  Cities are centers for the integration of capital and incubators of
invention, and attracting venture capital (VC) is of great importance for
cities to advance in innovative technology and business models towards a
sustainable and prosperous future. Yet we still lack a quantitative
understanding of the relationship between urban characteristics and VC
activities. In this paper, we find a clear nonlinear scaling relationship
between VC activities and the urban population of Chinese cities. In such
nonlinear systems, the widely applied linear per capita indicators would be
either biased to larger cities or smaller cities depends on whether it is
superlinear or sublinear, while the residual of cities relative to the
prediction of scaling law is a more objective and scale-invariant metric.
%(i.e., independent of the city size). Such a metric can distinguish the
effects of local dynamics and scaled growth induced by the change of population
size. The spatiotemporal evolution of such metrics on VC activities reveals
three distinct groups of cities, two of which stand out with increasing and
decreasing trends, respectively. And the taxonomy results together with spatial
analysis also signify different development modes between large urban
agglomeration regions. Besides, we notice the evolution of scaling exponents on
VC activities are of much larger fluctuations than on socioeconomic output of
cities, and a conceptual model that focuses on the growth dynamics of different
sized cities can well explain it, which we assume would be general to other
scenarios.
