  Growth of business firms or companies has been a subject of intensive
research over a century. However, there still remains controversy about the
basic mechanisms of their growth. Inspired by previous work on scaling laws in
other systems, here we extend the notion of size of firms from a scalar to a
vector in order to characterize in more detail the mechanisms of growth and
decay of firms. Based on a large scale dataset of Japanese firms covering over
two million firms for two decades (1994-2015), we compile the dataset of
vectors of three components, namely, annual sales, number of employee and
number of trading partners. We find that the number of employees is more
influential in determining firm sales compared to the number of trading
partners. This asymmetry is validated by regressions of sales against these
parameters and the analysis of growth rate correlations. We then explore
multi-variate dynamics of firms by elaborating an evolutionary flow diagram of
the averaged motion in the three-dimensional vector space. The flow diagram
indicates that firms which deviate from the balanced scaling relation tend to
return to this relation. We also find that firms with a chance of large sales
growth suffer the risk of high disappearance rate. These results could serve
for prediction and modeling of firms, and are relevant for theoretical
understanding of the general principles governing complex systems.
