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.

  • Article
  • Published:

Reaction–diffusion processes and metapopulation models in heterogeneous networks

Nature Physics volume 3, pages 276–282 (2007)Cite this article

Abstract

Dynamical reaction–diffusion processes and metapopulation models are standard modelling approaches for a wide array of phenomena in which local quantities—such as density, potentials and particles—diffuse and interact according to the physical laws. Here, we study the behaviour of the basic reaction–diffusion process (given by the reaction steps B→A and B+A→2B) defined on networks with heterogeneous topology and no limit on the nodes’ occupation number. We investigate the effect of network topology on the basic properties of the system’s phase diagram and find that the network heterogeneity sustains the reaction activity even in the limit of a vanishing density of particles, eventually suppressing the critical point in density-driven phase transitions, whereas phase transition and critical points independent of the particle density are not altered by topological fluctuations. This work lays out a theoretical and computational microscopic framework for the study of a wide range of realistic metapopulation and agent-based models that include the complex features of real-world networks.

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

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

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

Figure 1: Bosonic RD systems in heterogeneous networks.
Figure 2: Phase diagram and stationary densities for type-I processes.
Figure 3: Phase diagram and stationary densities for type-II processes.
Figure 4: Reaction activity in type-I and type-II processes: microscopic model and real-world examples.

Similar content being viewed by others

References

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

    Book  Google Scholar 

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

    MATH  Google Scholar 

  3. Murray, J. D. Mathematical Biology 3rd edn (Springer, Berlin, 2005).

    MATH  Google Scholar 

  4. Anderson, R. M. & May, R. M. Infectious Diseases of Humans: Dynamics and Control (Oxford Univ. Press, Oxford, 1992).

    Google Scholar 

  5. Anderson, R. M. & May, R. M. Spatial heterogeneity and the design of immunization programs. Math Biosci. 72, 83–111 (1984).

    Article  MathSciNet  Google Scholar 

  6. Bolker, B. M. & Grenfell, B. T. Space persistence and dynamics of measles epidemics. Phil. Trans. R. Soc. Lond. B 348, 309–320 (1995).

    Article  ADS  Google Scholar 

  7. Lloyd, A. L. & May, R. M. Spatial heterogeneity in epidemic models. J. Theor. Biol. 179, 1–11 (1996).

    Article  Google Scholar 

  8. Grenfell, B. T. & Bolker, B. M. Cities and villages: Infection hierarchies in a measles meta-population. Ecol. Lett. 1, 63–70 (1998).

    Article  Google Scholar 

  9. Keeling, M. J. & Rohani, P. Estimating spatial coupling in epidemiological systems: A mechanistic approach. Ecol. Lett. 5, 20–29 (1995).

    Article  Google Scholar 

  10. Ferguson, N. M. et al. Planning for smallpox outbreaks. Nature 425, 681–685 (2003).

    Article  ADS  Google Scholar 

  11. Liljeros, F., Edling, C. R., Amaral, L. A. N., Stanley, H. E. & Åberg, Y. The web of human sexual contacts. Nature 411, 907–908 (2001).

    Article  ADS  Google Scholar 

  12. Schneeberger, A. et al. Scale-free networks and sexually transmitted diseases: A description of observed patterns of sexual contacts in Britain and Zimbabwe. Sex. Transm. Dis. 31, 380–387 (2004).

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  14. Guimerà, R., Mossa, S., Turtschi, A. & Amaral, L. A. N. The worldwide air transportation network: Anomalous centrality, community structure, and cities’ global roles. Proc. Natl Acad. Sci. USA 102, 7794–7799 (2005).

    Article  ADS  MathSciNet  Google Scholar 

  15. Chowell, G., Hyman, J. M., Eubank, S. & Castillo-Chavez, C. Scaling laws for the movement of people between locations in a large city. Phys. Rev. E 68, 066102 (2003).

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  17. Newman, M. E. J. The structure and function of complex networks. SIAM Rev. 45, 167–256 (2003).

    Article  ADS  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

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

    Book  Google Scholar 

  20. Barrett, C. L. et al. TRANSIMS: Transportation Analysis Simulation System (Technical Report LA-UR-00-1725, Los Alamos National Laboratory, 2000).

