Cascading Failures in Interdependent Lattice Networks: The Critical Role of the Length of Dependency Links

We study the cascading failures in a system composed of two interdependent square lattice networks A and B placed on the same Cartesian plane, where each node in network A depends on a node in network B randomly chosen within a certain distance $r$ from the corresponding node in network A and vice versa. Our results suggest that percolation for small $r$ below $r_{\rm max}\approx 8$ (lattice units) is a second-order transition, and for larger $r$ is a first-order transition. For $r<r_{\rm max}$, the critical threshold increases linearly with $r$ from 0.593 at $r=0$ and reaches a maximum, 0.738 for $r=r_{\rm max}$ and then gradually decreases to 0.683 for $r=\infty$. Our analytical considerations are in good agreement with simulations. Our study suggests that interdependent infrastructures embedded in Euclidean space become most vulnerable when the distance between interdependent nodes is in the intermediate range, which is much smaller than the size of the system.

💡 Research Summary

The paper investigates how the spatial range of dependency links influences cascading failures in two mutually dependent square‑lattice networks that occupy the same two‑dimensional Euclidean plane. Each node in network A is linked to a node in network B (and vice‑versa) that lies within a prescribed Manhattan distance r; the partner is chosen uniformly at random among all nodes satisfying the distance constraint. Internally, each lattice retains its usual four‑nearest‑neighbour connectivity, thereby mimicking real‑world infrastructures (power, communication, transport) that are both locally wired and globally interdependent.

The authors study the system’s response to a random initial attack that removes a fraction 1 – p of nodes from both layers simultaneously. After the initial removal, any node whose dependency partner has failed is also removed, which can trigger further failures in a recursive “cascade.” The process continues until no more dependent nodes are left, and the size of the surviving giant component, P∞(p, r), is recorded. By varying p and r, the authors map out the percolation transition of the coupled system.

The central finding is a sharp change in the nature of the transition at a critical interaction range rmax ≈ 8 lattice units. For r < rmax the order parameter P∞ decays continuously to zero as p approaches a critical value pc(r); this is a second‑order (continuous) percolation transition. In this regime pc(r) grows linearly from pc(0) ≈ 0.593 to a maximum pc(rmax) ≈ 0.738. When r > rmax the transition becomes discontinuous: P∞ remains finite up to pc and then drops abruptly to zero, characteristic of a first‑order transition. In the limit r → ∞ (i.e., completely random inter‑layer matching) the critical threshold settles at pc ≈ 0.683, reproducing earlier results for interdependent random networks.

To explain these observations, the authors extend classical percolation theory by incorporating the finite‑range constraint on dependency links. They calculate the probability that a dependency partner lies within the same percolation cluster versus a different cluster, as a function of r and the cluster‑size distribution of a single lattice. When r is small, most partners belong to the same cluster, so failures remain localized and the cascade proceeds gradually, yielding a continuous transition. When r is large, partners are likely to belong to distinct large clusters; a small initial damage can therefore disconnect many clusters simultaneously, producing a catastrophic, abrupt collapse. By averaging over the random matching and integrating the distance‑dependent connection probability, they derive an analytical expression for pc(r) that matches the simulation data with high precision, including the location of rmax and the change in transition order.

The paper’s implications are practical as well as theoretical. Infrastructure systems that are physically embedded cannot ignore the geometry of interdependencies. If dependency links are too short, the system becomes vulnerable to localized failures that can percolate through tightly coupled neighborhoods. If they are too long, the system is prone to global cascades triggered by minor disturbances. The most dangerous configuration lies in the intermediate range—much smaller than the overall system size but large enough to connect distant clusters—where the critical threshold peaks. Consequently, designers of coupled infrastructures should carefully balance the spatial extent of inter‑layer dependencies to avoid the “sweet spot” of maximal fragility.

Overall, the study provides a comprehensive analytical and numerical framework for understanding cascading failures in spatially embedded interdependent networks. It demonstrates that the Euclidean distance between dependent nodes is a decisive control parameter that determines both the critical point and the order of the percolation transition. The results enrich the broader field of network robustness and offer concrete guidance for the planning and risk assessment of real‑world interdependent infrastructures.