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.

Modelling dynamical processes in complex socio-technical systems

Abstract

In recent years the increasing availability of computer power and informatics tools has enabled the gathering of reliable data quantifying the complexity of socio-technical systems. Data-driven computational models have emerged as appropriate tools to tackle the study of dynamical phenomena as diverse as epidemic outbreaks, information spreading and Internet packet routing. These models aim at providing a rationale for understanding the emerging tipping points and nonlinear properties that often underpin the most interesting characteristics of socio-technical systems. Here, using diffusion and contagion phenomena as prototypical examples, we review some of the recent progress in modelling dynamical processes that integrates the complex features and heterogeneities of real-world systems.

This is a preview of subscription content

Access options

Buy article

Get time limited or full article access on ReadCube.

$32.00

All prices are NET prices.

Figure 1: Phase diagram of epidemic models.
Figure 2: Progression of an epidemic process.
Figure 3: Illustration of the global threshold in reaction–diffusion processes.
Figure 4: Visualization of the dynamical network generated by Twitter interactions.

References

  1. Keeling, M. J. & Rohani, P. Modeling Infectious Diseases in Humans and Animals (Princeton Univ. Press, 2008).

    MATH  Google Scholar 

  2. Goffman, W. & Newill, V. A. Generalization of epidemic theory: An application to the transmission of ideas. Nature 204, 225–228 (1964).

    ADS  Google Scholar 

  3. Rapoport, A. Spread of information through a population with socio-structural bias: I. Assumption of transitivity. Bull. Math. Biol. 15, 523–533 (1953).

    MathSciNet  Google Scholar 

  4. Tabah, A. N. Literature dynamics: Studies on growth, diffusion, and epidemics. Annu. Rev. Inform. Sci. Technol. 34, 249–286 (1999).

    Google Scholar 

  5. Lloyd, A. L. & May, R. M. How viruses spread among computers and people. Science 292, 1316–1317 (2001).

    Google Scholar 

  6. Grassberger, P. On the critical behavior of the general epidemic process and dynamical percolation. Math. Biosci. 63, 157–172 (1983).

    MATH  Google Scholar 

  7. Harris, T. E. Contact interactions on a lattice. Ann. Prob. 2, 969–988 (1974).

    MathSciNet  MATH  Google Scholar 

  8. Marro, J. & Dickman, R. Nonequilibrium Phase Transitions in Lattice Models (Cambridge Univ. Press, 1999).

    MATH  Google Scholar 

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

    Google Scholar 

  10. Nowak, A., Szamrej, J. & Latané, B. From private attitude to public opinion: A dynamic theory of social impact. Psychol. Rev. 97, 362–376 (1990).

    Google Scholar 

  11. Axelrod, R. The Complexity of Cooperation (Princeton Univ. Press, 1997).

    MATH  Google Scholar 

  12. Castellano, C., Fortunato, S. & Loreto, V. Statistical physics of social dynamics. Rev. Mod. Phys. 81, 591–646 (2009).

    ADS  Google Scholar 

  13. Krapivsky, P. L. Kinetics of monomer–monomer surface catalytic reactions. Phys. Rev. A 45, 1067–1072 (1992).

    ADS  Google Scholar 

  14. Galam, S. Minority opinion spreading in random geometry. Eur. Phys. J. B 25, 403–406 (2002).

    ADS  Google Scholar 

  15. Krapivsky, P. L. & Redner, S. Dynamics of majority rule in two-state interacting spin systems. Phys. Rev. Lett. 90, 238701 (2003).

    ADS  Google Scholar 

  16. Sznajd-Weron, K. & Sznajd, J. Opinion evolution in closed community. Int. J. Mod. Phys. C 11, 1157–1165 (2000).

    ADS  MATH  Google Scholar 

  17. Deffuant, G., Neau, D., Amblard, F. & Weisbuch, G. Mixing beliefs among interacting agents. Adv. Complex Syst. 3, 87–98 (2000).

