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Network localization governs social contagion dynamics with macro-level reinforcement

Leyang Xue1,2,3, Kai-Cheng Yang4,5, Peng-Bi Cui1,2,3,*, and Zengru Di1,2,3

  • *Contact author: cuisir610@gmail.com
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Phys. Rev. Research 8, 013062 – Published 21 January, 2026

DOI: https://doi.org/10.1103/tdz7-r22b

Abstract

The spread of ideas, behaviors, and technologies generally depends on feedback mechanisms operating across multiple scales. Previous studies have extensively examined pairwise transmission and local reinforcement. However, the role of macro-level social influence—where widespread adoption enhances further adoption—remains understudied. Here, we focus on a contagion process that incorporates both pairwise interactions and macro-level reinforcement. We show that the contagion undergoes a shift from continuous to mixed-order transition as macro-level influence exceeds a reinforcement threshold. Simulations on various real-world networks indicate that network localization governs the contagion outcomes by determining the critical point and the reinforcement threshold. Building on this insight, we develop a structural metric linking network localization to contagion dynamics, revealing a key trade-off: Networks that facilitate weak contagion tend to experience slower diffusion and lower adoption rates, while networks that suppress weak contagions enable faster and more widespread adoption. These findings challenge the conventional belief that stronger local connectivity uniformly promotes contagion.

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References (88)

