Skip to main content

Thank you for visiting nature.com. 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.

Detecting and modelling real percolation and phase transitions of information on social media

Abstract

It is widely believed that information spread on social media is a percolation process, with parallels to phase transitions in theoretical physics. However, evidence for this hypothesis is limited, as phase transitions have not been directly observed in any social media. Here, through an analysis of 100 million Weibo and 40 million Twitter users, we identify percolation-like spread and find that it happens more readily than current theoretical models would predict. The lower percolation threshold can be explained by the existence of positive feedback in the coevolution between network structure and user activity level, such that more-active users gain more followers. Moreover, this coevolution induces an extreme imbalance in users’ influence. Our findings indicate that the ability of information to spread across social networks is higher than expected, with implications for many information-spread problems.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

Fig. 1: Unexpectedly low threshold of information cascades in Weibo.
Fig. 2: Empirical results and data-driven percolation model in Weibo.
Fig. 3: Results of the data-driven percolation model in Twitter.

Data availability

The data for this study are available at https://github.com/Jia-Rong-Xie/data-DMRP. The network is very large. You also can find detailed information on how to download the data at www.huyanqing.com. Source data are provided with this paper.

Code availability

The code for this study is available at https://github.com/Jia-Rong-Xie/code-DMRP.

References

  1. 1.

    Lazer, D. et al. Computational social science. Science 323, 721–723 (2009).

    CAS  PubMed  PubMed Central  Article  Google Scholar 

  2. 2.

    Kitsak, M. et al. Identification of influential spreaders in complex networks. Nat. Phys. 6, 888–893 (2010).

    CAS  Article  Google Scholar 

  3. 3.

    Del Vicario, M. et al. The spreading of misinformation online. Proc. Natl Acad. Sci. USA 113, 554–559 (2016).

    PubMed  Article  CAS  Google Scholar 

  4. 4.

    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).

    PubMed Central  PubMed  Google Scholar 

  5. 5.

    Wang, P., González, M. C., Hidalgo, C. A. & Barabási, A.-L. Understanding the spreading patterns of mobile phone viruses. Science 324, 1071–1076 (2009).

    CAS  PubMed  Article  Google Scholar 

  6. 6.

    Bakshy, E., Messing, S. & Adamic, L. Exposure to ideologically diverse news and opinion on Facebook. Science 348, 1130–1132 (2015).

    CAS  PubMed  Article  Google Scholar 

  7. 7.

    Vosoughi, S., Roy, D. & Aral, S. The spread of true and false news online. Science 359, 1146–1151 (2018).

    CAS  PubMed  PubMed Central  Article  Google Scholar 

  8. 8.

    Kempe, D., Kleinberg, J. & Tardos, É. Maximizing the spread of influence through a social network. in Proc. 9th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (eds Getoor, L., Senator, T., Domingos, P. & Faloutsos, C.) 137–146 (Association for Computing Machinery, 2003).

  9. 9.

    O’Keeffe, G. S. & Clarke-Pearson, K. The impact of social media on children, adolescents, and families. Pediatrics 127, 800–804 (2011).

    PubMed  Article  Google Scholar 

  10. 10.

    Centola, D. Social media and the science of health behavior. Circulation 127, 2135–2144 (2013).

    PubMed  Article  Google Scholar 

  11. 11.

    Onnela, J.-P. & Reed-Tsochas, F. Spontaneous emergence of social influence in online systems. Proc. Natl Acad. Sci. USA 107, 18375–18380 (2010).

    CAS  PubMed  Article  Google Scholar 

  12. 12.

    Bakshy, E., Rosenn, I., Marlow, C. & Adamic, L. The role of social networks in information diffusion. in Proc. 21st International Conference on World Wide Web (eds Mille, A., Gandon, F., Misselis, J., Rabinovich, M. & Staab, S.) 519–528 (Association for Computing Machinery, 2012).

  13. 13.

    Kane, G., Alavi, M., Labianca, G. & Borgatti, S. What’s different about social media networks? A framework and research agenda. MIS Q. 38, 274–304 (2014).

    Article  Google Scholar 

  14. 14.

    Tufekci, Z. & Wilson, C. Social media and the decision to participate in political protest: observations from Tahrir Square. J. Commun. 62, 363–379 (2012).

    Article  Google Scholar 

  15. 15.

    Lehmann, S. & Ahn, Y.-Y. in Complex Spreading Phenomena in Social Systems (eds Lehmann, S. & Ahn, Y.-Y.) 351–358 (Springer, 2018).