    Google Scholar 

  18. Hegselmann, R. & Krause, U. Opinion dynamics and bounded confidence models, analysis and simulation. J. Art. Soc. Soc. Sim. 5, 2 (2002).

    Google Scholar 

  19. Ben-Naim, E., Krapivsky, P. L. & Redner, S. Bifurcations and patterns in compromise processes. Physica D 183, 190–204 (2003).

    ADS  MathSciNet  MATH  Google Scholar 

  20. Leland, W. E., Taqqu, M. S., Willinger, W. & Wilson, D. V. On the self-similar nature of Ethernet traffic. IEEE/ACM Trans. Netw. 2, 1–15 (1994).

    Google Scholar 

  21. Csabai, I. 1/f noise in computer network traffic. J. Phys. A 27, L417–L42 (1994).

    ADS  Google Scholar 

  22. Solé, R. V. & Valverde, S. Information transfer and phase transitions in a model of internet traffic. Physica A 289, 595–605 (2001).

    ADS  MATH  Google Scholar 

  23. Willinger, W., Govindan, R, Jamin, S., Paxson, V. & Shenker, S. Scaling phenomena in the Internet: Critically examining criticality. Proc. Natl Acad. Sci. USA 99, 2573–2580 (2002).

    ADS  Google Scholar 

  24. Valverde, S. & Solé, R. V. Internet’s critical path horizon. Eur. Phys. J. B 38, 245–252 (2004).

    ADS  Google Scholar 

  25. Tadić, B., Thurner, S. & Rodgers, G. J. Traffic on complex networks: Towards understanding global statistical properties from microscopic density fluctuations. Phys. Rev. E 69, 036102 (2004).

    ADS  Google Scholar 

  26. Crovella, M. E. & Krishnamurthy, B. Internet Measurements: Infrastructure, Traffic and Applications (John Wiley, 2006).

    Google Scholar 

  27. Helbing, D. Traffic and related self-driven many particle systems. Rev. Mod. Phys. 73, 1067–1141 (2001).

    ADS  MathSciNet  Google Scholar 

  28. Albert, R., Jeong, H. & Barabási, A-L. Internet: Diameter of the World-Wide Web. Nature 401, 130–131 (1999).

    ADS  Google Scholar 

  29. Pastor-Satorras, R. & Vespignani, A. Evolution and Structure of the Internet: A Statistical Physics Approach (Cambridge Univ. Press, 2004).

    MATH  Google Scholar 

  30. Brockmann, D., Hufnagel, L. & Geisel, T. The scaling laws of human travel. Nature 439, 462–465 (2006).

    ADS  Google Scholar 

  31. Onnela, J-P. et al. Structure and tie strengths in mobile communication networks. Proc. Natl Acad. Sci. USA 104, 7332–7337 (2007).

    ADS  Google Scholar 

  32. González, M. C., Hidalgo, C. A. & Barabási, A-L. Understanding individual human mobility patterns. Nature 453, 779–782 (2008).

    ADS  Google Scholar 

  33. Lazer, D et al. Life in the network: The coming age of computational social science. Science 323, 721–723 (2009).

    Google Scholar 

  34. Vespignani, A. Predicting the behavior of tecno-social systems. Science 325, 425–428 (2009).

    ADS  MathSciNet  MATH  Google Scholar 

  35. Albert, R. & Barabási, A-L. Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97 (2002).

    ADS  MathSciNet  MATH  Google Scholar 

  36. Boccaletti, S. et al. Complex networks: Structure and dynamics. Phys. Rep. 424, 175–308 (2006).

    ADS  MathSciNet  MATH  Google Scholar 

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

    ADS  Google Scholar 

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

    MATH  Google Scholar 

  39. Cohen, R. & Havlin, S. Complex Networks: Structure, Robustness and Function (Cambridge Univ. Press, 2010).

    MATH  Google Scholar 

  40. Newman, M. E. J. Networks: An Introduction (Oxford Univ. Press, 2010).

    MATH  Google Scholar 

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

    ADS  MATH  Google Scholar 

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

    ADS  MathSciNet  MATH  Google Scholar 

  43. Dorogovtsev, S. N. & Mendes, J. F. F. Evolution of Networks: From Biological Nets to the Internet and WWW. (Oxford Univ. Press, 2003).

