Skip to main content

Exact probabilities for the indeterminacy of complex networks as perceived through press perturbations

Journal of Mathematical Biology volume 76, pages 877–909 (2018)Cite this article

Abstract

We consider the goal of predicting how complex networks respond to chronic (press) perturbations when characterizations of their network topology and interaction strengths are associated with uncertainty. Our primary result is the derivation of exact formulas for the expected number and probability of qualitatively incorrect predictions about a system’s responses under uncertainties drawn form arbitrary distributions of error. Additional indices provide new tools for identifying which links in a network are most qualitatively and quantitatively sensitive to error, and for determining the volume of errors within which predictions will remain qualitatively determinate (i.e. sign insensitive). Together with recent advances in the empirical characterization of uncertainty in networks, these tools bridge a way towards probabilistic predictions of network dynamics.

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

Access options

Buy single article

Instant access to the full article PDF.

39,95 €

Price includes VAT (Finland)

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

References

  • Allesina S, Tang S (2012) Stability criteria for complex ecosystems. Nature 483(7388):205–208

    Article  Google Scholar 

  • Barabás G, Pásztor L, Meszéna G, Ostling A (2014) Sensitivity analysis of coexistence in ecological communities: theory and application. Ecol Lett 17(12):1479–1494

    Article  Google Scholar 

  • Bender EA, Case TJ, Gilpin ME (1984) Perturbation experiments in community ecology: theory and practice. Ecology 65(1):1–13

    Article  Google Scholar 

  • Borrett SR, Patten BC (2003) Structure of pathways in ecological networks: relationships between length and number. Ecol Model 170(2–3):173–184

    Article  Google Scholar 

  • Carey MP, Levin PS, Townsend H, Minello TJ, Sutton GR, Francis TB, Harvey CJ, Toft JE, Arkema KK, Burke JL, Kim C-K, Guerry AD, Plummer M, Spiridonov G, Ruckelshaus M (2014) Characterizing coastal foodwebs with qualitative links to bridge the gap between the theory and the practice of ecosystem-based management. ICES J Mar Sci J Cons 71(3):713–724

    Article  Google Scholar 

  • Cortez MH, Abrams PA (2016) Hydra effects in stable communities and their implications for system dynamics. Ecology 97(5):1135–1145

    Article  Google Scholar 

  • da Fonseca C (2007) On the eigenvalues of some tridiagonal matrices. J Comput Appl Math 200(1):283–286

    MathSciNet  Article  MATH  Google Scholar 

  • Dambacher JM, Li HW, Rossignol PA (2002) Relevance of community structure in assessing indeterminacy of ecological predictions. Ecology 83(5):1372–1385

    Article  Google Scholar 

  • Dambacher JM, Li HW, Rossignol PA (2003) Qualitative predictions in model ecosystems. Ecol Model 161(1–2):79–93

    Article  Google Scholar 

  • Giordano G, Cuba Samaniego C, Franco E, Blanchini F (2016) Computing the structural influence matrix for biological systems. J Math Biol 72(7):1927–1958

    MathSciNet  Article  MATH  Google Scholar 

  • Holt RD (1977) Predation, apparent competition, and the structure of prey communities. Theor Popul Biol 12(2):197–229

    MathSciNet  Article  Google Scholar 

  • Horn R A, Johnson C R (2012) Matrix analysis. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Hosack GR, Hayes KR, Dambacher JM (2008) Assessing model structure uncertainty through an analysis of system feedback and Bayesian networks. Ecol Appl 18(4):1070–1082

    Article  Google Scholar 

  • Iles AC, Novak M (2016) Complexity increases predictability in allometrically constrained food webs. Am Nat 188(1):87–98

    Article  Google Scholar 

  • Ives AR, Carpenter SR (2007) Stability and diversity of ecosystems. Science 317(5834):58–62

    Article  Google Scholar 

  • Jerrum M, Sinclair A, Vigoda E (2004) A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries. JACM 51(4):671–697

    MathSciNet  Article  MATH  Google Scholar 

  • Lawlor LR (1979) Direct and indirect effects of n-species competition. Oecologia 43(3):355–364

    Article  Google Scholar 

  • Levins R (1968) Evolution in changing environments: some theoretical explorations. In: Levin SA, Horn HS (eds) Monographs in population biology. Princeton University Press, Princeton

  • Levins R (1974) The qualitative analysis of partially specified systems. Ann N Y Acad Sci 231:123–138

    Article  MATH  Google Scholar 

  • Lewis JW (1982) Inversion of tridiagonal matrices. Numer Math 38(3):333–345

    MathSciNet  Article  MATH  Google Scholar 

  • Marzloff MP, Melbourne-Thomas J, Hamon KG, Hoshino E, Jennings S, van Putten IE, Pecl GT (2016) Modelling marine community responses to climate-driven species redistribution to guide monitoring and adaptive ecosystem-based management. Glob Change Biol 22(7):2462–2474

    Article  Google Scholar 

  • Melbourne-Thomas J, Wotherspoon S, Raymond B, Constable A (2012) Comprehensive evaluation of model uncertainty in qualitative network analyses. Ecol Monogr 82(4):505–519

