Robustness of networks with topologies of dependency links

The robustness of complex networks with dependencies has been studied in recent years. However, previous studies focused on the robustness of networks composed of dependency links without network topology. In this study, we will analyze the percolation properties of a realistic network model where dependency links follow certain network topology. We perform the theoretical analysis and numerical simulations to show the critical effects of topology of dependency links on robustness of complex networks. For Erd"os-R'enyi (ER) connectivity network, we find that the system with dependency of RR topology is more vulnerable than system with dependency of ER topology. And RR-RR (i.e. random-regular (RR) network with dependency of RR topology) disintegrates in an abrupt transition. In particular, we find that the system of RR-ER shows different types of phase transitions. For system of different combinations, the type of percolation depends on the interaction between connectivity topology and dependency topology.

💡 Research Summary

This paper investigates how the topology of dependency links influences the robustness of complex networks. While previous studies have treated dependencies as isolated, randomly paired connections, the authors consider realistic scenarios where dependency links themselves form a network with a specific structure. The study focuses on percolation properties of a composite system consisting of a connectivity network and a dependency network, each of which can be either an Erdős–Rényi (ER) random graph or a Random‑Regular (RR) graph. Consequently, four combinations are examined: ER‑ER, ER‑RR, RR‑ER, and RR‑RR.

The authors develop an analytical framework that extends classical percolation theory to incorporate cascading failures induced by dependency links. For the connectivity network they use the standard generating‑function formalism, defining G₀(x) for the degree distribution P(k). For the dependency network they introduce a similar generating function G_d(x) based on its degree distribution P_d(k). The percolation process proceeds in two stages: (1) a fraction of nodes is removed from the connectivity layer, after which the size u of the remaining giant component is obtained; (2) any node belonging to a dependency cluster that contains at least one failed node triggers the simultaneous failure of all nodes in that cluster. This recursive failure mechanism yields a self‑consistent equation for the final giant component size S, which can be solved analytically for the four network‑pair configurations.

The theoretical predictions are validated by extensive Monte‑Carlo simulations. For the ER‑ER case, both layers have Poisson degree distributions, leading to a continuous (second‑order) percolation transition. The critical point is well approximated by the simple condition ⟨k⟩·⟨k_d⟩≈1, where ⟨k⟩ and ⟨k_d⟩ are the average degrees of the connectivity and dependency layers, respectively. In the ER‑RR configuration the dependency layer is regular, meaning every node has the same dependency degree. This eliminates degree heterogeneity in the dependency network, causing the system to become more fragile: the same average dependency degree results in a lower critical threshold and a sharper decline of the giant component.

The RR‑RR combination is the most vulnerable. Both layers are regular, so any initial removal quickly propagates through the dependency clusters, producing an abrupt (first‑order) collapse without a gradual approach to the critical point. The RR‑ER case exhibits the richest behavior. When the dependency layer is ER, its degree variance can buffer the cascade, but the connectivity layer’s regularity amplifies the effect of each failed node. As the average dependency degree ⟨k_d⟩ increases, the system undergoes a crossover from a continuous to a discontinuous transition. This demonstrates that the interplay between the homogeneity of the connectivity network and the heterogeneity of the dependency network determines the nature of the percolation transition.

The authors conclude that the topology of dependency links is a crucial factor for network resilience. Regular dependency structures (RR) dramatically reduce robustness, whereas random (ER) dependency structures, especially those with broader degree distributions, can mitigate cascading failures. The findings suggest practical design guidelines: to enhance robustness, one should avoid overly regular dependency patterns and, where possible, introduce heterogeneity or redundancy in the dependency layer. The paper also points to future work involving clustered or hierarchical dependency networks, time‑varying dependencies, and applications to real‑world infrastructures such as power‑communication interdependencies.