Although detecting and characterizing community structure is key in the study of networked systems, we still do not understand how community structure affects systemic resilience and stability. We use percolation theory to develop a framework for studying the resilience of networks with a community structure. We find both analytically and numerically that interlinks (the connections among communities) affect the percolation phase transition in a way similar to an external field in a ferromagnetic– paramagnetic spin system. We also study universality class by defining the analogous critical exponents δ and γ, and we find that their values in various models and in real-world coauthor networks follow the fundamental scaling relations found in physical phase transitions. The methodology and results presented here facilitate the study of network resilience and also provide a way to understand phase transitions under external fields. Significance Much work has focused on phase transitions in complex networks in which the system transitions from a resilient to a failed state. Furthermore, many of these networks have a community structure, whose effects on resilience have not yet been fully understood. Here, we show that the community structure can significantly affect the resilience of the system in that it removes the phase transition present in a single module, and the network remains resilient at this transition. In particular, we show that the effect of increasing interconnections is analogous to increasing external magnetic field in spin systems. Our findings provide insight into the resilience of many modular complex systems and clarify the important effects that community structure has on network resilience.

Rooftop photovoltaic (RPV) is often understood as a niche contribution to climate change mitigation. However, the global potential of RPVs to mitigate global warming is unknown. Here we map the global rooftop area at 1-km resolution, quantifying 286,393 km² of rooftops worldwide through geospatial data mining and artificial intelligence techniques. Using nine advanced Earth system models from the coupled model intercomparison project phase 6, we reveal that RPVs could substantially contribute to reducing global temperatures by 0.05–0.13 °C before 2050. Region-specific analysis underscores the variability in RPV potential and the necessity of tailored approaches to optimize RPV deployment, considering local solar resources, existing infrastructure and grid carbon intensity. Our findings reveal that leveraging RPV systems offers a viable and impactful strategy for reducing carbon footprints and combating climate change globally, while advocating targeted interventions to enhance the benefits of RPVs, particularly in areas with high solar radiation or rapid urbanization.

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. 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.