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The k-core as a predictor of structural collapse in mutualistic ecosystems

Nature Physics volume 15, pages 95–102 (2019)Cite this article

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Abstract

Collapses of dynamical systems into irrecoverable states are observed in ecosystems, human societies, financial systems and network infrastructures. Despite their widespread occurrence and impact, these events remain largely unpredictable. In searching for the causes for collapse and instability, theoretical investigations have so far been unable to determine quantitatively the influence of the structural features of the network formed by the interacting species. Here, we derive the condition for the stability of a mutualistic ecosystem as a constraint on the strength of the dynamical interactions between species and a topological invariant of the network: the k-core. Our solution predicts that when species located at the maximum k-core of the network go extinct, as a consequence of sufficiently weak interaction strengths, the system will reach the tipping point of its collapse. As a key variable involved in collapse phenomena, monitoring the k-core of the network may prove a powerful method to anticipate catastrophic events in the vast context that stretches from ecological and biological networks to finance.

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Fig. 1: k-core structure of a mutualistic network.
Fig. 2: Numerical solution to the fixed-point equations in weighted and directed networks.
Fig. 3: Solution scheme for the fixed point equations (4).
Fig. 4: Collapse of a plant–pollinator mutualistic network and the tipping line of the mutualistic ecosystem.
Fig. 5: Phase diagram of ecosystem stability.

Data availability

Data that support the findings of this study are publicly available at the Interaction Web Database at https://www.nceas.ucsb.edu/interactionweb/.

References

  1. May, R. M. Will a large complex system be stable? Nature 238, 413–414 (1972).

    Article  ADS  Google Scholar 

  2. Strogatz, S. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Westview Press, 2000).

  3. Caldarelli, G. & Vespignani, A. Large Scale Structure and Dynamics of Complex Networks: From Information Technology to Finance and Natural Science (World Scientific, Singapore, 2007).

    Book  Google Scholar 

  4. Buldyrev, S. V., Parshani, R., Paul, G., Stanley, H. E. & Havlin, S. Catastrophic cascade of failures in interdependent networks. Nature 464, 1025–1028 (2010).

    Article  ADS  Google Scholar 

  5. Scheffer, M. et al. Early-warning signals for critical transitions. Nature 461, 53–59 (2009).

    Article  ADS  Google Scholar 

  6. Scheffer, M. et al. Anticipating critical transitions. Science 338, 344–348 (2012).

    Article  ADS  Google Scholar 

  7. Allesina, S. & Tang, S. Stability criteria for complex ecosystems. Nature 483, 205–208 (2012).

    Article  ADS  Google Scholar 

  8. Gao, J., Barzel, B. & Barabási, A.-L. Universal resilience patterns in complex networks. Nature 530, 307–312 (2016).

    Article  ADS  Google Scholar 

  9. Bascompte, J., Jordano, P. & Olesen, J. M. Asymmetric coevolutionary networks facilitate biodiversity maintenance. Science 312, 431–433 (2006).

    Article  ADS  Google Scholar 

  10. Coyte, K. Z., Schluter, J. & Foster, K. R. The ecology of the microbiome: networks, competition, and stability. Science 350, 663–666 (2015).

    Article  ADS  Google Scholar 

  11. Haldane, A. G. & May, R. M. Systemic risk in banking ecosystems. Nature 469, 351–355 (2011).

    Article  ADS  Google Scholar 

  12. Battiston, S., Caldarelli, G., Georg, C.-P., May, R. M. & Stiglitz, J. Complex derivatives. Nat. Phys. 9, 123–125 (2013).

    Article  Google Scholar 

  13. Dai, L., Vorselen, D., Korolev, K. S. & Gore, J. Generic indicators for loss of resilience before a tipping point leading to population collapse. Science 336, 1175–1177 (2012).

    Article  ADS  Google Scholar 

  14. Seidman, S. B. Network structure and minimum degree. Soc. Networks 5, 269–287 (1983).

    Article  MathSciNet  Google Scholar 

  15. Pittel, B., Spencer, J. & Wormald, N. Sudden emergence of a giant k-core in a random graph. J. Comb. Theory B 67, 111–151 (1996).

    Article  MathSciNet  Google Scholar 

  16. Dorogovtsev, S. N., Goltsev, A. V. & Mendes, J. F. F. k-core organization of complex networks. Phys. Rev. Lett. 96, 040601 (2006).

    Article  ADS  Google Scholar 

  17. Carmi, S., Havlin, S., Kirkpatrick, S., Shavitt, Y. & Shir, E. A model of Internet topology using k-shell decomposition. Proc. Natl Acad. Sci. USA 104, 11150–11154 (2007).

    Article  ADS  Google Scholar 

  18. Alvarez-Hamelin, J. I., Dall’Asta, L., Barrat, A. & Vespignani, A. k-core decomposition of the Internet graphs: hierarchies, self-similarity and measurement biases. Netw. Heterog. Media 3, 371 (2008).

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  20. Hagmann, P. et al. Mapping the structural core of human cerebral cortex. PLoS Biol. 6, e159 (2008).

