Publicly traded companies are fundamental units of contemporary economies and markets and are important mechanisms through which humans interact with their environments. Understanding the general properties that underlie the processes of their growth has long been of interest, yet fundamental debates about the effects of firm size on growth have persisted. Here we develop a scaling framework that focuses on company size as the critical feature determining a variety of tradeoffs, and use this to reveal novel systematic behavior across the diversity of publicly-traded companies. Using a large database of 31,553 US companies over nearly 70 years, and 3,160 Chinese companies over 24 year, we show how the dynamics of companies expressed as scaling relationships leads to a quantitative, analytic theory for their growth. This theory produces several predictions that are in good agreement with data for both the US and China, whose markets have strikingly different histories and underlying structures. In both cases sales scale sublinearly with assets and exhibit nearly identical exponents leading, surprisingly and nontrivially, to assets that grow as a power law in time rather than exponentially, as often assumed. On the other hand, liabilities scale linearly in the US (exponent of 1.0) but superlinearly in China (exponent of 1.09). We show that such small differences in scaling exponents can have a significant impact on the character and long-term evolution of growth trajectories. These results illustrate that while companies are part of a larger class of growth phenomena driven by incomes and costs that scale with size, they are unique in that they grow following a temporal power-function which sets them apart from organisms, cities, nations, and markets, whose growth over time is often exponential. The perspective developed here highlights novel dynamics in the scaling and growth of human economies.

Predicting company growth carries significant implications for strategic adjustment, operational decision-making, risk assessment, and loan eligibility reviews. Traditional attempts to model company growth either overly focus on theory, neglecting practical forecasting, or rely on time-series forecasting techniques without incorporating the inherent mechanisms of corporate growth. A recent study established a first-principles growth model for enterprises based on scaling laws, providing predictions superior to baseline methods regarding average business growth. In this paper, we propose a hybrid prediction framework for company growth, blending the growth equation of enterprises with time-series forecasting technology. Specifically, we used a neural network model as a prediction method to make our framework more extensible. This model captures both the intrinsic growth mechanisms of enterprises and the fluctuations in the external environment, demonstrating superior predictive performance compared to methods using time-series forecasting technology alone. Moreover, we found that the advantages of our framework are more prominent for long-range prediction tasks.

Multiscale modeling of complex systems is crucial for understanding their intricacies. In recent years, data-driven multiscale modeling has emerged as a promising approach to tackle challenges associated with complex systems. Still, at present,this field is more focused on the prediction or control problems in specific fields, and there is no suitable framework to help us promote the establishment of complex system modeling theory. On the other hand, self-similarity is prevalent in complex systems, hinting that large-scale complex systems can be modeled at a reduced cost. In this paper, we introduce a multiscale neural network framework that incorporates self-similarity as prior knowledge, facilitating the modeling of self-similar dynamical systems. Our framework can discern whether the dynamics are self-similar to deterministic dynamics. For uncertain dynamics, it not only can judge whether it is self-similar or not, but also can compare and determine which parameter set is closer to self-similarity. The framework allows us to extract scale-invariant kernels from the dynamics for modeling at any scale. Moreover, our method can identify the power-law exponents in self-similar systems, providing valuable insights for the establishment of complex system modeling theory.