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Surfacic networks are structures built upon a 2D manifold. Many systems, including transportation networks and various urban networks, fall into this category. The fluctuations of node elevations imply significant deviations from typical plane networks and require specific tools to understand their impact. Here, we present such tools, including lazy paths that minimize elevation differences, graph arduousness which measures the tiring nature of shortest paths (SPs), and the excess effort, which characterizes positive elevation variations along SPs. We illustrate these measures using toy models of surfacic networks and empirically examine pedestrian networks in selected cities. Specifically, we examine how changes in elevation affect the spatial distribution of betweenness centrality. We also demonstrate that the excess effort follows a nontrivial power law distribution, with an exponent that is not universal, which illustrates that there is a significant probability of encountering steep slopes along SPs, regardless of the elevation difference between the starting point and the destination. These findings highlight the significance of elevation fluctuations in shaping network characteristics. Surfacic networks offer a promising framework for comprehensively analyzing and modeling complex systems that are situated on or constrained to a surface environment., (© The Author(s) 2025. Published by Oxford University Press on behalf of National Academy of Sciences.)

The association between residential greenness and allostatic load (AL), a marker of composite physiological burden and predictor of chronic disease, remains understudied. This study comprised 212,600 UK Biobank participants recruited over 2007 and 2010 at the baseline. Residential greenness was modeled as the normalized difference vegetation index (NDVI) from high spatial resolution (0.50 m) color infrared imagery and measured within a 0.5 km radial catchment. AL was measured as a composite index from 13 biomarkers comprising three physiological systems (metabolic, cardiovascular, and inflammatory systems) and two organ systems (liver and kidney). Multilevel mixed-effects generalized linear models with a random intercept for UK Biobank assessment centers were employed to examine the association between residential greenness and AL. Each interquartile range (IQR = 0.24) increment in NDVI greenness was associated with lower AL (beta (β) = -0.28, 95% confidence interval (CI) = -0.55, -0.01). Consistently, relative to the lowest NDVI greenness quintile, participants in the highest quintile had lower AL (β = -0.64, 95% CI = -1.02, -0.26). The proportion of the association between greenness and AL mediated by the physical activity was 3.2%. In conclusion, residential greenness was protectively associated with AL, a composite marker of wear and tear and general health.