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.
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References
Allesina S, Tang S (2012) Stability criteria for complex ecosystems. Nature 483(7388):205–208
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
Bender EA, Case TJ, Gilpin ME (1984) Perturbation experiments in community ecology: theory and practice. Ecology 65(1):1–13
Borrett SR, Patten BC (2003) Structure of pathways in ecological networks: relationships between length and number. Ecol Model 170(2–3):173–184
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
Cortez MH, Abrams PA (2016) Hydra effects in stable communities and their implications for system dynamics. Ecology 97(5):1135–1145
da Fonseca C (2007) On the eigenvalues of some tridiagonal matrices. J Comput Appl Math 200(1):283–286
Dambacher JM, Li HW, Rossignol PA (2002) Relevance of community structure in assessing indeterminacy of ecological predictions. Ecology 83(5):1372–1385
Dambacher JM, Li HW, Rossignol PA (2003) Qualitative predictions in model ecosystems. Ecol Model 161(1–2):79–93
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
Holt RD (1977) Predation, apparent competition, and the structure of prey communities. Theor Popul Biol 12(2):197–229
Horn R A, Johnson C R (2012) Matrix analysis. Cambridge University Press, Cambridge
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
Iles AC, Novak M (2016) Complexity increases predictability in allometrically constrained food webs. Am Nat 188(1):87–98
Ives AR, Carpenter SR (2007) Stability and diversity of ecosystems. Science 317(5834):58–62
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
Lawlor LR (1979) Direct and indirect effects of n-species competition. Oecologia 43(3):355–364
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
Lewis JW (1982) Inversion of tridiagonal matrices. Numer Math 38(3):333–345
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
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
Miller KS (1981) On the inverse of the sum of matrices. Math Mag 54(2):67–72
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
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
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
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
Prasolov VV (1994) Problems and theorems in linear algebra, vol 134. American Mathematical Socity, Providence
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
Rohr RP, Saavedra S, Bascompte J (2014) On the structural stability of mutualistic systems. Science 345(6195):1253497
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
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
Strogatz SH (2001) Exploring complex networks. Nature 410(6825):268–276
Takimoto G, Miki T, Kagami M (2007) Intraguild predation promotes complex alternative states along a productivity gradient. Theor Popul Biol 72(2):264–273
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
Usmani R (1994) Inversion of Jacobi’s tridiagonal matrix. Comput Math Appl 27(8):59–66
Usmani RA (1994) Inversion of a tridiagonal Jacobi matrix. Linear Algebra Appl 212:413–414
Wolf C, Novak M, Gitelman AI (2015) Bayesian characterization of uncertainty in species interaction strengths. PeerJ Preprints 3:e1717
Wootton JT, Emmerson M (2005) Measurement of interaction strength in nature. Annu Rev Ecol Evol Syst 36(1):419–444
Yodzis P (1988) The indeterminacy of ecological interactions as perceived through perturbation experiments. Ecology 69(2):508–515
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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
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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