    Article  Google Scholar 

  21. Morone, F., Burleson-Lesser, K., Vinutha, H. A., Sastry, S. & Makse, H. A. The jamming transition is a k-core percolation transition. Preprint at https://arXiv.org/abs/1804.07804 (2018).

  22. May, R. M. Mutualistic interactions among species. Nature 296, 803–804 (1982).

    Article  ADS  Google Scholar 

  23. Holland, J. N., DeAngelis, D. L. & Bronstein, J. L. Population dynamics and mutualism: functional responses of benefits and costs. Am. Nat. 159, 231–244 (2002).

    Article  Google Scholar 

  24. Bastolla, U. et al. The architecture of mutualistic networks minimizes competition and increases biodiversity. Nature 458, 1018–1020 (2009).

    Article  ADS  Google Scholar 

  25. Thebault, E. & Fontaine, C. Stability of ecological communities and the architecture of mutualistic and trophic networks. Science 329, 853–856 (2010).

    Article  ADS  Google Scholar 

  26. Alon, U. An Introduction to Systems Biology: Design Principles of Biological Circuits (CRC Press, Boca Raton, 2006).

  27. Amit, D. J. Modeling Brain Function: The World of Attractor Neural Networks (Cambridge Univ. Press, Cambridge, 1989).

  28. Sompolinsky, H., Crisanti, A. & Sommers, H. J. Chaos in random neural networks. Phys. Rev. Lett. 61, 259–262 (1988).

    Article  ADS  MathSciNet  Google Scholar 

  29. Okuyama, T. & Holland, J. N. Network structural properties mediate the stability of mutualistic communities. Ecol. Lett. 11, 208–216 (2008).

    Article  Google Scholar 

  30. Arroyo, M. T. K., Primack, R. B. & Armesto, J. J. Community studies in pollination ecology in the high temperate Andes of Central Chile. I. Pollination mechanisms and altitudinal variation. Amer. J. Bot. 69, 82–97 (1982).

    Article  Google Scholar 

  31. May, R. M. Simple mathematical models with very complicated dynamics. Nature 261, 459–467 (1976).

    Article  ADS  Google Scholar 

  32. Shen-Orr, S. S., Milo, R., Mangan, S. & Alon, U. Network motifs in the transcriptional regulation network of Escherichia coli. Nat. Genet. 31, 64–68 (2002).

    Article  Google Scholar 

  33. Kauffman, S. A. The Origins of Order: Self-organization and Selection in Evolution (Oxford Univ. Press, New York, 1993).

    Google Scholar 

  34. Glass, L. & Kauffman, S. A. The logical analysis of continuous, non-linear biochemical control networks. J. Theor. Biol. 38, 103–129 (1973).

    Article  Google Scholar 

  35. Rohr, R. P., Saavedra, S. & Bascompte, J. On the structural stability of mutualistic systems. Science 345, 1253497 (2014).

    Article  Google Scholar 

  36. Bascompte, J., Jordano, P., Melián, C. J. & Olesen, J. M. The nested assembly of plant-animal mutualistic networks. Proc. Natl Acad. Sci. USA 100, 9383–9387 (2003).

    Article  ADS  Google Scholar 

  37. Staniczenko, P. P. A., Kopp, J. C. & Allesina, S. The ghost of nestedness in ecological networks. Nat. Commun. 4, 1931 (2013).

    Article  Google Scholar 

  38. Bickle, A. Cores and shells of graphs. Math. Bohem. 138, 43–59 (2013).

    MathSciNet  MATH  Google Scholar 

  39. Berry, D. & Widder, S. Deciphering microbial interactions and detecting keystone species with co-occurrence networks. Front. Microbiol. 5, 00219 (2014).

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  41. Bucci, V., Bradde, S., Biroli, G. & Xavier, J. B. Social interaction, noise and antibiotic-mediated switches in the intestinal microbiota. PLoS. Comput. Biol. 8, e1002497 (2012).

    Article  ADS  MathSciNet  Google Scholar 

  42. Kato, M., Makutani, T., Inoue, T. & Itino, T. Insect–flower relationship in the primary beech forest of Ashu, Kyoto: an overview of the flowering phenology and seasonal pattern of insect visits. Contr. Biol. Lab. Kyoto Univ. 27, 309–375 (1990).

    Google Scholar 

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Acknowledgements

Research was sponsored by NSF-IIS 1515022, NIH-NIBIB R01EB022720, NIH-NCI U54CA137788/U54CA132378 and Army Research Laboratory under Cooperative Agreement W911NF-09-2-0053 (ARL Network Science CTA). We are grateful to S. Alarcón for discussions.

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  1. Levich Institute and Physics Department, City College of New York, New York, NY, USA

    Flaviano Morone, Gino Del Ferraro & Hernán A. Makse

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  1. Flaviano Morone
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  2. Gino Del Ferraro
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All authors contributed equally to all parts of the study.

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Correspondence to Hernán A. Makse.

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Morone, F., Del Ferraro, G. & Makse, H.A. The k-core as a predictor of structural collapse in mutualistic ecosystems. Nature Phys 15, 95–102 (2019). https://doi.org/10.1038/s41567-018-0304-8

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