In this study, we investigate how specific micro-interaction structures (motifs) affect the occurrence of tipping cascades on networks of stylized tipping elements. We compare the properties of cascades in Erdős–Rényi networks and an exemplary moisture recycling network of the Amazon rainforest. Within these networks, decisive small-scale motifs are the feed forward loop, the secondary feed forward loop, the zero loop, and the neighboring loop. Of all motifs, the feed forward loop motif stands out in tipping cascades since it decreases the critical coupling strength necessary to initiate a cascade more than the other motifs. We find that for this motif, the reduction of critical coupling strength is 11% less than the critical coupling of a pair of tipping elements. For highly connected networks, our analysis reveals that coupled feed forward loops coincide with a strong 90% decrease in the critical coupling strength. For the highly clustered moisture recycling network in the Amazon, we observe regions of a very high motif occurrence for each of the four investigated motifs, suggesting that these regions are more vulnerable. The occurrence of motifs is found to be one order of magnitude higher than in a random Erdős–Rényi network. This emphasizes the importance of local interaction structures for the emergence of global cascades and the stability of the network as a whole.
REFERENCES
1.
M. E.
Newman
, “The structure and function of complex networks
,” SIAM Rev.
45
, 167
–256
(2003
). 2.
W.
Zou
, D.
Senthilkumar
, M.
Zhan
, and J.
Kurths
, “Reviving oscillations in coupled nonlinear oscillators
,” Phys. Rev. Lett.
111
, 014101
(2013
). 3.
T.
Gross
, L.
Rudolf
, S. A.
Levin
, and U.
Dieckmann
, “Generalized models reveal stabilizing factors in food webs
,” Science
325
, 747
–750
(2009
). 4.
J.
Nitzbon
, P.
Schultz
, J.
Heitzig
, J.
Kurths
, and F.
Hellmann
, “Deciphering the imprint of topology on nonlinear dynamical network stability
,” New J. Phys.
19
, 033029
(2017
). 5.
T. M.
Lenton
, H.
Held
, E.
Kriegler
, J. W.
Hall
, W.
Lucht
, S.
Rahmstorf
, and H. J.
Schellnhuber
, “Tipping elements in the Earth’s climate system
,” Proc. Natl. Acad. Sci. U.S.A.
105
, 1786
–1793
(2008
). 6.
C. D.
Brummitt
, G.
Barnett
, and R. M.
D’Souza
, “Coupled catastrophes: Sudden shifts cascade and hop among interdependent systems
,” J. R. Soc. Interface
12
, 20150712
(2015
). 7.
E.
Kriegler
, J. W.
Hall
, H.
Held
, R.
Dawson
, and H. J.
Schellnhuber
, “Imprecise probability assessment of tipping points in the climate system
,” Proc. Natl. Acad. Sci. U.S.A.
106
, 5041
–5046
(2009
). 8.
Y.
Cai
, T. M.
Lenton
, and T. S.
Lontzek
, “Risk of multiple interacting tipping points should encourage rapid CO emission reduction
,” Nat. Clim. Change
6
, 520
–525
(2016
). 9.
J. C.
Rocha
, G.
Peterson
, Ö.
Bodin
, and S.
Levin
, “Cascading regime shifts within and across scales
,” Science
362
, 1379
–1383
(2018
). 10.
C.
Gaucherel
and V.
Moron
, “Potential stabilizing points to mitigate tipping point interactions in Earth’s climate
,” Int. J. Climatol.
37
, 399
–408
(2017
). 11.
M. M.
Dekker
, A. S.
Heydt
, and H. A.
Dijkstra
, “Cascading transitions in the climate system
,” Earth Syst. Dyn.
9
, 1243
–1260
(2018
). 12.
A. K.
Klose
, V.
Karle
, R.
Winkelmann
, and J. F.
Donges
, “Dynamic emergence of domino effects in systems of interacting tipping elements in ecology and climate,” arXiv:1910.12042 (2019).13.
W.
Steffen
, J.