  16. 16.

    Zhou, T. et al. Solving the apparent diversity–accuracy dilemma of recommender systems. Proc. Natl Acad. Sci. USA 107, 4511–4515 (2009).

    Article  Google Scholar 

  17. 17.

    Xiang, Z. & Gretzel, U. Role of social media in online travel information search. Tour. Manage. 31, 179–188 (2010).

    Article  Google Scholar 

  18. 18.

    De Vries, L., Gensler, S. & Leeflang, P. S. H. Popularity of brand posts on brand fan pages: an investigation of the effects of social media marketing. J. Interact. Mark. 26, 83–91 (2012).

    Article  Google Scholar 

  19. 19.

    Lazer, D., Kennedy, R., King, G. & Vespignani, A. The parable of Google flu: traps in big data analysis. Science 343, 1203–1205 (2014).

    CAS  PubMed  Article  Google Scholar 

  20. 20.

    Asur, S. & Huberman, B. A. Predicting the future with social media. in Proc. 2010 IEEE/WIC/ACM International Conference on Web Intelligence and Intelligent Agent Technology (eds Huang, X., King, I., Raghavan, V. & Rueger S.) 492–499 (IEEE Computer Society, 2010).

  21. 21.

    Backstrom, L. & Leskovec, J. Supervised random walks: predicting and recommending links in social networks. in Proc. 4th ACM International Conference on Web Search and Data Mining (eds King, I., Nejdl, W. & Li, H.) 635–644 (Association for Computing Machinery, 2011).

  22. 22.

    Conover, M. D. et al. Predicting the political alignment of Twitter users. in PASSAT and IEEE 3rd International Conference on Social Computing (eds Pentland, A., Clippinger, J. & Sweeney, L.) 192–199 (IEEE, 2011).

  23. 23.

    Gao, H., Barbier, H. & Goolsby, R. Harnessing the crowdsourcing power of social media for disaster relief. IEEE Intell. Syst. 26, 10–14 (2011).

    Article  Google Scholar 

  24. 24.

    Yates, D. & Paquette, S. Emergency knowledge management and social media technologies: a case study of the 2010 Haitian earthquake. Int. J. Inf. Manage. 31, 6–13 (2010).

    Article  Google Scholar 

  25. 25.

    Shirky, C. The political power of social media: technology, the public sphere, and political change. Foreign Aff. 90, 28–41 (2011).

    Google Scholar 

  26. 26.

    Gil de Zúñiga, H., Jung, N. & Valenzuela, S. Social media use for news and individuals’ social capital, civic engagement and political participation. J. Comput. Mediat. Commun. 17, 319–336 (2012).

    Article  Google Scholar 

  27. 27.

    Wang, D., Kaplan, L., Le, H. & Abdelzaher, T. On truth discovery in social sensing: a maximum likelihood estimation approach. in Proc. 11th ACM International Conference on Information Processing in Sensor Networks (eds Zhao, F., Terzis, A. & Whitehouse, K.) 233–244 (Association for Computing Machinery, 2012).

  28. 28.

    Pan, B., Zheng, Y., Wilkie, D. & Shahabi, C. Crowd sensing of traffic anomalies based on human mobility and social media. in Proc. 21st ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems (eds Knoblock, C., Schneider, M., Kröger, P., Krumm, J. & Widmayer, P.) 344–353 (Association for Computing Machinery, 2013).

  29. 29.

    Mocanu, D. et al. The Twitter of Babel: mapping world languages through microblogging platforms. PLoS ONE 8, e61981 (2013).

    CAS  PubMed  PubMed Central  Article  Google Scholar 

  30. 30.

    Kaplan, A. M. & Haenlein, M. Users of the world, unite! The challenges and opportunities of social media. Bus. Horiz. 53, 59–68 (2010).

    Article  Google Scholar 

  31. 31.

    Becatti, C., Caldarelli, G., Lambiotte, R. & Saracco, F. Extracting significant signal of news consumption from social networks: the case of Twitter in Italian political elections. Palgrave Commun. 5, 91 (2019).

    Article  Google Scholar 

  32. 32.

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

    CAS  PubMed  Article  Google Scholar 

  33. 33.

    Lü, L. et al. Vital nodes identification in complex networks. Phys. Rep. 650, 1–63 (2016).

    Article  Google Scholar 

  34. 34.

    Leskovec, J., Adamic, L. A. & Huberman, B. A. The dynamics of viral marketing. ACM Trans. Web 1, 5 (2007).

