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Emergence of scaling in complex substitutive systems

Nature Human Behaviour volume 3, pages 837–846 (2019)Cite this article

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Abstract

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.

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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/.

References

  1. Barrat, A., Barthelemy, M. & Vespignani, A. Dynamical Processes on Complex Networks (Cambridge Univ. Press, 2008).

  2. Ben-Avraham, D. & Havlin, S. Diffusion and Reactions in Fractals and Disordered Systems (Cambridge Univ. Press, 2000).

  3. Pastor-Satorras, R., Castellano, C., Van Mieghem, P. & Vespignani, A. Epidemic processes in complex networks. Rev. Mod. Phys. 87, 925 (2015).

    Article  Google Scholar 

  4. Rogers, E. M. Diffusion of Innovations (Simon and Schuster, 1962).

  5. Gladwell, M. The Tipping Point: How Little Things Can Make a Big Difference (Little Brown, 2006).

    Google Scholar 

  6. Anderson, R. M., May, R. M. & Anderson, B. Infectious Diseases of Humans: Dynamics and Control Vol. 28 (Wiley, 1992).

  7. Colizza, V., Barrat, A., Barthélemy, M. & Vespignani, A. The role of the airline transportation network in the prediction and predictability of global epidemics. Proc. Natl Acad. Sci. USA 103, 2015–2020 (2006).

    Article  CAS  Google Scholar 

  8. Brockmann, D. & Helbing, D. The hidden geometry of complex, network-driven contagion phenomena. Science 342, 1337–1342 (2013).

    Article  CAS  Google Scholar 

  9. Bass, F. M. A new product growth for model consumer durables. Manag. Sci. 15, 215–227 (1969).

    Article  Google Scholar 

  10. Fisher, J. C. & Pry, R. H. A simple substitution model of technological change. Technol. Forecast. Soc. Change 3, 75–88 (1972).

    Article  Google Scholar 

  11. Banerjee, A., Chandrasekhar, A. G., Duflo, E. & Jackson, M. O. The diffusion of microfinance. Science 341, 1236498 (2013).

    Article  Google Scholar 

  12. Karsai, M., Iñiguez, G., Kaski, K. & Kertész, J. Complex contagion process in spreading of online innovation. J. R. Soc. Interface 11, 20140694 (2014).

    Article  Google Scholar 

  13. Aral, S., Muchnik, L. & Sundararajan, A. Distinguishing influence-based contagion from homophily-driven diffusion in dynamic networks. Proc. Natl Acad. Sci. USA 106, 21544–21549 (2009).

    Article  CAS  Google Scholar 

  14. Weiss, C. H. et al. Adoption of a high-impact innovation in a homogeneous population. Phys. Rev. X 4, 041008 (2014).

    PubMed  PubMed Central  Google Scholar 

  15. Merton, R. K. The Sociology of Science: Theoretical and Empirical Investigations (Univ. Chicago Press, 1973).

  16. Evans, J. & Foster, J. Metaknowledge. Science 331, 721–725 (2011).

    Article  CAS  Google Scholar 

  17. Granovetter, M. S. The strength of weak ties. Am. J. Sociol. 78, 1360–1380 (1973).

    Article  Google Scholar 

  18. Onnela, J.-P. et al. Structure and tie strengths in mobile communication networks. Proc. Natl Acad. Sci. USA 104, 7332–7336 (2007).

    Article  CAS  Google Scholar 

  19. Pentland, A. Social Physics: How Social Networks Can Make Us Smarter (Penguin, 2015).

  20. Christakis, N. A. & Fowler, J. H. The spread of obesity in a large social network over 32 years. N. Engl. J. Med. 357, 370–379 (2007).

    Article  CAS  Google Scholar 

  21. Centola, D. The spread of behavior in an online social network experiment. Science 329, 1194–1197 (2010).

    Article  CAS  Google Scholar 

  22. Morone, F. & Makse, H. A. Influence maximization in complex networks through optimal percolation. Nature 524, 65–68 (2015).

    Article  CAS  Google Scholar 

  23. Castellano, C., Fortunato, S. & Loreto, V. Statistical physics of social dynamics. Rev. Mod. Phys. 81, 591 (2009).

    Article  Google Scholar 

  24. Kuhn, T. The Structure of Scientific Revolutions (Univ. Chicago Press, 1996).

  25. Jia, T., Wang, D. & Szymanski, B. K. Quantifying patterns of research-interest evolution. Nat. Hum. Behav. 1, 0078 (2017).