Rockström
, K.
Richardson
, T. M.
Lenton
, C.
Folke
, D.
Liverman
, C. P.
Summerhayes
, A. D.
Barnosky
, S. E.
Cornell
, M.
Crucifix
et al., “Trajectories of the Earth System in the Anthropocene
,” Proc. Natl. Acad. Sci. U.S.A.
115
, 8252
–8259
(2018
). 14.
J.
Krönke
, N.
Wunderling
, R.
Winkelmann
, A.
Staal
, B.
Stumpf
, O. A.
Tuinenburg
, and J. F.
Donges
, “Dynamics of tipping cascades on complex networks
,” Phys. Rev. E
(to be published).15.
Y.-H.
Eom
, “Resilience of networks to environmental stress: From regular to random networks
,” Phys. Rev. E
97
, 042313
(2018
). 16.
D. J.
Watts
, “A simple model of global cascades on random networks
,” Proc. Natl. Acad. Sci. U.S.A.
99
, 5766
–5771
(2002
). 17.
S. V.
Buldyrev
, R.
Parshani
, G.
Paul
, H. E.
Stanley
, and S.
Havlin
, “Catastrophic cascade of failures in interdependent networks
,” Nature
464
, 1025
(2010
). 18.
M.
Turalska
, K.
Burghardt
, M.
Rohden
, A.
Swami
, and R. M.
D’Souza
, “Cascading failures in scale-free interdependent networks
,” Phys. Rev. E
99
, 032308
(2019
). 19.
A.
Loppini
, S.
Filippi
, and H. E.
Stanley
, “Critical transitions in heterogeneous networks: Loss of low-degree nodes as an early warning signal
,” Phys. Rev. E
99
, 040301
(2019
). 20.
X.-Z.
Wu
, P. G.
Fennell
, A. G.
Percus
, K.
Lerman
et al., “Degree correlations amplify the growth of cascades in networks
,” Phys. Rev. E
98
, 022321
(2018
). 21.
X.
Liu
, L.
Pan
, H. E.
Stanley
, and J.
Gao
, “Multiple phase transitions in networks of directed networks
,” Phys. Rev. E
99
, 012312
(2019
). 22.
S.
Krishnagopal
, J.
Lehnert
, W.
Poel
, A.
Zakharova
, and E.
Schöll
, “Synchronization patterns: From network motifs to hierarchical networks
,” Philos. Trans. R. Soc. A
375
, 20160216
(2017
). 23.
O.
D’Huys
, R.
Vicente
, T.
Erneux
, J.
Danckaert
, and I.
Fischer
, “Synchronization properties of network motifs: Influence of coupling delay and symmetry
,” Chaos
18
, 037116
(2008
). 24.
L. V.
Gambuzza
, J.
Gómez-Gardeñes
, and M.
Frasca
, “Amplitude dynamics favors synchronization in complex networks
,” Sci. Rep.
6
, 24915
(2016
). 25.
R.
Milo
, S.
Shen-Orr
, S.
Itzkovitz
, N.
Kashtan
, D.
Chklovskii
, and U.
Alon
, “Network motifs: Simple building blocks of complex networks
,” Science
298
, 824
–827
(2002
). 26.
D. B.
Stouffer
, M.
Sales-Pardo
, M. I.
Sirer
, and J.
Bascompte
, “Evolutionary conservation of species’ roles in food webs
,” Science
335
, 1489
–1492
(2012
). 27.
V. Q.
Marinho
, G.
Hirst
, and D. R.
Amancio
, “Authorship attribution via network motifs identification,” in 2016 5th Brazilian Conference on Intelligent Systems (BRACIS) (IEEE, 2016), pp. 355–360.28.
U.
Alon
, “Network motifs: Theory and experimental approaches
,” Nat. Rev. Genet.
8
, 450
(2007
). 29.
G.
Lahav
, N.
Rosenfeld
, A.
Sigal
, N.
Geva-Zatorsky
, A. J.
Levine
, M. B.
Elowitz
, and U.
