Robustness of multilayer networks: A graph energy perspective

Real-world complex systems, encompassing domains like Social, Technological, and Infrastructure networks, exhibit interconnections and dependencies. These systems are susceptible to disruptions, stemming from failures in individual nodes or connections. While prevailing research often focuses on measuring robustness by examining the size of the largest connected component, this approach falls short of capturing the full spectrum of vulnerabilities. In situations where a fraction of edges fails due to cascading effects, the network can become sparse, yet the size of the largest connected component might remain proportional to the original network size. However, this seemingly robust scenario can be deceptive, as even minor disturbances can trigger a catastrophic breakdown in the network’s integrity. Thus, the current study delves deeper into evaluating the robustness of multilayer network systems by employing metrics such as Average Efficiency, Laplacian Energy, and Quantum Energies to better comprehend the system’s behavior during cascade failures, while still considering the size of the largest connected component. Simulation results reveal that during the cascade failure, Average efficiency and Laplacian energy keep on decreasing up to some initial instances, and the robustness index is 1 indicating that the multilayer network is robust. However, there is a critical point at which the system starts disintegrating for the given values of Average efficiency and Laplacian energy. For the Quantum energy scenario, the system is stable with minimum energy but during the cascade failure, the system begins to become unstable showing the increase in the Quantum energy at some critical point, and the robustness index begins to decrease. Hence, the robustness index (which depends only on network size independent of structure (connectivity pattern)) is not sufficient to evaluate the level of robustness of the network systems.