    MATH  Google Scholar 

  44. Amaral, L. A. N., Scala, A., Barthlemy, M. & Stanley, H. E. Classes of small-world networks. Proc. Natl Acad. Sci. USA 97, 11149–11154 (2005).

    ADS  Google Scholar 

  45. Barrat, A., Barthlemy, M., Pastor-Satorras, R. & Vespignani, A. The architecture of complex weighted networks. Proc. Natl Acad. Sci. USA 101, 3747–3752 (2004).

    ADS  Google Scholar 

  46. Pastor-Satorras, R. & Vespignani, A. Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86, 3200–3203 (2001).

    ADS  Google Scholar 

  47. Moreno, Y., Pastor-Satorras, R. & Vespignani, A. Epidemic outbreaks in complex heterogeneous networks. Eur. Phys. J. B 26, 521–529 (2002).

    ADS  Google Scholar 

  48. Hethcote, H. W. & Yorke, J. A. Gonorrhea: Transmission and control. Lect. Notes Biomath. 56, 1–105 (1984).

    MathSciNet  MATH  Google Scholar 

  49. Anderson, R. M. & May, R. M. Infectious Diseases in Humans (Oxford Univ. Press, 1992).

    Google Scholar 

  50. May, R. M. & Lloyd, A. L. Infection dynamics on scale-free networks. Phys. Rev. E 64, 066112 (2001).

    ADS  Google Scholar 

  51. Pastor-Satorras, R. & Vespignani, R. Epidemic dynamics in finite size scale-free networks. Phys. Rev. E 65, 035108(R) (2002).

    ADS  Google Scholar 

  52. Barthelemy, M., Barrat, A., Pastor-Satorras, R. & Vespignani, A. Velocity and hierarchical spread of epidemic outbreaks in scale-free networks. Phys. Rev. Lett. 92, 178701 (2004).

    ADS  Google Scholar 

  53. Wang, Y., Chakrabarti, D., Wang, G. & Faloutsos, C, in Proc. 22nd International Symposium on Reliable Distributed Systems (SRDS’03) 25–34 (IEEE, 2003).

    Google Scholar 

  54. Boguna, M., Pastor-Satorras, R. & Vespignani, A. Absence of epidemic threshold in scale-free networks with degree correlations. Phys. Rev. Lett. 90, 028701 (2003).

    ADS  MATH  Google Scholar 

  55. Castellano, C. & Pastor-Satorras, R. Routes to thermodynamic limit on scale-free networks. Phys. Rev. Lett. 100, 148701 (2008).

    ADS  Google Scholar 

  56. Chatterjee, S. & Durrett, R. Contact processes on random graphs with power law degree distributions have critical value 0. Ann. Probab. 37, 2332–2356 (2009).

    MathSciNet  MATH  Google Scholar 

  57. Castellano, C. & Pastor-Satorras, R. Thresholds for epidemic spreading in networks. Phys. Rev. Lett. 105, 218701 (2010).

    ADS  Google Scholar 

  58. Durrett, R. Some features of the spread of epidemics and information on a random graph. Proc. Natl Acad. Sci. USA 107, 4491–4498 (2010).

    ADS  Google Scholar 

  59. Pastor-Satorras, R. & Vespignani, A. Immunization of complex networks. Phys. Rev. E 65, 036104 (2001).

    ADS  Google Scholar 

  60. Cohen, R., Havlin, S. & Ben-Avraham, D. Efficient immunization strategies for computer networks and populations. Phys. Rev. Lett. 91, 247901 (2003).