  1. E. M. Rogers, Diffusion of Innovations (The Free Press, New York, 2003).
  2. J. Ugander, L. Backstrom, C. Marlow, and J. Kleinberg, Structural diversity in social contagion, Proc. Natl. Acad. Sci. USA 109, 5962 (2012).
  3. I. Iacopini, G. Petri, A. Barrat, and V. Latora, Simplicial models of social contagion, Nat. Commun. 10, 2485 (2019).
  4. R. A. Chatterjee and J. Eliashberg, The innovation diffusion process in a heterogeneous population: A micromodeling approach, Manag. Sci. 36, 1057 (1990).
  5. F. S. Berry and W. D. Berry, Innovation and diffusion models in policy research, theories of the policy process, in Theories of the Policy Process, edited by C. M. Weible and P. A. Sabatier, 4th ed. (Routledge, New York, 2018), p. 416.
  6. G. E. Kreindler and H. P. Young, Rapid innovation diffusion in social networks, Proc. Natl. Acad. Sci. USA 111, 10881 (2014).
  7. F. M. Bass, A new product growth for model consumer durables, Manag. Sci. 15, 215 (1969).
  8. E. Kiesling, M. Günther, C. Stummer, and L. M. Wakolbinger, Agent-based simulation of innovation diffusion: A review, Cent. Eur. J. Oper. Res. 20, 183 (2012).
  9. G. Cencetti, D. A. Contreras, M. Mancastroppa, and A. Barrat, Distinguishing simple and complex contagion processes on networks, Phys. Rev. Lett. 130, 247401 (2023).
  10. Y. Jiang and J. C. Jiang, Diffusion in social networks: A multiagent perspective, IEEE Trans. Syst. Man Cybern Syst. 45, 198 (2014).
  11. R. Pastor-Satorras, C. Castellano, P. Van Mieghem, and A. Vespignani, Epidemic processes in complex networks, Rev. Mod. Phys. 87, 925 (2015).
  12. L. M. A. Bettencourt, A. Cintrón-Arias, D. I. Kaiser, and C. Castillo-Chávez, The power of a good idea: Quantitative modeling of the spread of ideas from epidemiological models, Physica A 364, 513 (2006).
  13. D. Centola and M. Macy, Complex contagions and the weakness of long ties, Am. J. Sociol. 113, 702 (2007).
  14. D. Centola, The spread of behavior in an online social network experiment, Science 329, 1194 (2010).
  15. D. Centola, How Behavior Spreads: The Science of Complex Contagions (Princeton University Press, Princeton, NJ, 2018), Vol. 3.
  16. D. Guilbeault, J. Becker, and D. Centola, Complex Contagions: A Decade in Review (Springer International Publishing, Cham, 2018), pp. 3–25.
  17. D. J. Watts, A simple model of global cascades on random networks, Proc. Natl. Acad. Sci. USA 99, 5766 (2002).
  18. S. Chowdhary, A. Kumar, G. Cencetti, I. Iacopini, and F. Battiston, Simplicial contagion in temporal higher-order networks, J. Phys. Complexity 2, 035019 (2021).
  19. G. F. de Arruda, G. Petri, and Y. Moreno, Social contagion models on hypergraphs, Phys. Rev. Res. 2, 023032 (2020).
  20. P.-B. Cui, W. Wang, S.-M. Cai, T. Zhou, and Y.-C. Lai, Close and ordinary social contacts: How important are they in promoting large-scale contagion? Phys. Rev. E 98, 052311 (2018).
  21. Z. Su, W. Wang, L. Li, H. E. Stanley, and L. A. Braunstein, Optimal community structure for social contagions, New J. Phys. 20, 053053 (2018).
  22. G. St-Onge, I. Iacopini, V. Latora, A. Barrat, G. Petri, A. Allard, and L. Hébert-Dufresne, Influential groups for seeding and sustaining nonlinear contagion in heterogeneous hypergraphs, Commun. Phys. 5, 25 (2022).
  23. G. Ferraz de Arruda, G. Petri, P. M. Rodriguez, and Y. Moreno, Multistability, intermittency, and hybrid transitions in social contagion models on hypergraphs, Nat. Commun. 14, 1375 (2023).
  24. G. Ferraz de Arruda, A. Aleta, and Y. Moreno, Contagion dynamics on higher-order networks, Nat. Rev. Phys. 6, 468 (2024).
  25. J.-P. Onnela and F. Reed-Tsochas, Spontaneous emergence of social influence in online systems, Proc. Natl. Acad. Sci. USA 107, 18375 (2010).
  26. J. L. Toole, M. Cha, and M. C. González, Modeling the adoption of innovations in the presence of geographic and media influences, PLoS One 7, e29528 (2012).
  27. Z. Ma, Z. Yang, and M. Mourali, Consumer adoption of new products: Independent versus interdependent self-perspectives, J. Mark. 78, 101 (2014).
  28. W. Duan, B. Gu, and A. B. Whinston, Informational cascades and software adoption on the internet: An real-world investigation, MIS Q. 33, 23 (2009).
  29. H. Jo and Y. Bang, Analyzing ChatGPT adoption drivers with the TOEK framework, Sci. Rep. 13, 22606 (2023).
  30. H. Kwak, C. Lee, H. Park, and S. Moon, What is Twitter, a social network or a news media? in Proceedings of the 19th International Conference on World Wide Web (ACM, New York, 2010), pp. 591–600.
  31. F. Duarte, Number of ChatGPT users, Exploding topics, 12 November 2024, https://explodingtopics.com/blog/chatgpt-users, accessed 13 November 2024.
