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Self-Reinforcing Cascades: A Spreading Model for Beliefs or Products of Varying Intensity or Quality

Laurent Hébert-Dufresne1,2,3, Juniper Lovato1,2, Giulio Burgio1, James P. Gleeson4, S. Redner3,1, and P. L. Krapivsky5,3

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Phys. Rev. Lett. 135, 087401 – Published 21 August, 2025

DOI: https://doi.org/10.1103/5mph-sws5

Abstract

Models of how things spread often assume that transmission mechanisms are fixed over time. However, social contagions—the spread of ideas, beliefs, innovations—can lose or gain in momentum as they spread: ideas can get reinforced, beliefs strengthened, products refined. We study the impacts of such self-reinforcement mechanisms in cascade dynamics. We use different mathematical modeling techniques to capture the recursive, yet changing nature of the process. We find a critical regime with a range of power-law cascade size distributions with nonuniversal scaling exponents. This regime clashes with classic models, where criticality requires fine-tuning at a precise critical point. Self-reinforced cascades produce critical-like behavior over a wide range of parameters, which may help explain the ubiquity of power-law distributions in empirical social data.

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Supplemental Material

References (31)

  1. Daniele Notarmuzi, Claudio Castellano, Alessandro Flammini, Dario Mazzilli, and Filippo Radicchi, Universality, criticality and complexity of information propagation in social media, Nat. Commun. 13, 1308 (2022).
  2. Theodore Edward Harris et al., The Theory of Branching Processes (Springer, Berlin, 1963), Vol. 6.
  3. Filippo Radicchi, Claudio Castellano, Alessandro Flammini, Miguel A. Muñoz, and Daniele Notarmuzi, Classes of critical avalanche dynamics in complex networks, Phys. Rev. Res. 2, 033171 (2020).
  4. Márton Karsai, Kimmo Kaski, Albert-László Barabási, and János Kertész, Universal features of correlated bursty behaviour, Sci. Rep. 2, 397 (2012).
  5. Ryosuke Nishi, Taro Takaguchi, Keigo Oka, Takanori Maehara, Masashi Toyoda, Ken-ichi Kawarabayashi, and Naoki Masuda, Reply trees in twitter: Data analysis and branching process models, Social Network Anal. Mining 6, 1 (2016).
  6. Karol Wegrzycki, Piotr Sankowski, Andrzej Pacuk, and Piotr Wygocki, Why do cascade sizes follow a power-law?, in Proceedings of the 26th International Conference on World Wide Web (2017), pp. 569–576.
  7. Javier Borge-Holthoefer, Alejandro Rivero, Iñigo García, Elisa Cauhé, Alfredo Ferrer, Darío Ferrer, David Francos, David Iniguez, María Pilar Pérez, Gonzalo Ruiz et al., Structural and dynamical patterns on online social networks: The Spanish May 15th Movement as a case study, PLoS One 6, e23883 (2011).
  8. Filippo Zimmaro and Henrik Olsson, A meta-model of belief dynamics with personal, expressed and social beliefs, arXiv:2502.14362.
  9. Mirta Galesic, Henrik Olsson, Jonas Dalege, Tamara Van Der Does, and Daniel L. Stein, Integrating social and cognitive aspects of belief dynamics: Towards a unifying framework, J. R. Soc. Interface 18, 20200857 (2021).
  10. Linus Nyman, Understanding Code Forking in Open Source Software: An Examination of Code Forking, Its Effect on Open Source Software, and How it is Viewed and Practiced by Developers (Hanken School of Economics, Helsinki, Finland, 2015).
  11. Robert Viseur, Forks impacts and motivations in free and open source projects, Int. J. Adv. Comput. Sci. Appl. 3, 117 (2012).
  12. Jonas Gamalielsson and Björn Lundell, Sustainability of open source software communities beyond a fork: How and why has the LibreOffice project evolved?, J. Syst. Software 89, 128 (2014).
  13. Jailton Coelho and Marco Tulio Valente, Why modern open source projects fail, in Proceedings of the 2017 11th Joint Meeting on Foundations of Software Engineering (2017), pp. 186–196.
  14. S. Redner, L. Hébert-Dufresne, and P. L. Krapivsky, Long-range fire propagation (to be published).
  15. Daniele Notarmuzi, Claudio Castellano, Alessandro Flammini, Dario Mazzilli, and Filippo Radicchi, Percolation theory of self-exciting temporal processes, Phys. Rev. E 103, L020302 (2021).
  16. James P. Gleeson, Jonathan A. Ward, Kevin P. O’Sullivan, and William T. Lee, Competition-induced criticality in a model of meme popularity, Phys. Rev. Lett. 112, 048701 (2014).
  17. Harry Kesten, Branching Brownian motion with absorption, Stoch. Proc. Appl. 7, 9 (1978).
  18. E. F. Aidékon, Y. Hu, and O. Zindy, The precise tail behavior of the total progeny of a killed branching random walk, Ann. Probab. 41, 3786 (2013).
  19. Herbert S. Wilf, Generating Functionology (Academic Press, New York, 1994).
  20. James P. Gleeson, Tomokatsu Onaga, Peter Fennell, James Cotter, Raymond Burke, and David J. P. O’Sullivan, Branching process descriptions of information cascades on Twitter, J. Complex Netw. 8, cnab002 (2020).
  21. M. E. J. Newman, S. H. Strogatz, and D. J. Watts, Random graphs with arbitrary degree distributions and their applications, Phys. Rev. E 64, 026118 (2001).
  22. See Supplemental Material at http://link.aps.org/supplemental/10.1103/5mph-sws5 for additional analyses.
  23. See notebook at http://bit.ly/SRCnotebook to reproduce the calculations.
  24. Michael Molloy and Bruce Reed, A critical point for random graphs with a given degree sequence, Random Struct. Algorithms 6, 161 (1995).
  25. Satya N. Majumdar and P. L. Krapivsky, Extremal paths on a random Cayley tree, Phys. Rev. E 62, 7735 (2000).
  26. Maury D. Bramson, Maximal displacement of branching Brownian motion, Commun. Pure Appl. Math. 31, 531 (1978).
  27. Éric Brunet and Bernard Derrida, Shift in the velocity of a front due to a cutoff, Phys. Rev. E 56, 2597 (1997).
  28. Éric Brunet, Some aspects of the Fisher-KPP equation and the branching Brownian motion, Habilitation à diriger des recherches, UPMC, 2016, https://theses.hal.science/tel-01417420.
  29. Aaron Clauset, Cosma Rohilla Shalizi, and Mark E. J. Newman, Power-law distributions in empirical data, SIAM Rev. 51, 661 (2009).
  30. M. E. J. Newman, Power laws, Pareto distributions and Zipf’s law, Contemp. Phys. 46, 323 (2005).
  31. , Self-reinforcing cascades notebook (2025), https://bit.ly/SRCnotebook.

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Phys. Rev. Lett. 135, 087401– Published 21 August, 2025
  • Received 12 January 2025
  • Revised 14 July 2025
  • Accepted 18 July 2025
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