Spatial association and convergence of the water–energy–food nexus in the Yellow River Basin using a graph-convolution-assisted spatial econometric model

Water, energy, and food security are jointly constrained by ecological pressure, resource scarcity, and uneven regional development in the Yellow River Basin. This study examines the spatial association and convergence of the water–energy–food nexus across nine provincial regions from 2011 to 2021. A composite development index is constructed by combining CRITIC and entropy weights. Geographical proximity and pairwise dependence between provincial WEF index series are then integrated into an equally weighted hybrid spatial matrix. Graph convolution is used as a transparent neighborhood aggregation step that transforms standardized regional covariates before they enter a spatial Durbin specification and a spatial absolute beta-convergence model. Global Moran’s I increases from 0.235 to 0.318, indicating persistent and strengthening positive spatial dependence. The reported point estimates show that indirect effects account for 30.4% to 31.3% of the total effects of the examined factors. The basin-wide beta coefficient is negative 0.423 and the spatial lag coefficient is positive 0.286, which is consistent with faster growth in regions with lower initial WEF development and with positive neighborhood association. Three development-profile groups display different initial levels and adjustment speeds. These groups are interpreted as exploratory regional profiles rather than formal Phillips-Sul convergence clubs. The findings support basin-wide coordination while also showing that policy design should reflect differences in resource endowment, internal variation, and regional development stage.