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Optimal resilience of modular interacting networks

Significance In real-world scenarios, many subnetworks interact with other subnetworks to form a modular interacting network system. In previous models, it was assumed that all subnetworks may be linked pairwise; however, in real systems, only some pairs of subnetworks are connected. Understanding how to develop theoretical frameworks and study system resilience of general coupling patterns of the subnetworks is of significant importance for designing and optimizing economic, social, and infrastructure networks. Here we present two frameworks for such systems having specific coupling patterns to investigate their resilience. Specifically, we find an optimal level of interaction between subnetworks, which maximizes the system’s resilience to failures. Our findings highlight the need to consider real-world coupling patterns and possible optimizations for designing resilient systems. Coupling between networks is widely prevalent in real systems and has dramatic effects on their resilience and functional properties. However, current theoretical models tend to assume homogeneous coupling where all the various subcomponents interact with one another, whereas real-world systems tend to have various different coupling patterns. We develop two frameworks to explore the resilience of such modular networks, including specific deterministic coupling patterns and coupling patterns where specific subnetworks are connected randomly. We find both analytically and numerically that the location of the percolation phase transition varies nonmonotonically with the fraction of interconnected nodes when the total number of interconnecting links remains fixed. Furthermore, there exists an optimal fraction r* of interconnected nodes where the system becomes optimally resilient and is able to withstand more damage. Our results suggest that, although the exact location of the optimal r* varies based on the coupling patterns, for all coupling patterns, there exists such an optimal point. Our findings provide a deeper understanding of network resilience and show how networks can be optimized based on their specific coupling patterns.