  Is there a general economic pathway recapitulated by individual cities over
and over? Identifying such evolution structure, if any, would inform models for
the assessment, maintenance, and forecasting of urban sustainability and
economic success as a quantitative baseline. This premise seems to contradict
the existing body of empirical evidences for path-dependent growth shaping the
unique history of individual cities. And yet, recent empirical evidences and
theoretical models have amounted to the universal patterns, mostly
size-dependent, thereby expressing many of urban quantities as a set of simple
scaling laws. Here, we provide a mathematical framework to integrate repeated
cross-sectional data, each of which freezes in time dimension, into a frame of
reference for longitudinal evolution of individual cities in time. Using data
of over 100 millions employment in thousand business categories between 1998
and 2013, we decompose each city's evolution into a pre-factor and relative
changes to eliminate national and global effects. In this way, we show the
longitudinal dynamics of individual cities recapitulate the observed
cross-sectional regularity. Larger cities are not only scaled-up versions of
their smaller peers but also of their past. In addition, our model shows that
both specialization and diversification are attributed to the distribution of
industry's scaling exponents, resulting a critical population of 1.2 million at
which a city makes an industrial transition into innovative economies.
