Time evolution of companies towards a stable scaling curve obtained from flow diagrams in three-dimensional phase space

Abstract

Although scaling relations of power law forms have been revealed in a wide variety of complex systems, the origin of scaling relations in real systems remains mostly unsolved. Based on a long tradition in physics, we here explore the phase space dynamics of business companies, whose different measures of size have been found to exhibit multiple scaling relations. Using a large scale dataset of Japanese companies covering over two million companies for over two decades (1994-2015), we compile the data of three measures of size, namely, annual sales, number of employee and number of trading partners. Tracking the historical time evolution of companies, we first show that there exists a stable region that attracts most of the data points in the long time scale, although the company dynamics in the three-dimensional phase space looks random in a shorter time scale. We then elaborate on 'evolving flow diagrams' of the averaged motion in the 3D space. Remarkably, the flow diagrams indicate that a 3D curve which represents the scaling relations between the three size measures can be regarded as an approximate attractor of the averaged flows. The results could serve for better modeling and understanding of companies dynamics and our method can be applied to other dynamics such as social and biological phenomena.