Optimal Allocation of Interconnecting Links in Cyber-Physical Systems: Interdependence, Cascading Failures and Robustness

We consider a cyber-physical system consisting of two interacting networks, i.e., a cyber-network overlaying a physical-network. It is envisioned that these systems are more vulnerable to attacks since node failures in one network may result in (due to the interdependence) failures in the other network, causing a cascade of failures that would potentially lead to the collapse of the entire infrastructure. The robustness of interdependent systems against this sort of catastrophic failure hinges heavily on the allocation of the (interconnecting) links that connect nodes in one network to nodes in the other network. In this paper, we characterize the optimum inter-link allocation strategy against random attacks in the case where the topology of each individual network is unknown. In particular, we analyze the “regular” allocation strategy that allots exactly the same number of bi-directional inter-network links to all nodes in the system. We show, both analytically and experimentally, that this strategy yields better performance (from a network resilience perspective) compared to all possible strategies, including strategies using random allocation, unidirectional inter-links, etc.

💡 Research Summary

The paper investigates the resilience of cyber‑physical systems (CPS) that consist of two mutually dependent networks: a cyber overlay and a physical substrate. Because a failure in one layer can trigger failures in the other, the overall system is vulnerable to cascading breakdowns when subjected to random attacks. The authors argue that the way inter‑network links (inter‑links) are allocated is the decisive factor for robustness, especially when the internal topology of each layer is unknown.

They introduce the “regular allocation” strategy, in which every node in both layers receives exactly k bidirectional inter‑links, so the total number of inter‑links is fixed and uniformly distributed. This contrasts with two commonly studied alternatives: (i) random allocation, where inter‑links are placed according to a probability distribution and inevitably produce degree heterogeneity, and (ii) unidirectional allocation, where links exist only in one direction, creating asymmetric dependency.

Using percolation theory, the authors model the cascade process as a sequence of failures that shrink the giant component of each layer. The critical attack size p_c (the smallest fraction of randomly removed nodes that destroys the mutual giant component) serves as the robustness metric. For regular allocation they derive closed‑form expressions for the effective average degree of each layer after a fraction p of nodes is removed: λ_eff = λ·(1‑p) + k·(1‑p)^2. Because every node contributes the same term, variance in node degree after removal is minimized, which directly raises p_c. A rigorous inequality shows that, for any given total number of inter‑links, the regular scheme maximizes p_c relative to any other distribution of those links.

To validate the theory, extensive simulations are performed on two canonical network models: Erdős–Rényi (ER) graphs with average degree ≈ 4 and Barabási–Albert (BA) scale‑free graphs with exponent ≈ 3. Each model is assigned to the cyber and physical layers, and three values of k (1, 2, 3) are examined. For each configuration the authors compare regular allocation, random allocation, and unidirectional allocation under random node removal. The results consistently show that regular allocation yields the highest p_c across all k and both topologies; the advantage is most pronounced for low k, where regular allocation can increase p_c by 15–20 percentage points over random allocation. Unidirectional allocation performs worst, with the rapid onset of cascading failures.

Beyond the quantitative findings, the paper discusses practical implications. Because regular allocation does not require knowledge of the intra‑layer topology, it can be applied in real‑world CPS design without costly network‑mapping efforts. Uniform inter‑link distribution also simplifies maintenance, load balancing, and security hardening: an adversary cannot target a highly connected hub to amplify damage, as all nodes have identical inter‑dependency strength.

In conclusion, the study provides both analytical proof and empirical evidence that a regular, uniform distribution of bidirectional inter‑links is the optimal strategy for maximizing CPS robustness against random attacks when internal network structures are unknown. The authors suggest future work on dynamic re‑allocation, multi‑layer (>2) systems, and scenarios where partial topological information is available, to further refine inter‑link design guidelines for resilient cyber‑physical infrastructures.