Alon
, “Dynamics of the p53-Mdm2 feedback loop in individual cells
,” Nat. Genet.
36
, 147
(2004
). 30.
E.
Anastasiadou
, L. S.
Jacob
, and F. J.
Slack
, “Non-coding RNA networks in cancer
,” Nat. Rev. Cancer
18
, 5
(2018
). 31.
S. S.
Shen-Orr
, R.
Milo
, S.
Mangan
, and U.
Alon
, “Network motifs in the transcriptional regulation network of Escherichia coli
,” Nat. Genet.
31
, 64
(2002
). 32.
S.
Mangan
, S.
Itzkovitz
, A.
Zaslaver
, and U.
Alon
, “The incoherent feed-forward loop accelerates the response-time of the gal system of Escherichia coli
,” J. Mol. Biol.
356
, 1073
–1081
(2006
). 33.
T. E.
Gorochowski
, C. S.
Grierson
, and M.
di Bernardo
, “Organization of feed-forward loop motifs reveals architectural principles in natural and engineered networks
,” Sci. Adv.
4
, eaap9751
(2018
). 34.
E. H.
van Nes
, W. J.
Rip
, and M.
Scheffer
, “A theory for cyclic shifts between alternative states in shallow lakes
,” Ecosystems
10
, 17
(2007
). 35.
M.
Scheffer
and E.
Jeppesen
, “Regime shifts in shallow lakes
,” Ecosystems
10
, 1
–3
(2007
). 36.
M.
Scheffer
, S.
Carpenter
, J. A.
Foley
, C.
Folke
, and B.
Walker
, “Catastrophic shifts in ecosystems
,” Nature
413
, 591
(2001
). 37.
38.
D. C.
Zemp
, C.-F.
Schleussner
, H. M.
Barbosa
, M.
Hirota
, V.
Montade
, G.
Sampaio
, A.
Staal
, L.
Wang-Erlandsson
, and A.
Rammig
, “Self-amplified Amazon forest loss due to vegetation-atmosphere feedbacks
,” Nat. Commun.
8
, 14681
(2017
). 39.
M.
Hirota
, M.
Holmgren
, E. H.
van Nes
, and M.
Scheffer
, “Global resilience of tropical forest and savanna to critical transitions
,” Science
334
, 232
–235
(2011
). 40.
A.
Staal
, O. A.
Tuinenburg
, J. H.
Bosmans
, M.
Holmgren
, E. H.
van Nes
, M.
Scheffer
, D. C.
Zemp
, and S. C.
Dekker
, “Forest-rainfall cascades buffer against drought across the Amazon
,” Nat. Clim. Change
8
, 539
–543
(2018
). 41.
C. A.
Nobre
, G.
Sampaio
, L. S.
Borma
, J. C.
Castilla-Rubio
, J. S.
Silva
, and M.
Cardoso
, “Land-use and climate change risks in the Amazon and the need of a novel sustainable development paradigm
,” Proc. Natl. Acad. Sci. U.S.A.
113
, 10759
–10768
(2016
). 42.
E. H.
van Nes
, M.
Hirota
, M.
Holmgren
, and M.
Scheffer
, “Tipping points in tropical tree cover: Linking theory to data
,” Global Change Biol.
20
, 1016
–1021
(2014
). 43.
A.
Staal
, S. C.
Dekker
, M.
Hirota
, and E. H.
van Nes
, “Synergistic effects of drought and deforestation on the resilience of the south-eastern Amazon rainforest
,” Ecol. Complex.
22
, 65
–75
(2015
). 44.
D. C.
Zemp
, C.-F.
Schleussner
, H.
Barbosa
, R.
Van der Ent
, J. F.
Donges
, J.
Heinke
, G.
Sampaio
, and A.
Rammig
, “On the importance of cascading moisture recycling in South America
,” Atmos. Chem. Phys.
14
, 13337
–13359
(2014
). © 2020 Author(s).
2020
Author(s)
You do not currently have access to this content.
