The stability–complexity relationship at age 40: a random matrix perspective
Since the work of Robert May in 1972, the local asymptotic stability of large ecological systems has been a focus of theoretical ecology. Here we review May’s work in the light of random matrix theory, the field of mathematics devoted to the study of large matrices whose coefficients are randomly sampled from distributions with given characteristics. We show how May’s celebrated “stability criterion” can be derived using random matrix theory, and how extensions of the so-called circular law for the limiting distribution of the eigenvalues of large random matrix can further our understanding of ecological systems. Our goal is to present the more technical material in an accessible way, and to provide pointers to the primary mathematical literature on this subject. We conclude by enumerating a number of challenges, whose solution is going to greatly improve our ability to predict the stability of large ecological networks.
KeywordsComplexity Eigenvalue Food web Random matrix Stability
SA and ST funded by NSF #1148867. Thanks to G. Barabás for comments. D. Gravel and an anonymous reviewer provided valuable suggestions.
- Anderson GW, Guionnet A, Zeitouni O (2010) An introduction to random matrices. Cambridge University Press, CambridgeGoogle Scholar
- Backstrom L, Boldi P, Rosa M, Ugander J, Vigna S (2012) Four degrees of separation. In: Proceedings of the 3rd annual ACM web science conference. ACM, New York, pp 33–42Google Scholar
- Bai Z, Silverstein JW (2009) Spectral analysis of large dimensional random matrices. Springer, New YorkGoogle Scholar
- Hiai F, Petz D (2000) The semicircle law, free random variables and entropy, vol 77. American Mathematical Society, ProvidenceGoogle Scholar
- Levins R (1968) Evolution in changing environments: some theoretical explorations. Princeton University Press, PrincetonGoogle Scholar
- May RM (2001) Stability and complexity in model ecosystems. Princeton University Press, PrincetonGoogle Scholar
- Metha M (1967) Random matrices and the statistical theory of energy levels. Academic, New YorkGoogle Scholar
- Naumov A (2012) Elliptic law for real random matrices. arXiv:1201.1639
- Nguyen H, O’Rourke S (2012) The elliptic law. arXiv:1208.5883
- Sinha S, Sinha S (2005) Evidence of universality for the May–Wigner stability theorem for random networks with local dynamics. Phys Rev E 71(020):902Google Scholar
- Van Mieghem P, Cator E (2012) Epidemics in networks with nodal self-infection and the epidemic threshold. Phys Rev E 86(016):116Google Scholar
- Wang Y, Chakrabarti D, Wang C, Faloutsos C (2003) Epidemic spreading in real networks: an eigenvalue viewpoint. In: Proceedings of 22nd international symposium on reliable distributed systems. IEEE, New York, pp 25–34Google Scholar