  21. Cohen, R., Erez, K., ben-Avraham, D. & Havlin, S. Resilience of the Internet to random breakdowns. Phys. Rev. Lett. 85, 4626–4628 (2000).

    Article  ADS  Google Scholar 

  22. Callaway, D. S., Newman, M. E. J., Strogatz, S., H. & Watts, D. J. Network robustness and fragility: Percolation on random graphs. Phys. Rev. Lett. 85, 5468–5471 (2000).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  24. Cardy, J. L. & Grassberger, P. Epidemics models and percolation. J. Phys. A 18, L267–L271 (1985).

    Article  ADS  MathSciNet  Google Scholar 

  25. de Freitas, J. E., Lucena, L. S., da Silva, L. R. & Hilhorst, H. J. Critical behavior of a two-species reaction-diffusion problem. Phys. Rev. E 61, 6330–6336 (2000).

    Article  ADS  Google Scholar 

  26. Pastor-Satorras, R. & Vespignani, A. Field theory of absorbing phase transitions with a nondiffusive conserved field. Phys. Rev. E 62, R5875–R5878 (2000).

    Article  ADS  Google Scholar 

  27. Kree, R., Schaub, B. & Schmittman, B. Effects of pollution on critical population dynamics. Phys. Rev. A 39, 2214–2221 (1989).

    Article  ADS  Google Scholar 

  28. van Wijland, F., Oerding, K. & Hilhorst, H. J. Wilson renormalization of a reaction-diffusion process. Physica A 251, 179–201 (1998).

    Article  ADS  Google Scholar 

  29. Rvachev, L. A. & Longini, I. M. A mathematical model for the global spread of influenza. Math. Biosci. 75, 3–22 (1985).

    Article  MathSciNet  Google Scholar 

  30. 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 

  31. Colizza, V., Barrat, A., Barthélemy, 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 

  32. Pastor-Satorras, R., Vázquez, A. & Vespignani, A. Dynamical and correlation properties of the internet. Phys. Rev. Lett. 87, 258701 (2001).

    Article  ADS  Google Scholar 

  33. Newman, M. E. J. Assortative mixing in networks. Phys. Rev. Lett. 89, 208701 (2002).

    Article  ADS  Google Scholar 

  34. Catanzaro, M., Boguna, M. & Pastor-Satorras, R. Generation of uncorrelated random scale-free networks. Phys. Rev. E 71, 027103 (2005).

    Article  ADS  Google Scholar 

  35. Molloy, M. & Reed, B. A critical point for random graphs with a given degree sequence. Random Struct. Algorithms 6, 161–179 (1995).

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

A.V. is partially supported by the NSF award IIS-0513650. R.P.-S. acknowledges financial support from the Spanish MEC (FEDER), under project No. FIS2004-05923-C02-01 and additional support from the DURSI, Generalitat de Catalunya (Spain).

Author information

Authors and Affiliations

  1. Complex Networks Lagrange Laboratory, Institute for Scientific Interchange (ISI), Torino 10133, Italy

    Vittoria Colizza & Alessandro Vespignani

  2. School of Informatics and Department of Physics, Indiana University, Bloomington 47406 Indiana, USA

    Vittoria Colizza & Alessandro Vespignani

  3. Departament de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya, Campus Nord B4, 08034 Barcelona, Spain

    Romualdo Pastor-Satorras

Authors
  1. Vittoria Colizza
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Romualdo Pastor-Satorras
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alessandro Vespignani
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

V.C., R.P.-S. and A.V. designed the study, analysed the data and contributed to writing the paper.

Corresponding authors

Correspondence to Vittoria Colizza or Alessandro Vespignani.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Rights and permissions

Reprints and permissions

About this article

Cite this article

Colizza, V., Pastor-Satorras, R. & Vespignani, A. Reaction–diffusion processes and metapopulation models in heterogeneous networks. Nature Phys 3, 276–282 (2007). https://doi.org/10.1038/nphys560

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nphys560

This article is cited by

Search

Advanced 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