    Article  Google Scholar 

  26. Barabási, A.-L. Network Science (Cambridge Univ. Press, 2016).

  27. Zang, C., Cui, P. & Faloutsos, C. Beyond sigmoids: the nettide model for social network growth, and its applications. In Proc. 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining 2015–2024 (ACM, 2016).

  28. Song, C., Qu, Z., Blumm, N. & Barabási, A.-L. Limits of predictability in human mobility. Science 327, 1018–1021 (2010).

    Article  CAS  Google Scholar 

  29. Kooti, F. et al. Portrait of an online shopper: understanding and predicting consumer behavior. In Proc. Ninth ACM International Conference on Web Search and Data Mining 205–214 (ACM, 2016).

  30. Chowell, G., Viboud, C., Hyman, J. M. & Simonsen, L. The western Africa Ebola virus disease epidemic exhibits both global exponential and local polynomial growth rates. PLoS Curr. https://doi.org/10.1371/currents.outbreaks.8b55f4bad99ac5c5db3663e916803261 (2015).

  31. Chowell, G., Sattenspiel, L., Bansal, S. & Viboud, C. Mathematical models to characterize early epidemic growth: a review. Phys. Life Rev. 18, 66–97 (2016).

    Article  Google Scholar 

  32. Chowell, G., Viboud, C., Simonsen, L., Merler, S. & Vespignani, A. Perspectives on model forecasts of the 2014–2015 Ebola epidemic in West Africa: lessons and the way forward. BMC Med. 15, 42 (2017).

    Article  Google Scholar 

  33. Danon, L. & Brooks-Pollock, E. The need for data science in epidemic modelling: comment on: “Mathematical models to characterize early epidemic growth: a review” by Gerardo Chowell et al. Phys. Life Rev. 18, 102–104 (2016).

    Article  Google Scholar 

  34. Chowell, G., Sattenspiel, L., Bansal, S. & Viboud, C. Early sub-exponential epidemic growth: simple models, nonlinear incidence rates, and additional mechanisms: reply to comments on “Mathematical models to characterize early epidemic growth: a review.” Phys. Life Rev. 18, 114–117 (2016).

    Article  Google Scholar 

  35. Wu, F. & Huberman, B. A. Novelty and collective attention. Proc. Natl Acad. Sci. USA 104, 17599–17601 (2007).

    Article  CAS  Google Scholar 

  36. Crane, R. & Sornette, D. Robust dynamic classes revealed by measuring the response function of a social system. Proc. Natl Acad. Sci. USA 105, 15649–15653 (2008).

    Article  CAS  Google Scholar 

  37. Iribarren, J. L. & Moro, E. Impact of human activity patterns on the dynamics of information diffusion. Phys. Rev. Lett. 103, 038702 (2009).

    Article  Google Scholar 

  38. Gleeson, J. P., O’Sullivan, K. P., Baños, R. A. & Moreno, Y. Effects of network structure, competition and memory time on social spreading phenomena. Phys. Rev. X 6, 021019 (2016).

    Google Scholar 

  39. Gleeson, J. P., Cellai, D., Onnela, J.-P., Porter, M. A. & Reed-Tsochas, F. A simple generative model of collective online behavior. Proc. Natl Acad. Sci. USA 111, 10411–10415 (2014).

    Article  CAS  Google Scholar 

  40. Shen, H. -W., Wang, D., Song, C. & Barabási, A. -L. Modeling and predicting popularity dynamics via reinforced Poisson processes. In Proc. 28th AAAI Conference on Artificial Intelligence 14, 291–297 (2014).

    Google Scholar 

  41. Wang, D., Song, C. & Barabási, A.-L. Quantifying long-term scientific impact. Science 342, 127–132 (2013).

    Article  CAS  Google Scholar 

  42. Serrano, M. Á., Boguná, M. & Vespignani, A. Extracting the multiscale backbone of complex weighted networks. Proc. Natl Acad. Sci. USA 106, 6483–6488 (2009).

