  Understanding quantitative relationships between urban elements is crucial
for a wide range of applications. The observation at the macroscopic level
demonstrates that the aggregated urban quantities (e.g., gross domestic
product) scale systematically with population sizes across cities, also known
as urban scaling laws. However, at the mesoscopic level, we lack an
understanding of whether the simple scaling relationship holds within cities,
which is a fundamental question regarding the spatial origin of scaling in
urban systems. Here, by analyzing four extensive datasets covering millions of
mobile phone users and urban facilities, we investigate the scaling phenomena
within cities. We find that the mesoscopic infrastructure volume and
socioeconomic activity scale sub- and super-linearly with the active
population, respectively. For a same scaling phenomenon, however, the exponents
vary in cities of similar population sizes. To explain these empirical
observations, we propose a conceptual framework by considering the
heterogeneous distributions of population and facilities, and the spatial
interactions between them. Analytical and numerical results suggest that,
despite the large number of complexities that influence urban activities, the
simple interaction rules can effectively explain the observed regularity and
heterogeneity in scaling behaviors within cities.