    Article  Google Scholar 

  35. 35.

    Braunstein, A., Dall’Asta, L., Semerjian, G. & Zdeborová, L. Network dismantling. Proc. Natl Acad. Sci. USA 113, 12368–12373 (2016).

    CAS  PubMed  Article  Google Scholar 

  36. 36.

    Mugisha, S. & Zhou, H.-J. Identifying optimal targets of network attack by belief propagation. Phys. Rev. E 94, 012305 (2016).

    PubMed  Article  CAS  Google Scholar 

  37. 37.

    Clusella, P., Grassberger, P., Pérez-Reche, F. J. & Politi, A. Immunization and targeted destruction of networks using explosive percolation. Phys. Rev. Lett. 117, 208301 (2016).

    PubMed  Article  Google Scholar 

  38. 38.

    Watts, D. J. & Strogatz, S. H. Collective dynamics of ‘small-world’ networks. Nature 393, 440–442 (1998).

    CAS  PubMed  Article  Google Scholar 

  39. 39.

    Barabási, A.-L. & Albert, R. Emergence of scaling in random networks. Science 286, 509–512 (1999).

    PubMed  Article  Google Scholar 

  40. 40.

    Dorogovtsev, S. N., Mendes, J. F. F. & Samukhin, A. N. Giant strongly connected component of directed networks. Phys. Rev. E 64, 025101 (2001).

    CAS  Article  Google Scholar 

  41. 41.

    Hu, Y. et al. Local structure can identify and quantify influential global spreaders in large scale social networks. Proc. Natl Acad. Sci. USA 115, 7468–7472 (2018).

    CAS  PubMed  Article  Google Scholar 

  42. 42.

    Newman, M. E. J. Spread of epidemic disease on networks. Phys. Rev. E 66, 016128 (2002).

    CAS  Article  Google Scholar 

  43. 43.

    Kwak, H. et al. What is Twitter, a social network or a news media? in Proc. 19th International Conference on World Wide Web (eds Rappa, M., Jones, P., Freire, J. & Chakrabarti, S.) 591–600 (Association for Computing Machinery, 2010).

  44. 44.

    Weng, L. et al. The role of information diffusion in the evolution of social networks. in Proc. 19th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (eds Ghani, R., et al.) 356–364 (Association for Computing Machinery, 2013).

  45. 45.

    Antoniades, D. & Dovrolis, C. Co-evolutionary dynamics in social networks: a case study of Twitter. Comput. Soc. Netw. 2, 14 (2015).

    Article  Google Scholar 

  46. 46.

    Myers, S. A. & Leskovec, J. The bursty dynamics of the Twitter information network. in Proc. 23rd International Conference on World Wide Web (eds Chung, C., Broder, A., Shim, K. & Suel, T.) 913–924 (Association for Computing Machinery, 2014).

  47. 47.

    Newman, M. E. J., Strogatz, S. H. & Watts, D. J. Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E 64, 026118 (2001).

    CAS  Article  Google Scholar 

  48. 48.

    Dorogovtsev, S. N., Goltsev, A. V. & Mendes, J. F. F. Critical phenomena in complex networks. Rev. Mod. Phys. 80, 1275–1335 (2008).

    Article  Google Scholar 

  49. 49.

    Pastor-Satorras, R., Castellano, C., Van Mieghem, P. & Vespignani, A. Epidemic processes in complex networks. Rev. Mod. Phys. 87, 925–979 (2015).

    Article  Google Scholar 

  50. 50.

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

    CAS  PubMed  Article  Google Scholar 

  51. 51.

    Romero, D. M., Meeder, B. & Kleinberg, J. Differences in the mechanics of information diffusion across topics: idioms, political hashtags, and complex contagion on Twitter. in Proc. 20th International Conference on World Wide Web (eds Sadagopan, S., et al.) 695–704 (Association for Computing Machinery, 2011).

  52. 52.

    Cha, M., Haddadi, H., Benevenuto, F. & Gummadi, K. P. Measuring user influence in Twitter: the million follower fallacy. in Proc. 4th International AAAI Conference on Weblogs and Social Media (eds Hearst, M., Cohen, W. & Gosling, S.) 10–17 (AAAI Press, 2010).

  53. 53.

    Wu, S., Hofman, J. M., Mason, W. A. & Watts, D. J. Who says what to whom on Twitter. in Proc. 20th International Conference on World Wide Web (eds Sadagopan, S., et al.) 705–714 (Association for Computing Machinery, 2011).

