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Analyzing 9,000 urban areas' street networks, we identify properties, including extreme betweenness centrality heterogeneity, that typical spatial network models fail to explain. Accordingly we propose a universal, parsimonious, generative model based on a two-step mechanism that begins with a spanning tree as a backbone then iteratively adds edges to match empirical degree distributions. Controlled by a single parameter representing lattice-equivalent node density, it accurately reproduces key universal properties to bridge the gap between empirical observations and generative models.

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