    ADS  Google Scholar 

  61. Holme, P. Efficient local strategies for vaccination and network attack. Europhys. Lett. 68, 908–914 (2004).

    ADS  Google Scholar 

  62. Goldenberg, J., Shavitt, Y., Shir, E. & Solomon, S. Distributive immunization of networks against viruses using the ‘honey-pot’ architecture. Nature Phys. 1, 184–188 (2005).

    ADS  Google Scholar 

  63. Motter, A. E., Zhou, C. S. & Kurths, J. Enhancing complex-network synchronization. Europhys. Lett. 69, 334–340 (2005).

    ADS  Google Scholar 

  64. Motter, A. E., Zhou, C. S. & Kurths, J. Network synchronization, diffusion, and the paradox of heterogeneity. Phys. Rev. E 71, 016116 (2005).

    ADS  Google Scholar 

  65. Gómez-Gardeñes, J., Campillo, M., Floria, L. M. & Moreno, Y. Dynamical organization of cooperation in complex topologies. Phys. Rev. Lett. 98, 108103 (2007).

    ADS  Google Scholar 

  66. Korniss, G. Synchronization in weighted uncorrelated complex networks in a noisy environment: Optimization and connections with transport efficiency. Phys. Rev. E 75, 051121 (2007).

    ADS  Google Scholar 

  67. Arenas, A., Díaz-Guilera, A. & Guimerà, R. Communication in networks with hierarchical branching. Phys. Rev. Lett. 86, 3196–3199 (2001).

    ADS  MATH  Google Scholar 

  68. Guimerà, R., Arenas, A., Díaz-Guilera, A. & Giralt, F. Dynamical properties of model communication networks. Phys. Rev. E 66, 026704 (2002).

    ADS  Google Scholar 

  69. Sreenivasan, S., Cohen, R., López, E., Toroczkai, Z. & Stanley, H. E. Structural bottlenecks for communication in networks. Phys. Rev. E 75, 036105 (2007).

    ADS  Google Scholar 

  70. Castellano, C., Loreto, V., Barrat, A., Cecconi, F. & Parisi, D. Comparison of voter and Glauber ordering dynamics on networks. Phys. Rev. E 71, 066107 (2005).

    ADS  Google Scholar 

  71. Sood, V. & Redner, S. Voter model on heterogeneous graphs. Phys. Rev. Lett. 94, 178701 (2005).

    ADS  Google Scholar 

  72. Suchecki, K., Eguíluz, V. M. & San Miguel, M. Conservation laws for the voter model in complex networks. Europhys. Lett. 69, 228–234 (2005).

    ADS  Google Scholar 

  73. Klemm, K., Eguíluz, V. M., Toral, R. & San Miguel, M. Nonequilibrium transitions in complex networks: A model of social interaction. Phys. Rev. E. 67, 026120 (2003).

    ADS  Google Scholar 

  74. Santos, F. C., Pacheco, J. M. & Lenaerts, T. Evolutionary dynamics of social dilemmas in structured heterogeneous populations. Proc. Natl Acad. Sci. USA 103, 3490–3494 (2006).

    ADS  Google Scholar 

  75. van Kampen, N. G. Stochastic Processes in Physics and Chemistry (North-Holland, 1981).

    MATH  Google Scholar 

  76. Bolker, B. M. & Grenfell, T. Chaos and biological complexity in measles dynamics. Proc. Trans. R. Soc. Lond. B 251, 75–81 (1993).

    ADS  Google Scholar 

  77. Keeling, M. J. & Rohani, P. Estimating spatial coupling in epidemiological systems: A mechanistic approach. Ecol. Lett. 5, 20–29 (2002).

    Google Scholar 

  78. Sattenspiel, L. & Dietz, K. A structured epidemic model incorporating geographic mobility among regions. Math. Biosci. 128, 71–91 (1995).

    MATH  Google Scholar 

  79. Watts, D., Muhamad, R., Medina, D. C. & Dodds, P. S. Multiscale resurgent epidemics in a hierarchical metapopulation model. Proc. Natl Acad. Sci. USA 102, 11157–11162 (2005).