  54. 54.

    Bond, R. M. et al. A 61-million-person experiment in social influence and political mobilization. Nature 489, 295–298 (2012).

    CAS  PubMed  Article  Google Scholar 

  55. 55.

    Huang, X., Gao, J., Buldyrev, S. V., Havlin, S. & Stanley, H. E. Robustness of interdependent networks under targeted attack. Phys. Rev. E 83, 065101 (2011).

    Article  CAS  Google Scholar 

  56. 56.

    Efron, B. & Tibshirani, R. J. An Introduction to the Bootstrap (CRC, 1994).

Download references

Acknowledgements

We thank L. Feng and W. Liu for their very helpful discussions. This work was supported by the Natural Science Foundation of Guangdong for Distinguished Youth Scholar, Guangdong Provincial Department of Science and Technology (grant no. 2020B1515020052), Guangdong High-Level Personnel of Special Support Program, Young TopNotch Talents in Technological Innovation (grant no. 2019TQ05X138), the National Natural Science Foundation of China (grant nos 61903385, 61773412, U1911201, U1711265 and 61971454) and the National Key R&D Program of China (grant no. 2018AAA0101203). The funders had no role in study design, data collection and analysis, decision to publish or preparation of the manuscript.

Author information

Affiliations

Authors

Contributions

Y.H. conceived the project. Y.H., J.X. and G.Y. designed the experiments. J.X. performed the experiments and numerical modelling. J.X. and Y.H. solved the model. J.X., F.M., J.S., X.M., G.Y. and Y.H. discussed and analysed the results. J.X., G.Y. and Y.H. wrote the manuscript.

Corresponding author

Correspondence to Yanqing Hu.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature Human Behaviour thanks Michael Danziger, Maksim Kitsak and the other, anonymous, reviewer(s) 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.

Extended data

Extended Data Fig. 1 Relative error of estimated retweet probability \(\frac{{\hat \beta - \beta }}{\beta }\) as a function of cascade size P.

Each dot represents one simulation of data-driven model M2 in Weibo network 2014. \(\hat \beta\) represents the estimated retweet probability by \(\hat \beta = \frac{{P_\infty }}{{P_e}}\) and the real retweet probability is β=0.001 for all dots. Both axis are plotted in log scale. Source data

Extended Data Fig. 2 Global state and localized state of information spreading.

a, Schematic of information cascade in directed network. The spreading in undirected network is a specific scenario in which PGIN(β)=PGOUT(β)=PGSCC(β). b, The cascade sizes are obtained from simulations using data-driven percolation model in Weibo network. The x-axis is plotted in log scale. Each blue point represents one realization. The box region with dashed purple line is most useful for observing phase transitions. Source data

Extended Data Fig. 3 Cascade size distribution ps of uniform models in Weibo.

a-c, The cascade size distribution ps with different retweet probabilities β or λ. The red crosses and yellow circles represent simulation results. The theoretical results are numerical solutions with quadruple-precision floating-point format. d, The probability of large cascade size fluctuation below the percolation threshold. The values of probability that P happens in the boxes are theoretical results. Each blue dot represents a real cascade. βc=2.64×10−3 is the percolation threshold of uniform site percolation, which equals to the threshold of bond model. We use log scale for both axis in A-C, and set only x-axis as log scale in D. Source data

Extended Data Fig. 4 Distributions of relative error between empirical cascades and models.

a, Distribution of relative error in data-driven percolation model. b, Distribution of relative error in uniform percolation model. Source data

Supplementary information

Supplementary Information

Supplementary Methods 1–5, Supplementary Results 1–6, Supplementary Discussion 1–8, Supplementary Figs. 1–32, Supplementary Tables 1–11 and Supplementary References.

Reporting Summary

Source data

Source Data Fig. 1

Statistical source data and simulation results.

Source Data Fig. 2

Statistical source data, simulation results and theoretical results.

Source Data Fig. 3

Statistical source data, simulation results and theoretical results.

Source Data Extended Data Fig. 1

Simulation results.

Source Data Extended Data Fig. 2

Simulation results.

Source Data Extended Data Fig. 3

Statistical source data, simulation results and theoretical results.

Source Data Extended Data Fig. 4

Statistical source data and simulation results.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Xie, J., Meng, F., Sun, J. et al. Detecting and modelling real percolation and phase transitions of information on social media. Nat Hum Behav (2021). https://doi.org/10.1038/s41562-021-01090-z

Download citation

Search

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