  32. S. Vosoughi, D. Roy, and S. Aral, The spread of true and false news online, Science 359, 1146 (2018).
  33. V. Kumar, A. R. Ashraf, and W. Nadeem, AI-powered marketing: What, where, and how? Int. J. Inf. Manag. 77, 102783 (2024).
  34. D. McCoy and S. Lyons, Consumer preferences and the influence of networks in electric vehicle diffusion: An agent-based microsimulation in Ireland, Energy Res. Soc. Sci. 3, 89 (2014).
  35. H. P. Young, Innovation diffusion in heterogeneous populations: Contagion, social influence, and social learning, Am. Econ. Rev. 99, 1899 (2009).
  36. C.-W. Chen, H.-Y. Chang, J.-H. Chen, and R. Weng, Elucidating the role of conformity in innovative smartphones, Int. J. Mob. Commun. 14, 56 (2016).
  37. M. L. Katz and C. Shapiro, Product introduction with network externalities, J. Ind. Econ. 40, 55 (1992).
  38. D. Schoder, Forecasting the success of telecommunication services in the presence of network effects, Inf. Econ. Policy 12, 181 (2000).
  39. J. H. Rohlfs, Bandwagon Effects in High-Technology Industries (MIT Press, Cambridge, 2003).
  40. F. M. Scherer and D. R. Ross, Industrial Market Structure and Economic Performance (Houghton Mifflin, 1990).
  41. E. K. Browning and M. A. Zupan, Microeconomics: Theory and Applications (John Wiley & Sons, New York, 2020).
  42. F. A. Buttle, Word of mouth: Understanding and managing referral marketing, J. Strateg. Mark. 6, 241 (1998).
  43. C. S. E. Bale, N. J. McCullen, T. J. Foxon, A. M. Rucklidge, and W. F. Gale, Harnessing social networks for promoting adoption of energy technologies in the domestic sector, Energy Policy 63, 833 (2013).
  44. S. A. Delre, W. Jager, T. H. A. Bijmolt, and M. A. Janssen, Will it spread or not? The effects of social influences and network topology on innovation diffusion, J. Prod. Innov. Manage. 27, 267 (2010).
  45. See Supplemental Material at https://link.aps.org/supplemental/10.1103/tdz7-r22b for details on the network datasets and additional numerical results [74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88].
  46. See Supplemental Material at https://link.aps.org/supplemental/10.1103/tdz7-r22b for details on the mixed-order (hybrid) phase transition and the associated scaling analyses (Sec. B 2).
  47. See Supplemental Material at https://link.aps.org/supplemental/10.1103/tdz7-r22b for details on the effect of macro-level reinforcement on the intrinsic threshold βc (Sec. B 3).
  48. S. Boccaletti, J. A. Almendral, S. Guan, I. Leyva, Z. Liu, I. Sendiña-Nadal, Z. Wang, and Y. Zou, Explosive transitions in complex networks’ structure and dynamics: Percolation and synchronization, Phys. Rep. 660, 1 (2016).
  49. R. M. D’ Souza, J. Gómez-Gardenes, J. Nagler, and A. Arenas, Explosive phenomena in complex networks, Adv. Phys. 68, 123 (2019).
  50. See Supplemental Material at https://link.aps.org/supplemental/10.1103/tdz7-r22b for details on the implications of the reinforcement threshold αc (Sec. B 1).
  51. T. Martin, X. Zhang, and M. E. J. Newman, Localization and centrality in networks, Phys. Rev. E 90, 052808 (2014).
  52. A. Koher, H. H. K. Lentz, J. P. Gleeson, and P. Hövel, Contact-based model for epidemic spreading on temporal networks, Phys. Rev. X 9, 031017 (2019).
  53. R. Pastor-Satorras and C. Castellano, The localization of non-backtracking centrality in networks and its physical consequences, Sci. Rep. 10, 21639 (2020).
  54. W. Cai, L. Chen, F. Ghanbarnejad, and P. Grassberger, Avalanche outbreaks emerging in cooperative contagions, Nat. Phys. 11, 936 (2015).
  55. M. Shrestha, S. V. Scarpino, and C. Moore, Message-passing approach for recurrent-state epidemic models on networks, Phys. Rev. E 92, 022821 (2015).
  56. G. Timár, S. N. Dorogovtsev, and J. F. F. Mendes, Localization of nonbacktracking centrality on dense subgraphs of sparse networks, Phys. Rev. E 107, 014301 (2023).
  57. P. Bonacich, Factoring and weighting approaches to status scores and clique identification, J. Math. Sociol. 2, 113 (1972).
  58. R. B. Bendel, S. S. Higgins, J. E. Teberg, and D. A. Pyke, Comparison of skewness coefficient, coefficient of variation, and Gini coefficient as inequality measures within populations, Oecologia 78, 394 (1989).
  59. See Supplemental Material at https://link.aps.org/supplemental/10.1103/tdz7-r22b for a comparison of different approaches to quantify the localization strength and their performance in predicting αc (Sec. B 4).
  60. M. L. Sheng, S.-Y. Chang, T. Teo, and Y.-F. Lin, Knowledge barriers, knowledge transfer, and innovation competitive advantage in healthcare settings, Manag. Decis. 51, 461 (2013).