    Article  Google Scholar 

  • Miller KS (1981) On the inverse of the sum of matrices. Math Mag 54(2):67–72

    MathSciNet  Article  MATH  Google Scholar 

  • Novak M, Wootton JT, Doak DF, Emmerson M, Estes JA, Tinker MT (2011) Predicting community responses to perturbations in the face of imperfect knowledge and network complexity. Ecology 92(4):836–846

    Article  Google Scholar 

  • Novak M, Yeakel J, Noble AE, Doak DF, Emmerson M, Estes JA, Jacob U, Tinker MT, Wootton JT (2016) Characterizing species interactions to understand press perturbations: what is the community matrix? Annu Rev Ecol Evol Syst 47:409–432

    Article  Google Scholar 

  • Petchey OL, Pontarp M, Massie TM, Kéfi S, Ozgul A, Weilenmann M, Palamara GM, Altermatt F, Matthews B, Levine JM, Childs DZ, McGill BJ, Schaepman ME, Schmid B, Spaak P, Beckerman AP, Pennekamp F, Pearse IS (2015) The ecological forecast horizon, and examples of its uses and determinants. Ecol Lett 18(7):597–611

    Article  Google Scholar 

  • Poisot T, Cirtwill AR, Cazelles K, Gravel D, Fortin M-J, Stouffer DB (2016) The structure of probabilistic networks. Methods Ecol Evol 7(3):303–312

    Article  Google Scholar 

  • Prasolov VV (1994) Problems and theorems in linear algebra, vol 134. American Mathematical Socity, Providence

    MATH  Google Scholar 

  • Raymond B, McInnes J, Dambacher JM, Way S, Bergstrom DM (2011) Qualitative modelling of invasive species eradication on subantarctic Macquarie Island. J Appl Ecol 48(1):181–191

    Article  Google Scholar 

  • Rohr RP, Saavedra S, Bascompte J (2014) On the structural stability of mutualistic systems. Science 345(6195):1253497

    Article  Google Scholar 

  • Scheffer M, Carpenter SR, Lenton TM, Bascompte J, Brock W, Dakos V, van de Koppel J, van de Leemput IA, Levin SA, van Nes EH, Pascual M, Vandermeer J (2012) Anticipating critical transitions. Science 338(6105):344–348

    Article  Google Scholar 

  • Stouffer DB, Camacho J, Jiang W, Amaral LAN (2007) Evidence for the existence of a robust pattern of prey selection in food webs. Proc R Soc B Biol Sci 274(1621):1931–1940

    Article  Google Scholar 

  • Strogatz SH (2001) Exploring complex networks. Nature 410(6825):268–276

    Article  MATH  Google Scholar 

  • Takimoto G, Miki T, Kagami M (2007) Intraguild predation promotes complex alternative states along a productivity gradient. Theor Popul Biol 72(2):264–273

    Article  MATH  Google Scholar 

  • Travis J, Coleman FC, Auster PJ, Cury PM, Estes JA, Orensanz J, Peterson CH, Power ME, Steneck RS, Wootton JT (2014) Integrating the invisible fabric of nature into fisheries management. Proc Nat Acad Sci 111(2):581–584

    Article  Google Scholar 

  • Usmani R (1994) Inversion of Jacobi’s tridiagonal matrix. Comput Math Appl 27(8):59–66

    MathSciNet  Article  MATH  Google Scholar 

  • Usmani RA (1994) Inversion of a tridiagonal Jacobi matrix. Linear Algebra Appl 212:413–414

    MathSciNet  Article  MATH  Google Scholar 

  • Wolf C, Novak M, Gitelman AI (2015) Bayesian characterization of uncertainty in species interaction strengths. PeerJ Preprints 3:e1717

    Google Scholar 

  • Wootton JT, Emmerson M (2005) Measurement of interaction strength in nature. Annu Rev Ecol Evol Syst 36(1):419–444

    Article  Google Scholar 

  • Yodzis P (1988) The indeterminacy of ecological interactions as perceived through perturbation experiments. Ecology 69(2):508–515

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Department, Oregon State University, Corvallis, OR, USA

    David Koslicki

  2. Department of Untegrative Biology, Oregon State University, Corvallis, OR, USA

    Mark Novak

Authors
  1. David Koslicki
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mark Novak
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David Koslicki.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Koslicki, D., Novak, M. Exact probabilities for the indeterminacy of complex networks as perceived through press perturbations. J. Math. Biol. 76, 877–909 (2018). https://doi.org/10.1007/s00285-017-1163-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00285-017-1163-0

Keywords

  • Press perturbations
  • Net effects
  • Loop analysis
  • Sign sensitivity
  • Qualitative indeterminacy
  • Ecosystem-based management
  • Community matrix
  • Intraguild predation
  • Trophic chain
  • Sherman–Morrison
  • Matrix perturbation
  • Inverse
  • Sign pattern

Mathematics Subject Classification

  • Primary 92B05
  • Secondary 15A09
  • 92D25