<p>In this paper, a new model of Binary Interdependent Networks with Weak Coupling (BINWC) is proposed based on the interdependence of nodes and edges in realistic complex networks. Unlike traditional models that focus solely on node-coupled or edge-coupled interdependent networks, this model introduces a mechanism where nodes depend on edges in heterogeneous networks. In this framework, a node failure will directly cause the failure of its dependent edge. Conversely, the failure of an edge affects the stability of its dependent node with a certain probability. This approach aligns more closely with the robustness characteristics observed in real networks. This paper presents a mathematical analysis framework based on percolation theory and the generating function method to study the percolation behavior of BINWC under random attacks. The accuracy of the theoretical analysis is verified through numerical solutions and computer simulations. Furthermore, we investigate the relationship between network robustness and the failure probabilities of both nodes and edges. It is demonstrated that the robustness of BINWC increases as the probability of both nodes and edges failure decreases. However, this improvement in robustness is less significant in BINWC with higher average degrees. Additionally, for both Random Regular (RR) and Erdős–Rényi (ER) networks, a first-order phase transition occurs at the average degree <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\langle k \rangle = 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">⟨</mo> <mi>k</mi> <mo stretchy="false">⟩</mo> <mo>=</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, while a second-order phase transition is observed when <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\langle k \rangle \ne 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">⟨</mo> <mi>k</mi> <mo stretchy="false">⟩</mo> <mo>≠</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. This study provides theoretical support for understanding binary interdependent networks and their robustness in realistic networks.</p>

Robustness Analysis of Binary Interdependent Networks with Weak Coupling

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发表日期：

2026年7月31日

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Yan-Li Gao,
Wei-Nan Xu,
Qiu-Yu Tang,
Lei Zhang,
Chao Feng,
Hai-Bo Yu

摘要

In this paper, a new model of Binary Interdependent Networks with Weak Coupling (BINWC) is proposed based on the interdependence of nodes and edges in realistic complex networks. Unlike traditional models that focus solely on node-coupled or edge-coupled interdependent networks, this model introduces a mechanism where nodes depend on edges in heterogeneous networks. In this framework, a node failure will directly cause the failure of its dependent edge. Conversely, the failure of an edge affects the stability of its dependent node with a certain probability. This approach aligns more closely with the robustness characteristics observed in real networks. This paper presents a mathematical analysis framework based on percolation theory and the generating function method to study the percolation behavior of BINWC under random attacks. The accuracy of the theoretical analysis is verified through numerical solutions and computer simulations. Furthermore, we investigate the relationship between network robustness and the failure probabilities of both nodes and edges. It is demonstrated that the robustness of BINWC increases as the probability of both nodes and edges failure decreases. However, this improvement in robustness is less significant in BINWC with higher average degrees. Additionally, for both Random Regular (RR) and Erdős–Rényi (ER) networks, a first-order phase transition occurs at the average degree $\langle k \rangle = 3$ $⟨ k ⟩ = 3$ , while a second-order phase transition is observed when $\langle k \rangle \ne 3$ $⟨ k ⟩ \neq 3$ . This study provides theoretical support for understanding binary interdependent networks and their robustness in realistic networks.