    Article  CAS  Google Scholar 

  43. Lotka, A. J. Contribution to the theory of periodic reactions. J. Phys. Chem. 14, 271–274 (1910).

    Article  CAS  Google Scholar 

  44. Volterra, V. Variations and fluctuations of the number of individuals in animal species living together. J. Cons. Int. Explor. Mer. 3, 3–51 (1928).

    Article  Google Scholar 

  45. Salganik, M. J., Dodds, P. S. & Watts, D. J. Experimental study of inequality and unpredictability in an artificial cultural market. Science 311, 854–856 (2006).

    Article  CAS  Google Scholar 

  46. van de Rijt, A., Kang, S. M., Restivo, M. & Patil, A. Field experiments of success-breeds-success dynamics. Proc. Natl Acad. Sci. USA 111, 6934–6939 (2014).

    Article  Google Scholar 

  47. Watts, D. J. Everything is Obvious:* Once You Know the Answer (Crown Business, 2011).

  48. Eagle, N., Pentland, A. S. & Lazer, D. Inferring friendship network structure by using mobile phone data. Proc. Natl Acad. Sci. USA 106, 15274–15278 (2009).

    Article  CAS  Google Scholar 

  49. Valera, I. & Gomez-Rodriguez, M. Modeling adoption and usage of competing products. In 2015 IEEE International Conference on Data Mining (ICDM) 409–418 (IEEE, 2015).

  50. Dasgupta, K. et al. Social ties and their relevance to churn in mobile telecom networks. In Proc. 11th International Conference on Extending Database Technology: Advances in Database Technology 668–677 (ACM, 2008).

  51. Sundsøy, P. R., Bjelland, J., Canright, G., Engø-Monsen, K. & Ling, R. Product adoption networks and their growth in a large mobile phone network. In 2010 International Conference on Advances in Social Networks Analysis and Mining 208–216 (IEEE, 2010).

  52. Deville, P. et al. Scaling identity connects human mobility and social interactions. Proc. Natl Acad. Sci. USA 113, 7047–7052 (2016).

    Article  CAS  Google Scholar 

  53. Hèbert-Dufresne, L. & Althouse, B. M. Complex dynamics of synergistic coinfections on realistically clustered networks. Proc. Natl Acad. Sci. USA 112, 10551–10556 (2015).

    Article  Google Scholar 

  54. Scarpino, S. V. et al. Epidemiological and viral genomic sequence analysis of the 2014 Ebola outbreak reveals clustered transmission. Clin. Infect. Dis. 60, 1079–1082 (2014).

    Article  Google Scholar 

  55. Scarpino, S. V., Allard, A. & Hèbert-Dufresne, L. The effect of a prudent adaptive behaviour on disease transmission. Nat. Phys. 12, 1042–1046 (2016).

    Article  CAS  Google Scholar 

  56. Viboud, C., Simonsen, L. & Chowell, G. A generalized-growth model to characterize the early ascending phase of infectious disease outbreaks. Epidemics 15, 27–37 (2016).

    Article  Google Scholar 

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Acknowledgements

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.

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Author notes

  1. These authors contributed equally: Ching Jin, Chaoming Song.

Authors and Affiliations

  1. Northwestern Institute on Complex Systems, Northwestern University, Evanston, IL, USA

    Ching Jin & Dashun Wang

  2. Kellogg School of Management, Northwestern University, Evanston, IL, USA

    Ching Jin & Dashun Wang

  3. Center for Complex Network Research, Northeastern University, Boston, MA, USA

    Ching Jin

  4. Department of Physics, University of Miami, Coral Gables, FL, USA

    Chaoming Song

  5. Telenor Research and Development, Fornebu, Norway

    Johannes Bjelland & Geoffrey Canright

  6. McCormick School of Engineering, Northwestern University, Evanston, IL, USA

    Dashun Wang

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  1. Ching Jin
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Contributions

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.

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Correspondence to Dashun Wang.

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The authors declare no competing interests.

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Peer review information: Primary Handling Editor: Stavroula Kousta.

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Supplementary Information

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

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Jin, C., Song, C., Bjelland, J. et al. Emergence of scaling in complex substitutive systems. Nat Hum Behav 3, 837–846 (2019). https://doi.org/10.1038/s41562-019-0638-y

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