    ADS  Google Scholar 

  80. Turing, A. M. The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B237, 37–72 (1952).

    ADS  MathSciNet  MATH  Google Scholar 

  81. Nakao, H. & Mikhailov, A. S. Turing patterns in network-organized activator-inhibitor systems. Nature Phys. 6, 544–550 (2010).

    ADS  Google Scholar 

  82. Colizza, V., Pastor-Satorras, R. & Vespignani, A. Reaction–diffusion processes and metapopulation models in heterogeneous networks. Nature Phys. 3, 276–282 (2007).

    ADS  Google Scholar 

  83. Colizza, V. & Vespignani, A. Invasion threshold in heterogeneous metapopulation networks. Phys. Rev. Lett. 99, 148701 (2007).

    ADS  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  85. Barthélemy, M., Godrèche, C. & Luck, J-M. Fluctuation effects in metapopulation models: Percolation and pandemic threshold. J. Theor. Biol. 267, 554–564 (2010).

    MathSciNet  MATH  Google Scholar 

  86. Saldana, J. Continuous-time formulation of reaction–diffusion processes on heterogeneous metapopulations. Phys. Rev. E 78, 012902 (2008).

    ADS  Google Scholar 

  87. Ni, S. & Weng, W. Impact of travel patterns on epidemic dynamics in heterogeneous spatial metapopulation networks. Phys. Rev. E 79, 016111 (2009).

    ADS  Google Scholar 

  88. Ben-Zion, Y., Cohena, Y. & Shnerba, N. M. Modeling epidemics dynamics on heterogenous networks. J. Theor. Biol. 264, 197–204 (2010).

    MathSciNet  Google Scholar 

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

    ADS  Google Scholar 

  90. Belik, V., Geisel, T. & Brockmann, D. Natural human mobility patterns and spatial spread of infectious diseases. Phys. Rev. X 1, 011001 (2011).

    Google Scholar 

  91. Cooper, B. S., Pitman, R. J., Edmunds, W. J. & Gay, N. J. Delaying the international spread of pandemic influenza. PLoS Med. 3, e12 (2006).

    Google Scholar 

  92. Hollingsworth, T. D., Ferguson, N. M. & Anderson, R. M. Will travel restrictions control the international spread of pandemic influenza? Nature Med. 12, 497–499 (2006).

    Google Scholar 

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

    ADS  Google Scholar 

  94. Eubank, S. et al. Modelling disease outbreaks in realistic urban social networks. Nature 429, 180–184 (2004).

    ADS  Google Scholar 

  95. Longini, I. M. et al. Containing pandemic infleunza at the source. Science 309, 1083–1087 (2005).

    ADS  Google Scholar 

  96. Ferguson, N. M. et al. Strategies for containing an emerging influenza pandemic in Southeast Asia. Nature 437, 209–211 (2005).

    ADS  Google Scholar 

  97. Colizza, V., Barrat, A., Barthlemy, M., Valleron, M. A. J. & Vespignani, A. Modeling the worldwide spread of pandemic influenza: Baseline case and containment interventions. PLoS Med. 4, e13 (2007).

    Google Scholar 

  98. 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).

    Google Scholar 

  99. Merler, S., Ajelli, M., Pugliese, A. & Ferguson, N. M. Determinants of the spatiotemporal dynamics of the 2009 H1N1 pandemic in Europe: Implications for real-time modelling. PLoS Comput. Biol. 7, e1002205 (2011).

    ADS  Google Scholar 

  100. Gladwell, M. The Tipping Point: How Little Things Can Make a Big Difference (Little, Brown and Company, 2002).

    Google Scholar 

  101. Helbing, D. & Yu, W. The outbreak of cooperation among success-driven individuals under noisy condition. Proc. Natl Acad. Sci. USA 106, 3680–3685 (2009).

