Emergence of scaling in complex substitutive systems

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Diffusion processes are central to human interactions. One common prediction of the current modelling frameworks is that initial spreading dynamics follow exponential growth. Here we find that, for subjects ranging from mobile handsets to automobiles and from smartphone 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 analysing unique data tracing 3.6 million individuals substituting 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.

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Fig. 1: Power-law growth patterns in substitutive systems.
Fig. 2: Empirical substitution network.
Fig. 3: Empirical substitution patterns.
Fig. 4: Universal impact dynamics.

Data availability

Data necessary to reproduce the results in the manuscript are available. The datasets for automobiles, smartphone apps and scientific fields are publicly available at https://chingjin.github.io/substitution/. The mobile phone dataset is not publicly available due to commercially sensitive information contained, but is available from the corresponding author on reasonable request.

Code availability

The custom code used is available at https://chingjin.github.io/substitution/.


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We thank B. Uzzi, J. Colyvas, J. Chu, M. Kouchaki, Q. Zhang, Z. Ma and all members of the Northwestern Institute on Complex Systems (NICO) for helpful comments. We are indebted to A.-L. Barabási for initial collaboration on this project and invaluable feedback on the manuscript. This work was supported by the Air Force Office of Scientific Research under award number FA9550-15-1-0162 and FA9550-17-1-0089, Northwestern University’s Data Science Initiative, and National Science Foundation grant SBE 1829344. C.S. was supported by the National Science Foundation (IBSS-L-1620294) and by a Convergence Grant from the College of Arts & Sciences, University of Miami. The funders had no role in study design, data collection and analysis, decision to publish or preparation of the manuscript.

Author information

All authors designed the research. C.J., C.S. and D.W. conducted the analytical and numerical calculations. C.J., C.S., J.B. and D.W. analysed the empirical data. D.W. was the lead writer of the manuscript.

Correspondence to Dashun Wang.

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Supplementary Notes 1–4, Supplementary Figs. 1–30, Supplementary Tables 1–3, and Supplementary References.

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