcal{MR}(D)$ is NP-complete and inapproximable, and develops heuristic algorithms for evaluation, showing bidirectional interdependency enhances robustness and linking robust network design to expander graph properties.

ABSTRACT

We study the robustness of interdependent networks where two networks are said to be interdependent if the operation of one network depends on the operation of the other one, and vice versa. In this paper, we propose a model for analyzing bidirectional interdependent networks with known topology. We define the metric MR(D) to be the minimum number of nodes that should be removed from one network to cause the failure of D nodes in the other network due to cascading failures. We prove that evaluating this metric is not only NP-complete, but also inapproximable. Next, we propose heuristics for evaluating this metric and compare their performances using simulation results. Finally, we introduce two closely related definitions for robust design of interdependent networks; propose algorithms for explicit design, and demonstrate the relation between robust interdependent networks and expander graphs.

Motivation & Objective

• To develop a deterministic framework for analyzing interdependent networks with known topologies, moving beyond asymptotic random graph models.
• To define and evaluate the metric $\mathcal{MR}(D)$, representing the minimum node removals in one network that cause $D$ failures in the other due to cascading effects.
• To prove that computing $\mathcal{MR}(D)$ is NP-complete and inapproximable, establishing fundamental computational hardness.
• To propose heuristic algorithms (greedy, randomized rounding, simulated annealing) for approximating $\mathcal{MR}(D)$ and evaluate their performance via simulation.
• To introduce two robust design definitions—lexicographic and relative—and demonstrate their connection to expander graphs for explicit network construction.

Proposed method

• Proposes a deterministic model for bidirectional interdependent networks where each node in one network depends on a corresponding node in the other, with failure propagation governed by connectivity and interdependency links.
• Introduces $\mathcal{MR}(D)$ as the minimum number of nodes to remove from network $A$ to cause $D$ nodes to fail in network $B$ due to cascading failures.
• Proves NP-completeness and inapproximability of $\mathcal{MR}(D)$ via a reduction from the set cover problem, using a construction that maps parent and child nodes to simulate failure propagation.
• Employs greedy, randomized rounding, and simulated annealing heuristics to approximate $\mathcal{MR}(D)$, comparing their performance in simulations.
• Constructs a bidirectional interdependent network with star topology and uses a parent-child node mapping to simulate cascading failures, proving equivalence in failure propagation between unidirectional and bidirectional cases under specific constructions.
• Demonstrates that bidirectional interdependency leads to higher robustness than unidirectional by showing stronger failure propagation resilience.

Experimental results

Research questions

• RQ1 What is the computational complexity of determining the minimum number of node removals required to cause $D$ failures in a dependent interdependent network?
• RQ2 How can robustness metrics like $\mathcal{MR}(D)$ be effectively approximated in practice for networks with known topologies?
• RQ3 To what extent does bidirectional interdependency enhance network robustness compared to unidirectional interdependency?
• RQ4 How can interdependent networks be explicitly designed to be robust against cascading failures?
• RQ5 What structural properties—such as expander graph characteristics—underlie robust interdependent network designs?

Key findings

• The metric $\mathcal{MR}(D)$, representing the minimum node removals to cause $D$ failures in the dependent network, is proven to be NP-complete and inapproximable.
• Heuristic algorithms based on greedy selection, randomized rounding, and simulated annealing effectively approximate $\mathcal{MR}(D)$, with simulated annealing showing superior performance in simulations.
• Bidirectional interdependent networks are inherently more robust than unidirectional ones, as they allow for more resilient failure propagation and recovery mechanisms.
• The optimal design of robust interdependent networks is linked to expander graph properties, suggesting that high expansion in interdependency links enhances system resilience.
• A constructive proof shows that failure propagation in bidirectional networks with tree topology cannot always be reduced to unidirectional star topologies, indicating limitations in generalizing from simplified models.