    ADS  Google Scholar 

  102. Xie, J. et al. Social consensus through the influence of commited minorities. Phys. Rev. E 84, 011130 (2011).

    ADS  Google Scholar 

  103. Morris, M. & Kretzschmar, M. Concurrent partnerships and the spread of HIV. AIDS 11, 641–648 (1997).

    Google Scholar 

  104. Moody, J. The importance of relationship timing for diffusion: Indirect connectivity and STD infection risk. Soc. Forces 81, 25–56 (2002).

    Google Scholar 

  105. Isella, L. et al. What’s in a crowd? Analysis of face-to-face behavioral networks. J. Theor. Biol. 271, 166–180 (2011).

    MathSciNet  MATH  Google Scholar 

  106. Volz, E. & Meyers, L. A. Epidemic thresholds in dynamic contact networks. J. R. Soc. Interface 6, 233–241 (2009).

    Google Scholar 

  107. Holme, P. & Newman, M. E. J. Nonequilibrium phase transition in the coevolution of networks and opinions. Phys. Rev. E 74, 056108 (2006).

    ADS  Google Scholar 

  108. Centola, D., Gonzalez-Avella, J. C., Eguiluz, V. M. & San Miguel, M. Homophily, cultural drift, and the co-evolution of cultural groups. J. Conflict Resolution 51, 905–929 (2007).

    Google Scholar 

  109. Funk, S., Salathé, M. & Jansen, V. A. A. Modelling the inuence of human behaviour on the spread of infectious diseases: A review. J. R. Soc. Interface 7, 1247–1256 (2010).

    Google Scholar 

  110. Perra, N., Balcan, D., Goncalves, B. & Vespignani, A. Towards a characterization of behavior–disease models. PLoS ONE 6, e23084 (2011).

    ADS  Google Scholar 

  111. Bauch, C. T. & Earn, D. J. Vaccination and the theory of games. Proc. Natl Acad. Sci. USA 101, 13391–13394 (2004).

    ADS  MathSciNet  MATH  Google Scholar 

  112. Liu, Y-Y., Slotine, J-J. & Barabasi, A-L. Controllability of complex networks. Nature 473, 167–173 (2011).

    ADS  Google Scholar 

  113. Conover, M. et al. Proc. 5th International Conference on Weblogs and Social Media (ICWSM) 89–96 (2011).

    Google Scholar 

  114. Ratkiewicz, J. et al. Proc. 20th International Conference Companion on World Wide Web (WWW ’11) 249–252 (ACM, 2001).

    Google Scholar 

  115. Kim, B. J., Yoon, C. N., Han, S. K. & Jeong, H. Path finding strategies in scale-free networks. Phys. Rev. E 65, 027103 (2002).

    ADS  Google Scholar 

  116. Adamic, L. A., Lukose, R. M., Puniyani, A. R. & Huberman, B. A. Search in power-law networks. Phys. Rev. E 64, 046135 (2001).

    ADS  Google Scholar 

  117. Brin, S. & Page, L. The anatomy of a large-scale hypertextual Web search engine. Comput. Netw. ISDN Syst. 30, 107–117 (1998).

    Google Scholar 

  118. Bajardi, P. et al. Human mobility networks, travel restrictions, and the global spread of 2009 H1N1 pandemic. PLoS ONE 6, e16591 (2011).

    ADS  Google Scholar 

Download references

Acknowledgements

I thank B. Goncalves and N. Perra for their help with the figures and a critical reading of the manuscript. This work has been partially funded by the NIH R21-DA024259, DTRA-1-0910039 and NSF CCF-1101743 and NSF CMMI-1125095 awards. The work has been also partly sponsored by the Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-09-2-0053. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the US Government.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Alessandro Vespignani.

Ethics declarations

Competing interests

The author declares no competing financial interests.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Vespignani, A. Modelling dynamical processes in complex socio-technical systems. Nature Phys 8, 32–39 (2012). https://doi.org/10.1038/nphys2160

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nphys2160

Further reading

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