Diffusion processes are central to human interactions. One common prediction of the current modeling frameworks is that initial spreading dynamics follow exponential growth. Here, we find that, ranging from mobile handsets to automobiles, from smart-phone apps to scientific fields, early growth patterns follow a power law with non-integer exponents. We test the hypothesis that mechanisms specific to substitution dynamics may play a role, by analyzing a unique data tracing 3.6M individuals substituting for different mobile handsets. We uncover three generic ingredients governing substitutions, allowing us to develop a minimal substitution model, which not only explains the power-law growth, but also collapses diverse growth trajectories of individual constituents into a single curve. These results offer a mechanistic understanding of power-law early growth patterns emerging from various domains and demonstrate that substitution dynamics are governed by robust self-organizing principles that go beyond the particulars of individual systems.
We compiled four large-scaled datasets in this paper:
D1 traces 3.6 Million individuals choosing among different types of mobile handsets, recorded by a Northern European telecommunication company from January 2006 to November 2014.
D2 captures monthly transaction records of 126 automobiles sold in the North America between 2010 and 2016.
D3 records number of daily downloads for 2,672 new popular smartphone apps in the iOS systems from November to December 2016.
D4 documents 246,630 scientists substituting for 6,399 scientific fields from 1980 to 2018.
The automobile, smartphone apps and scientific fields datasets necessary to reproduce the results in the manuscript are available here. Along with the data, we also provide the related computer code below. The mobile phone dataset is not public available due to commercially sensitive information contained.