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


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.

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Data that support the findings of this study are publicly available at the Interaction Web Database at


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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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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).

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