Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Phase transitions in information spreading on structured populations


Mathematical models of social contagion that incorporate networks of human interactions have become increasingly popular, however, very few approaches have tackled the challenges of including complex and realistic properties of socio-technical systems. Here, we define a framework to characterize the dynamics of the Maki–Thompson rumour spreading model in structured populations, and analytically find a previously uncharacterized dynamical phase transition that separates the local and global contagion regimes. We validate our threshold prediction through extensive Monte Carlo simulations. Furthermore, we apply this framework in two real-world systems, the European commuting and transportation network and the Digital Bibliography and Library Project collaboration network. Our findings highlight the importance of the underlying population structure in understanding social contagion phenomena and have the potential to define new intervention strategies aimed at hindering or facilitating the diffusion of information in socio-technical systems.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Rent or buy this article

Prices vary by article type



Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Types of structured population considered in the modelling framework.
Fig. 2: Results from numerical simulations of the rumour spreading process in homogeneous structured populations.
Fig. 3: Results from numerical simulations of a rumour spreading in real-world networks.

Data availability

The data represented in Fig. 3b are available through the Stanford Network Analysis Project (SNAP)48. All other data that supports the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

Code availability

Code is available upon request from the corresponding author.


  1. Hufnagel, L., Brockmann, D. & Geisel, T. Forecast and control of epidemics in a globalized world. Proc. Natl Acad. Sci. USA 101, 15124–15129 (2004).

    Article  ADS  Google Scholar 

  2. Colizza, V., Barrat, A., Barthelemy, 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  ADS  Google Scholar 

  3. Halloran, M. E. et al. Modeling targeted layered containment of an influenza pandemic in the United States. Proc. Natl Acad. Sci. USA 105, 4639–4644 (2008).

    Article  ADS  Google Scholar 

  4. Marathe, M. & Vullikanti, A. K. S. Computational epidemiology. Commun. ACM 56, 88–96 (2013).

    Article  Google Scholar 

  5. Heesterbeek, H. et al. Modeling infectious disease dynamics in the complex landscape of global health. Science 347, aaa4339 (2015).

    Article  Google Scholar 

  6. Liu, Q.-H. et al. Measurability of the epidemic reproduction number in data-driven contact networks. Proc. Natl Acad. Sci. USA 115, 12680–12685 (2018).

    Article  Google Scholar 

  7. Goffman, W. Mathematical approach to the spread of scientific ideas—the history of mast cell research. Nature 212, 449–452 (1966).

    Article  ADS  Google Scholar 

  8. Moreno, Y., Nekovee, M. & Pacheco, A. F. Dynamics of rumor spreading in complex networks. Phys. Rev. E 69, 066130 (2004).

    Article  ADS  Google Scholar 

  9. Bettencourt, L. M., Cintrón-Arias, A., Kaiser, D. I. & Castillo-Chávez, C. The power of a good idea: quantitative modeling of the spread of ideas from epidemiological models. Physica A 364, 513–536 (2006).

    Article  ADS  Google Scholar 

  10. GLEAMviz: The Global Epidemic and Mobility Model;

  11. Centola, D. How Behavior Spreads: The Science of Complex Contagions Vol. 3 (Princeton Univ. Press, 2018).

  12. Guilbeault, D., Becker, J. & Centola, D. in Complex Spreading Phenomena in Social Systems (eds Lehmann, S. & Ahn, Y.-Y) 3–25 (Springer, 2018).

  13. Lehmann, S. & Ahn, Y.-Y. (eds) Complex Spreading Phenomena in Social Systems (Springer, 2018).

  14. Axelrod, R. The dissemination of culture: a model with local convergence and global polarization. J. Conflict Resolution 41, 203–226 (1997).

    Article  Google Scholar 

  15. Baronchelli, A., Felici, M., Loreto, V., Caglioti, E. & Steels, L. Sharp transition towards shared vocabularies in multi-agent systems. J. Stat. Mech. 2006, P06014 (2006).

    Article  Google Scholar 

  16. Centola, D., Becker, J., Brackbill, D. & Baronchelli, A. Experimental evidence for tipping points in social convention. Science 360, 1116–1119 (2018).

    Article  ADS  Google Scholar 

  17. Moreno, Y., Nekovee, M. & Vespignani, A. Efficiency and reliability of epidemic data dissemination in complex networks. Phys. Rev. E 69, 055101(R) (2004).

    Article  ADS  Google Scholar 

  18. 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 

  19. Volkening, A. Linder, D. F. Porter, M. A. & Rempala, G. A. Forecasting elections using compartmental models of infection. Preprint at (2019).

  20. Centola, D. & Macy, M. Complex contagions and the weakness of long ties. Am. J. Sociol. 113, 702–734 (2007).

    Article  Google Scholar 

  21. Granovetter, M. Threshold models of collective behavior. J. Am. Sociol. 83, 1420–1443 (1978).

    Article  Google Scholar 

  22. Watts, D. J. & Dodds, P. in The Oxford Handbook of Analytical Sociology (eds Bearman, P. & Hedström, P.) 475–497 (Oxford University Press, 2017).

  23. Baronchelli, A. The emergence of consensus: a primer. R. Soc. Open Sci. 5, 172189 (2018).

    Article  ADS  MathSciNet  Google Scholar 

  24. Daley, D. J. & Kendall, D. G. Epidemics and rumours. Nature 204, 1118 (1964).

    Article  ADS  Google Scholar 

  25. Maki, D. P. & Thompson, M. Mathematical Models and Applications: With Emphasis on the Social Life, and Management Sciences (Prentice-Hall, 1973).