  61. R. Xulvi-Brunet and I. M. Sokolov, Reshuffling scale-free networks: From random to assortative, Phys. Rev. E 70, 066102 (2004).
  62. See Supplemental Material at https://link.aps.org/supplemental/10.1103/tdz7-r22b for details on tuning the localization strength of networks (Sec. B 7).
  63. See Supplemental Material at https://link.aps.org/supplemental/10.1103/tdz7-r22b for an analysis of the relation between the predicted critical threshold βc and the reinforcement threshold αc (Sec. B 6).
  64. F. Malizia, A. Guzmán, I. Iacopini, and I. Z. Kiss, Disentangling the role of heterogeneity and hyperedge overlap in explosive contagion on higher-order networks, Phys. Rev. Lett. 135, 207401 (2025).
  65. S. Lamata-Otín, A. Reyna-Lara, D. Soriano-Paños, V. Latora, and J. Gómez-Gardeñes, Collapse transition in epidemic spreading subject to detection with limited resources, Phys. Rev. E 108, 024305 (2023).
  66. A. V. Goltsev, S. N. Dorogovtsev, J. G. Oliveira, and J. F. F. Mendes, Localization and spreading of diseases in complex networks, Phys. Rev. Lett. 109, 128702 (2012).
  67. https://data.mendeley.com/datasets/d848h7rcdg.
  68. See Supplemental Material at https://link.aps.org/supplemental/10.1103/tdz7-r22b for detailed descriptions of the data used in this study.
  69. See Supplemental Material at https://link.aps.org/supplemental/10.1103/tdz7-r22b for details on identifying the class of phase transitions (Sec. B 5).
  70. R. A Horn and C. R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 2012).
  71. B. Karrer and M. E. J. Newman, Message passing approach for general epidemic models, Phys. Rev. E 82, 016101 (2010).
  72. A. Y. Lokhov, M. Mézard, H. Ohta, and L. Zdeborová, Inferring the origin of an epidemic with a dynamic message-passing algorithm, Phys. Rev. E 90, 012801 (2014).
  73. See Supplemental Material at http://link.aps.org/supplemental/10.1103/tdz7-r22b for details on the network models (Sec. A 2).
  74. R. A. Rossi and N. K. Ahmed, The network data repository with interactive graph analytics and visualization, in Proceedings of the AAAI Conference on Artificial Intelligence (AAAI Press, Palo Alto, CA, 2015).
  75. K. J. Cios and L. A. Kurgan, Trends in data mining and knowledge discovery, in Advanced Techniques in Knowledge Discovery and Data Mining, edited by N. R. Pal and L. Jain (Springer, London, 2005), pp. 1–26.
  76. V. Batagelj and A. Mrvar, Analysis of large networks, in Proceedings of Pajek Workshop at XXVI Sunbelt Conference (University of Ljubljana, Institute of Mathematics, Physics and Mechanics, Department of Mathematics, Ljubljana, Slovenia, 2006).
  77. J. Leskovec and A. Krevl, SNAP datasets: Stanford large network dataset collection (2014), https://snap.stanford.edu/data.
  78. A.-L. Barabási, Network Science (Cambridge University Press, Cambridge, 2016).
  79. K.-I. Goh, B. Kahng, and D. Kim, Universal behavior of load distribution in scale-free networks, Phys. Rev. Lett. 87, 278701 (2001).
  80. B. Gross, I. Bonamassa, and S. Havlin, Fractal fluctuations at mixed-order transitions in interdependent networks, Phys. Rev. Lett. 129, 268301 (2022).
  81. I. Bonamassa, B. Gross, J. Kertész, and S. Havlin, Hybrid universality classes of systemic cascades, Nat. Commun. 16, 1415 (2025).
  82. J. Korbel, S. Havlin, and S. Thurner, Microscopic origin of abrupt mixed-order phase transitions, Nat. Commun. 16, 2628 (2025).
  83. Fractals and Disordered Systems, edited by A. Bunde and S. Havlin (Springer, Berlin, Heidelberg, 2012).
  84. A. Bar and D. Mukamel, Mixed-order phase transition in a one-dimensional model, Phys. Rev. Lett. 112, 015701 (2014).
  85. D. Stauffer and A. Aharony, Introduction to Percolation Theory (Taylor & Francis, London, 2018).
  86. D. Zhou, A. Bashan, R. Cohen, Y. Berezin, N. Shnerb, and S. Havlin, Simultaneous first- and second-order percolation transitions in interdependent networks, Phys. Rev. E 90, 012803 (2014).
  87. B. Gross, I. Volotsenko, Y. Sallem, N. Yadid, I. Bonamassa, S. Havlin, and A. Frydman, The random cascading origin of abrupt transitions in interdependent systems, Nat. Commun. 16, 5869 (2025).
  88. D. Dilmoney, B. Gross, S. Havlin, and N. M. Shnerb, Dynamics of critical cascades in interdependent networks, Phys. Rev. Lett. 134, 187401 (2025).

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Phys. Rev. Research 8, 013062– Published 21 January, 2026
  • Received 29 January 2023
  • Accepted 15 December 2025
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