  26. Zanette, D. H. Dynamics of rumor propagation on small-world networks. Phys. Rev. E 65, 041908 (2002).

    Article  ADS  Google Scholar 

  27. Kempe, D., Kleinberg, J. & Tardos, É. Maximizing the spread of influence through a social network. In Proc. Ninth ACM SIGKDD Int. Conf. Knowledge Discovery and Data Mining KDD03 137 (ACM Press, 2003).

  28. Kosfeld, M. Rumours and markets. J. Math. Econ. 41, 646–664 (2005).

    Article  MathSciNet  Google Scholar 

  29. Liu, Z. & Bambi, H. U. Epidemic spreading in community networks. Europhys. Lett. 72, 315–321 (2005).

    Article  ADS  Google Scholar 

  30. Nekovee, M., Moreno, Y., Bianconi, G. & Marsili, M. Theory of rumour spreading in complex social networks. Physica A 374, 457–470 (2007).

    Article  ADS  Google Scholar 

  31. Borge-Holthoefer, J., Meloni, S., Gonçalves, B. & Moreno, Y. Emergence of influential spreaders in modified rumor models. J. Stat. Phys. 151, 383–393 (2013).

    Article  ADS  MathSciNet  Google Scholar 

  32. Nematzadeh, A., Rodriguez, N., Flammini, A. & Ahn, Y.-Y. in Complex Spreading Phenomena in Social Systems (eds Lehmann, S. & Ahn, Y.-Y.) 97–107 (Springer, 2018).

  33. Vespignani, A. Modeling dynamical processes in complex socio-technical systems. Nat. Phys. 8, 32–30 (2012).

    Article  MathSciNet  Google Scholar 

  34. Karsai, M., Perra, N. & Vespignani, A. Time varying networks and the weakness of strong ties. Sci. Rep. 4, 4001 (2014).

    Article  ADS  Google Scholar 

  35. Fumanelli, L., Ajelli, M., Manfredi, P., Vespignani, A. & Merler, S. Inferring the structure of social contacts from demographic data in the analysis of infectious diseases spread. PLoS Comput. Biol. 8, e1002673 (2012).

    Article  ADS  MathSciNet  Google Scholar 

  36. Daley, D. J. & Gani, J. Epidemic Modelling: An Introduction (Cambridge Univ. Press, 1999).

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

  38. Levins, R. Some demographic and genetic consequences of environmental heterogeneity for biological control. Bull. Entomol. Soc. Am. 15, 237–240 (1969).

    Google Scholar 

  39. Keeling, M. J. Metapopulation moments: coupling, stochasticity and persistence. J. Anim. Ecol. 69, 725–736 (2000).

    Article  Google Scholar 

  40. Murrell, D. J. & Law, R. Beetles in fragmented woodlands: a formal framework for dynamics of movement in ecological landscapes. J. Anim. Ecol. 69, 471–483 (2000).

    Article  Google Scholar 

  41. Colizza, V. & Vespignani, A. Epidemic modeling in metapopulation systems with heterogeneous coupling pattern: theory and simulations. J. Theor. Biol. 251, 450–467 (2008).

    Article  MathSciNet  Google Scholar 

  42. Perra, N., Gonçalves, B., Pastor-Satorras, R. & Vespignani, A. Activity driven modeling of time varying networks. Sci. Rep. 2, 469 (2012).

    Article  ADS  Google Scholar 

  43. Nadini, M. et al. Epidemic spreading in modular time-varying networks. Sci. Rep. 8, 2352 (2018).

    Article  ADS  Google Scholar 

  44. Balcan, D. & Vespignani, A. Phase transitions in contagion processes mediated by recurrent mobility patterns. Nat. Phys. 7, 581–586 (2011).

    Article  Google Scholar 

  45. Balcan, D. et al. Seasonal transmission potential and activity peaks of the new influenza A(H1N1): a Monte Carlo likelihood analysis based on human mobility. BMC Med. 7, 45 (2009).

    Article  Google Scholar 

  46. Balcan, D. et al. Modeling the spatial spread of infectious diseases: the global epidemic and mobility computational model. J. Comput. Sci. 1, 132–145 (2010).

    Article  Google Scholar 

  47. Yang, J. & Leskovec, J. Defining and evaluating network communities based on ground-truth. Knowl. Inf. Syst. 42, 181–213 (2015).

    Article  Google Scholar 

  48. Leskovec, J. & Krevl, A. SNAP Datasets: Stanford Large Network Dataset Collection;

Download references


N.P. was supported in part by the US Army Research Laboratory and the US Army Research Office under contract/grant number W911NF-18-1-0376. Y.M. acknowledges partial support from the Government of Aragón, Spain through grant E36-17R, by MINECO and FEDER funds (grant FIS2017-87519-P) and from Intesa Sanpaolo Innovation Center. The funder had no role in study design, data collection, and analysis, decision to publish, or preparation of the manuscript.

Author information

Authors and Affiliations



J.T.D., N.P., Y.M. and A.V. designed the research. J.T.D., A.V. and Q.Z. performed research and analysed data. All authors wrote the manuscript.

Corresponding author

Correspondence to Alessandro Vespignani.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature Physics thanks Damon Centola, Chris Danforth and Hawoong Jeong for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Source data

Source Data Fig. 2

Data to remake Fig. 2 plots.

Source Data Fig. 3

Data to remake Fig. 3a,b.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Davis, J.T., Perra, N., Zhang, Q. et al. Phase transitions in information spreading on structured populations. Nat. Phys. 16, 590–596 (2020).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing