We introduce the pyunicorn (Pythonic unified complex network and recurrence analysis toolbox) open source software package for applying and combining modern methods of data analysis and modeling from complex network theory and nonlinear time series analysis. pyunicorn is a fully object-oriented and easily parallelizable package written in the language Python. It allows for the construction of functional networks such as climate networks in climatology or functional brain networks in neuroscience representing the structure of statistical interrelationships in large data sets of time series and, subsequently, investigating this structure using advanced methods of complex network theory such as measures and models for spatial networks, networks of interacting networks, node-weighted statistics, or network surrogates. Additionally, pyunicorn provides insights into the nonlinear dynamics of complex systems as recorded in uni- and multivariate time series from a non-traditional perspective by means of recurrence quantification analysis, recurrence networks, visibility graphs, and construction of surrogate time series. The range of possible applications of the library is outlined, drawing on several examples mainly from the field of climatology.

1.
R.
Albert
and
A. L.
Barabási
, “
Statistical mechanics of complex networks
,”
Rev. Mod. Phys.
74
,
47
97
(
2002
).
2.
M. E. J.
Newman
, “
The structure and function of complex networks
,”
SIAM Rev.
45
,
167
256
(
2003
).
3.
S.
Boccaletti
,
V.
Latora
,
Y.
Moreno
,
M.
Chavez
, and
D.-U.
Hwang
, “
Complex networks: Structure and dynamics
,”
Phys. Rep.
424
,
175
308
(
2006
).
4.
R.
Cohen
and
S.
Havlin
,
Complex Networks: Structure, Robustness and Function
(
Cambridge University Press
,
Cambridge
,
2010
).
5.
M. E. J.
Newman
,
Networks: An Introduction
(
Oxford University Press
,
Oxford
,
2010
).
6.
H. D. I.
Abarbanel
,
Analysis of Observed Chaotic Data
(
Springer
,
New York
,
1996
).
7.
J. C.
Sprott
,
Chaos and Time-Series Analysis
(
Oxford University Press
,
Oxford
,
2003
).
8.
H.
Kantz
and
T.
Schreiber
,
Nonlinear Time Series Analysis
, 2nd ed. (
Cambridge University Press
,
Cambridge
,
2004
).
9.
G.
Csárdi
and
T.
Nepusz
, “
The igraph software package for complex network research
,”
InterJ. Complex Syst.
CX.18
,
1695
(
2006
).
10.
D. A.
Schult
and
P. J.
Swart
, “
Exploring network structure, dynamics, and function using NetworkX
,” in
Proceedings of the 7th Python in Science Conferences (SciPy 2008)
(2008), Vol.
2008
, pp.
11
16
.
11.
R.
Hegger
,
H.
Kantz
, and
T.
Schreiber
, “
Practical implementation of nonlinear time series methods: The TISEAN package
,”
Chaos
9
,
413
435
(
1999
).
12.
C.
Zhou
,
L.
Zemanová
,
G.
Zamora
,
C. C.
Hilgetag
, and
J.
Kurths
, “
Hierarchical organization unveiled by functional connectivity in complex brain networks
,”
Phys. Rev. Lett.
97
,
238103
(
2006
).
13.
C.
Zhou
,
L.
Zemanová
,
G.
Zamora-Lopez
,
C. C.
Hilgetag
, and
J.
Kurths
, “
Structure–function relationship in complex brain networks expressed by hierarchical synchronization
,”
New J. Phys.
9
,
178
(
2007
).
14.
E.
Bullmore
and
O.
Sporns
, “
Complex brain networks: Graph theoretical analysis of structural and functional systems
,”
Nat. Rev. Neurosci.
10
,
186
198
(
2009
).
15.
A. A.
Tsonis
and
P. J.
Roebber
, “
The architecture of the climate network
,”
Physica A
333
,
497
504
(
2004
).
16.
A. A.
Tsonis
and
K. L.
Swanson
, “
Topology and predictability of El Niño and La Niña networks
,”
Phys. Rev. Lett.
100
,
228502
(
2008
).
17.
K.
Yamasaki
,
A.
Gozolchiani
, and
S.
Havlin
, “
Climate networks around the globe are significantly affected by El Niño
,”
Phys. Rev. Lett.
100
,
228501
(
2008
).
18.
J. F.
Donges
,
Y.
Zou
,
N.
Marwan
, and
J.
Kurths
, “
Complex networks in climate dynamics—Comparing linear and nonlinear network construction methods
,”
Eur. Phys. J. ST
174
,
157
179
(
2009
).
19.
J. F.
Donges
,
Y.
Zou
,
N.
Marwan
, and
J.
Kurths
, “
The backbone of the climate network
,”
Europhys. Lett.
87
,
48007
(
2009
).
20.
J. F.
Donges
,
I.
Petrova
,
A.
Loew
,
N.
Marwan
, and
J.
Kurths
, “
How complex climate networks complement eigen techniques for the statistical analysis of climatological data
,”
Clim. Dyn.
(published online 2015).
21.
W.-Q.
Huang
,
X.-T.
Zhuang
, and
S.
Yao
, “
A network analysis of the Chinese stock market
,”
Physica A
388
,
2956
2964
(
2009
).
22.
R. V.
Donner
,
M.
Small
,
J. F.
Donges
,
N.
Marwan
,
Y.
Zou
,
R.
Xiang
, and
J.
Kurths
, “
Recurrence-based time series analysis by means of complex network methods
,”
Int. J. Bifurcation Chaos
21
,
1019
1046
(
2011
).
23.
X.
Xu
,
J.
Zhang
, and
M.
Small
, “
Superfamily phenomena and motifs of networks induced from time series
,”
Proc. Natl. Acad. Sci. U.S.A.
105
,
19601
19605
(
2008
).
24.
N.
Marwan
,
J. F.
Donges
,
Y.
Zou
,
R. V.
Donner
, and
J.
Kurths
, “
Complex network approach for recurrence analysis of time series
,”
Phys. Lett. A
373
,
4246
4254
(
2009
).
25.
R. V.
Donner
,
Y.
Zou
,
J. F.
Donges
,
N.
Marwan
, and
J.
Kurths
, “
Recurrence networks—A novel paradigm for nonlinear time series analysis
,”
New J. Phys.
12
,
033025
(
2010
).
26.
J. F.
Donges
,
J.
Heitzig
,
R. V.
Donner
, and
J.
Kurths
, “
Analytical framework for recurrence network analysis of time series
,”
Phys. Rev. E
85
,
046105
(
2012
).
27.
G.
Nicolis
,
A.
Garciá Cantú
, and
C.
Nicolis
, “
Dynamical aspects of interaction networks
,”
Int. J. Bifurcation Chaos
15
,
3467
(
2005
).
28.
L.
Lacasa
,
B.
Luque
,
F.
Ballesteros
,
J.
Luque
, and
J. C.
Nuno
, “
From time series to complex networks: The visibility graph
,”
Proc. Natl. Acad. Sci. U.S.A.
105
,
4972
4975
(
2008
).
29.
R. V.
Donner
and
J. F.
Donges
, “
Visibility graph analysis of geophysical time series: Potentials and possible pitfalls
,”
Acta Geophys.
60
,
589
623
(
2012
).
30.
J. F.
Donges
,
R. V.
Donner
, and
J.
Kurths
, “
Testing time series irreversibility using complex network methods
,”
Europhys. Lett.
102
,
10004
(
2013
).
31.
N. P.
Subramaniyam
and
J.
Hyttinen
, “
Characterization of dynamical systems under noise using recurrence networks: Application to simulated and EEG data
,”
Phys. Lett. A
378
,
3464
3474
(
2014
).
32.
N. P.
Subramaniyam
,
J. F.
Donges
, and
J.
Hyttinen
, “
Signatures of chaotic and stochastic dynamics uncovered with ε-recurrence networks
,”
Proc. R. Soc. A—Math. Phys.
(in press).
33.
See supplemental material at http://dx.doi.org/10.1063/1.4934554 for a comprehensive pyunicorn API documentation and exemplary code.
34.
T. E.
Oliphant
, “
Python for scientific computing
,”
Comput. Sci. Eng.
9
,
10
20
(
2007
).
35.
K. J.
Millman
and
M.
Aivazis
, “
Python for scientists and engineers
,”
Comput. Sci. Eng.
13
,
9
12
(
2011
).
36.
F.
Pérez
and
B. E.
Granger
, “
Ipython: A system for interactive scientific computing
,”
Comput. Sci. Eng.
9
,
21
29
(
2007
).
37.
S.
van der Walt
,
S. C.
Colbert
, and
G.
Varoquaux
, “
The numpy array: A structure for efficient numerical computation
,”
Comput. Sci. Eng.
13
,
22
30
(
2011
).
38.
E.
Jones
,
T.
Oliphant
, and
P.
Peterson
 et al., “
SciPy: Open source scientific tools for Python
,” 2001, Online, http://www.scipy.org/ (accessed May 30,
2015
).
39.
J. D.
Hunter
, “
Matplotlib: A 2d graphics environment
,”
Comput. Sci. Eng.
9
,
90
95
(
2007
).
40.
T.
Nocke
,
S.
Buschmann
,
J. F.
Donges
,
N.
Marwan
,
H.-J.
Schulz
, and
C.
Tominski
, “
Review: Visual analytics of climate networks
,”
Nonlinear Proc. Geophys.
22
,
545
570
(
2015
).
41.
C.
Tominski
,
J.
Abello
, and
H.
Schumann
, “
CGV–an interactive graph visualization system
,”
Comput. Graph.
33
,
660
678
(
2009
).
42.
C.
Tominski
,
J. F.
Donges
, and
T.
Nocke
, “
Information visualization in climate research
,” in
15th International Conference on Information Visualisation (IV)
(IEEE,
2011
), pp.
298
305
.
43.
S.
Behnel
,
R.
Bradshaw
,
C.
Citro
,
L.
Dalcin
,
D. S.
Seljebotn
, and
K.
Smith
, “
Cython: The best of both worlds
,”
Comput. Sci. Eng.
13
,
31
39
(
2011
).
44.
M. E. J.
Newman
, “
A measure of betweenness centrality based on random walks
,”
Soc. Networks
27
,
39
54
(
2005
).
45.
A.
Arenas
,
A.
Cabrales
,
A.
Díaz-Guilera
,
R.
Guimerà
, and
F.
Vega-Redondo
, “
Search and congestion in complex networks
,” in
Statistical Mechanics of Complex Networks
, Lecture Notes in Physics Vol.
625
, edited by
R.
Pastor-Satorras
,
M.
Rubi
, and
A.
Díaz-Guilera
(
Springer
,
Berlin/Heidelberg
,
2003
), pp.
175
194
.
46.
U.
Brandes
,
M.
Eiglsperger
,
I.
Herman
,
M.
Himsolt
, and
M. S.
Marshall
, “
GraphML progress report. Structural layer proposal
,” in
Proceedings 9th International Symposium on Graph Drawing (GD'01)
, edited by Department of Computer & Information Science, University of Konstanz, Germany (Springer,
2002
), pp.
501
512
.
47.
M.
Bastian
,
S.
Heymann
, and
M.
Jacomy
, “
Gephi: An open source software for exploring and manipulating networks
,” in
Proceedings of the International AAAI Conference on Weblogs and Social Media
(
2009
).
48.
M.
Barthélemy
, “
Spatial networks
,”
Phys. Rep.
499
,
1
101
(
2011
).
49.
A. A.
Tsonis
,
K. L.
Swanson
, and
G.
Wang
, “
On the role of atmospheric teleconnections in climate
,”
J. Clim.
21
,
2990
3001
(
2008
).
50.
M.
Wiedermann
,
J. F.
Donges
,
J.
Kurths
, and
R. V.
Donner
, “
Spatial network surrogates for disentangling complex system structure from spatial embedding of nodes
,” preprint arXiv:150909293 (
2015
).
51.
M. T.
Gastner
and
M. E. J.
Newman
, “
The spatial structure of networks
,”
Eur. Phys. J. B
49
,
247
252
(
2006
).
52.
J. F.
Donges
,
H. C. H.
Schultz
,
N.
Marwan
,
Y.
Zou
, and
J.
Kurths
, “
Investigating the topology of interacting networks—Theory and application to coupled climate subnetworks
,”
Eur. Phys. J. B
84
,
635
652
(
2011
).
53.
M.
Wiedermann
,
J. F.
Donges
,
J.
Heitzig
, and
J.
Kurths
, “
Node-weighted interacting network measures improve the representation of real-world complex systems
,”
Europhys. Lett.
102
,
28007
(
2013
).
54.
S. V.
Buldyrev
,
R.
Parshani
,
G.
Paul
,
H. E.
Stanley
, and
S.
Havlin
, “
Catastrophic cascade of failures in interdependent networks
,”
Nature
464
,
1025
1028
(
2010
).
55.
J.
Gao
,
S. V.
Buldyrev
,
H. E.
Stanley
, and
S.
Havlin
, “
Networks formed from interdependent networks
,”
Nat. Phys.
8
,
40
48
(
2012
).
56.
S.
Boccaletti
,
G.
Bianconi
,
R.
Criado
,
C.
Del Genio
,
J.
Gómez-Gardeñes
,
M.
Romance
,
I.
Sendina-Nadal
,
Z.
Wang
, and
M.
Zanin
, “
The structure and dynamics of multilayer networks
,”
Phys. Rep.
544
,
1
122
(
2014
).
57.
M.
Girvan
and
M. E. J.
Newman
, “
Community structure in social and biological networks
,”
Proc. Natl. Acad. Sci. U.S.A.
99
,
7821
7826
(
2002
).
58.
M. E. J.
Newman
, “
Modularity and community structure in networks
,”
Proc. Natl. Acad. Sci. U.S.A.
103
,
8577
8582
(
2006
).
59.
S.
Fortunato
, “
Community detection in graphs
,”
Phys. Rep.
486
,
75
174
(
2010
).
60.
P.
Erdős
and
A.
Rényi
, “
On random graphs I
,”
Publ. Math. Debrecen
6
,
290
297
(
1959
).
61.
W. W.
Zachary
, “
An information flow model for conflict and fission in small groups
,”
J. Anthropol. Res.
33
,
452
473
(
1977
).
62.
M.
Wiedermann
,
J. F.
Donges
,
D.
Handorf
,
J.
Kurths
, and
R. V.
Donner
, “
Hierarchical structures in northern hemispheric extratropical winter ocean-atmosphere interactions
,” preprint arXiv:150606634 (
2015
).
63.
J. H.
Feldhoff
,
R. V.
Donner
,
J. F.
Donges
,
N.
Marwan
, and
J.
Kurths
, “
Geometric detection of coupling directions by means of inter-system recurrence networks
,”
Phys. Lett. A
376
,
3504
3513
(
2012
).
64.
J.
Heitzig
,
J. F.
Donges
,
Y.
Zou
,
N.
Marwan
, and
J.
Kurths
, “
Node-weighted measures for complex networks with spatially embedded, sampled, or differently sized nodes
,”
Eur. Phys. J. B
85
,
38
(
2012
).
65.
A.
Rheinwalt
,
N.
Marwan
,
J.
Kurths
,
P.
Werner
, and
F.-W.
Gerstengarbe
, “
Boundary effects in network measures of spatially embedded networks
,”
Europhys. Lett.
100
,
28002
(
2012
).
66.
A.
Radebach
,
R. V.
Donner
,
J.
Runge
,
J. F.
Donges
, and
J.
Kurths
, “
Disentangling different types of El Niño episodes by evolving climate network analysis
,”
Phys. Rev. E
88
,
052807
(
2013
).
67.
N.
Molkenthin
,
K.
Rehfeld
,
V.
Stolbova
,
L.
Tupikina
, and
J.
Kurths
, “
On the influence of spatial sampling on climate networks
,”
Nonlinear Proc. Geophys.
21
,
651
657
(
2014
).
68.
D. C.
Zemp
,
M.
Wiedermann
,
J.
Kurths
,
A.
Rammig
, and
J. F.
Donges
, “
Node-weighted measures for complex networks with directed and weighted edges for studying continental moisture recycling
,”
Europhys. Lett.
107
,
58005
(
2014
).
69.
D. C.
Zemp
,
C.-F.
Schleussner
,
H. M. J.
Barbosa
,
R. J.
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
).
70.
J. H.
Feldhoff
,
S.
Lange
,
J.
Volkholz
,
J. F.
Donges
,
J.
Kurths
, and
F.-W.
Gerstengarbe
, “
Complex networks for climate model evaluation with application to statistical versus dynamical modeling of South American climate
,”
Clim. Dyn.
44
,
1567
1581
(
2015
).
71.
S.
Lange
,
J. F.
Donges
,
J.
Volkholz
, and
J.
Kurths
, “
Local difference measures between complex networks for dynamical system model evaluation
,”
PLoS ONE
10
,
e0118088
(
2015
).
72.
A.
Rheinwalt
,
N.
Boers
,
N.
Marwan
,
J.
Kurths
,
P.
Hoffmann
,
F.-W.
Gerstengarbe
, and
P.
Werner
, “
Non-linear time series analysis of precipitation events using regional climate networks for Germany
,”
Clim. Dyn.
(published online
2015
).
73.
N.
Molkenthin
,
K.
Rehfeld
,
N.
Marwan
, and
J.
Kurths
, “
Networks from flows—From dynamics to topology
,”
Sci. Rep.
4
,
4119
(
2014
).
74.
T. M.
Cover
and
J. A.
Thomas
,
Elements of Information Theory
(
John Wiley & Sons
,
Hoboken
,
2006
).
75.
C. E.
Shannon
, “
A Mathematical Theory of Communication
,”
Bell Syst. Tech. J.
27
,
379
423
(
1948
).
76.
M.
Paluš
, “
Coarse-grained entropy rates for characterization of complex time series
,”
Physica D
93
,
64
77
(
1996
).
77.
K.
Hlaváčková-Schindler
,
M.
Paluš
,
M.
Vejmelka
, and
J.
Bhattacharya
, “
Causality detection based on information-theoretic approaches in time series analysis
,”
Phys. Rep.
441
,
1
46
(
2007
).
78.
A.
Kraskov
,
H.
Stögbauer
, and
P.
Grassberger
, “
Estimating mutual information
,”
Phys. Rev. E
69
,
066138
(
2004
).
79.
J.
Runge
,
J.
Heitzig
,
N.
Marwan
, and
J.
Kurths
, “
Quantifying causal coupling strength: A lag-specific measure for multivariate time series related to transfer entropy
,”
Phys. Rev. E
86
,
061121
(
2012
).
80.
J.
Hlinka
,
D.
Hartman
,
M.
Vejmelka
,
J.
Runge
,
N.
Marwan
,
J.
Kurths
, and
M.
Palus
, “
Reliability of inference of directed climate networks using conditional mutual information
,”
Entropy
15
,
2023
2045
(
2013
).
81.
T.
Schreiber
, “
Measuring information transfer
,”
Phys. Rev. Lett.
85
,
461
464
(
2000
).
82.
B.
Pompe
and
J.
Runge
, “
Momentary information transfer as a coupling measure of time series
,”
Phys. Rev. E
83
,
051122
(
2011
).
83.
S.
Frenzel
and
B.
Pompe
, “
Partial mutual information for coupling analysis of multivariate time series
,”
Phys. Rev. Lett.
99
,
204101
(
2007
).
84.
J.
Runge
,
V.
Petoukhov
, and
J.
Kurths
, “
Quantifying the strength and delay of climatic interactions: The ambiguities of cross correlation and a novel measure based on graphical models
,”
J. Clim.
27
,
720
739
(
2014
).
85.
M.
Eichler
, “
Graphical modelling of multivariate time series
,”
Probab. Theory Relat. Fields
153
,
233
268
(
2012
).
86.
J.
Runge
,
J.
Heitzig
,
V.
Petoukhov
, and
J.
Kurths
, “
Escaping the curse of dimensionality in estimating multivariate transfer entropy
,”
Phys. Rev. Lett.
108
,
258701
(
2012
).
87.
C.-F.
Schleussner
,
J.
Runge
,
J.
Lehmann
, and
A.
Levermann
, “
The role of the North Atlantic overturning and deep ocean for multi-decadal global-mean-temperature variability
,”
Earth Syst. Dyn.
5
,
103
115
(
2014
).
88.
G.
Balasis
,
R. V.
Donner
,
S. M.
Potirakis
,
J.
Runge
,
C.
Papadimitriou
,
I.
Daglis
,
K.
Eftaxis
, and
J.
Kurths
, “
Statistical mechanics and information-theoretic perspectives on complexity in the Earth system
,”
Entropy
15
,
4844
4888
(
2013
).
89.
J.
Runge
,
V.
Petoukhov
,
J. F.
Donges
,
J.
Hlinka
,
N.
Jajcay
,
M.
Vejmelka
,
D.
Hartman
,
N.
Marwan
,
M.
Paluš
, and
J.
Kurths
, “
Identifying causal gateways and mediators in complex spatio-temporal systems
,”
Nat. Commun.
6
,
8502
(
2015
).
90.
J.
Runge
, “
Quantifying information transfer and mediation along causal pathways in complex systems
,” preprint arXiv:150803808 (
2015
).
91.
J. F.
Donges
, “
Functional network macroscopes for probing past and present Earth system dynamics: Complex hierarchical interactions, tipping points, and beyond
,” Ph.D. dissertation,
Humboldt University, Berlin, Germany
,
2012
.
92.
J.
Ludescher
,
A.
Gozolchiani
,
M. I.
Bogachev
,
A.
Bunde
,
S.
Havlin
, and
H. J.
Schellnhuber
, “
Improved El Nino forecasting by cooperativity detection
,”
Proc. Natl. Acad. Sci. U.S.A.
110
,
11742
11745
(
2013
).
93.
J.
Ludescher
,
A.
Gozolchiani
,
M. I.
Bogachev
,
A.
Bunde
,
S.
Havlin
, and
H. J.
Schellnhuber
, “
Very early warning of next El Niño
,”
Proc. Natl. Acad. Sci. U.S.A.
111
,
2064
2066
(
2014
).
94.
H.
Ihshaish
,
A.
Tantet
,
J. C. M.
Dijkzeul
, and
H. A.
Dijkstra
, “
Par@Graph—A parallel toolbox for the construction and analysis of large complex climate networks
,”
Geosci. Model Dev.
8
,
3321
3331
(
2015
).
95.
Q. Y.
Feng
and
H. A.
Dijkstra
, “
Are North Atlantic multidecadal SST anomalies westward propagating?
Geophys. Res. Lett.
41
,
541
546
, doi: (
2014
).
96.
M.
Mheen
,
H. A.
Dijkstra
,
A.
Gozolchiani
,
M.
den Toom
,
J.
Feng
,
Q.
Kurths
, and
E.
Hernandez-Garcia
, “
Interaction network based early warning indicators for the Atlantic MOC collapse
,”
Geophys. Res. Lett.
40
,
2714
2719
, doi: (
2013
).
97.
Q. Y.
Feng
,
J. P.
Viebahn
, and
H. A.
Dijkstra
, “
Deep ocean early warning signals of an Atlantic MOC collapse
,”
Geophys. Res. Lett.
41
,
6009
6015
, doi: (
2014
).
98.
E.
Hawkins
,
R. S.
Smith
,
L. C.
Allison
,
J. M.
Gregory
,
T. J.
Woollings
,
H.
Pohlmann
, and
B.
de Cuevas
, “
Bistability of the Atlantic overturning circulation in a global climate model and links to ocean freshwater transport
,”
Geophys. Res. Lett.
38
,
L10605
, doi: (
2011
).
99.
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
).
100.
F. O.
Bryan
, “
High-latitude salinity effects and interhemispheric thermohaline circulations
,”
Nature
323
,
301
304
(
1986
).
101.
S.
Rahmstorf
, “
The thermohaline circulation: a system with dangerous thresholds?
,”
Clim. Change
46
,
247
256
(
2000
).
102.
K.
Rehfeld
,
N.
Marwan
,
S. F. M.
Breitenbach
, and
J.
Kurths
, “
Late Holocene Asian Summer Monsoon dynamics from small but complex networks of palaeoclimate data
,”
Clim. Dyn.
41
,
3
19
(
2013
).
103.
A.
Gozolchiani
,
K.
Yamasaki
,
O.
Gazit
, and
S.
Havlin
, “
Pattern of climate network blinking links follows El Niño events
,”
Europhys. Lett.
83
,
28005
(
2008
).
104.
N.
Malik
,
B.
Bookhagen
,
N.
Marwan
, and
J.
Kurths
, “
Analysis of spatial and temporal extreme monsoonal rainfall over South Asia using complex networks
,”
Clim. Dyn.
39
,
971
987
(
2012
).
105.
V.
Stolbova
,
P.
Martin
,
B.
Bookhagen
,
N.
Marwan
, and
J.
Kurths
, “
Topology and seasonal evolution of the network of extreme precipitation over the Indian subcontinent and Sri Lanka
,”
Nonlinear Proc. Geophys.
21
,
901
917
(
2014
).
106.
L.
Tupikina
,
K.
Rehfeld
,
N.
Molkenthin
,
V.
Stolbova
,
N.
Marwan
, and
J.
Kurths
, “
Characterizing the evolution of climate networks
,”
Nonlinear Proc. Geophys.
21
,
705
711
(
2014
).
107.
G. J.
Huffman
,
D. T.
Bolvin
,
E. J.
Nelkin
,
D. B.
Wolff
,
R. F.
Adler
,
G.
Gu
,
Y.
Hong
,
K. P.
Bowman
, and
E. F.
Stocker
, “
The TRMM multisatellite precipitation analysis (TMPA): Quasi-global, multiyear, combined-sensor precipitation estimates at fine scales
,”
J. Hydrometeorol.
8
,
38
55
(
2007
).
108.
TRMM
, “
TRMM data set
,”
2012
, Online, http://disc.sci.gsfc.nasa.gov/precipitation/documentation/ (accessed February 25, 2014).
109.
R.
Quian Quiroga
,
T.
Kreuz
, and
P.
Grassberger
, “
Event synchronization: A simple and fast method to measure synchronicity and time delay patterns
,”
Phys. Rev. E
66
,
041904
(
2002
).
110.
N.
Boers
,
A.
Rheinwalt
,
B.
Bookhagen
,
H. M.
Barbosa
,
N.
Marwan
,
J.
Marengo
, and
J.
Kurths
, “
The South American rainfall dipole: A complex network analysis of extreme events
,”
Geophys. Res. Lett.
41
,
7397
7405
, doi: (
2014
).
111.
R.
Kistler
,
E.
Kalnay
,
W.
Collins
,
S.
Saha
,
G.
White
,
J.
Woollen
,
M.
Chelliah
,
W.
Ebisuzaki
,
M.
Kanamitsu
,
V.
Kousky
,
H. V. D.
Dool
,
R.
Jenne
, and
M.
Fiorino
, “
The NCEP–NCAR 50–year reanalysis: Monthly means CD–ROM and documentation
,”
Bull. Am. Meteorol. Soc.
82
,
247
268
(
2001
).
112.
P.
Holme
and
J.
Saramäki
, “
Temporal networks
,”
Phys. Rep.
519
,
97
125
(
2012
).
113.
Y.
Berezin
,
A.
Gozolchiani
,
O.
Guez
, and
S.
Havlin
, “
Stability of climate networks with time
,”
Sci. Rep.
2
,
666
(
2012
).
114.
J.-S.
Kug
,
S.-I.
Choi
,
J. A.
An
,
F.-F.
Jin
, and
A. T.
Wittenberg
, “
Warm pool and cold tongue El Niño events as simulated by the GFDL 2.1 coupled GCM
,”
J. Clim.
23
,
1226
1239
(
2010
).
115.
A.
Feng
,
Z.
Gong
,
Q.
Wang
, and
G.
Feng
, “
Three-dimensional air–sea interactions investigated with bilayer networks
,”
Theor. Appl. Climatol.
109
,
635
643
(
2012
).
116.
N. A.
Rayner
,
D. E.
Parker
,
E. B.
Horton
,
C. K.
Folland
,
L. V.
Alexander
,
D. P.
Rowell
,
E. C.
Kent
, and
A.
Kaplan
, “
Global analyses of sea surface temperature, sea ice, and night marine air temperature since the late nineteenth century
,”
J. Geophys. Res.
108
,
4407
, doi: (
2003
).
117.
S. M.
Uppala
,
P. W.
Kållberg
,
A. J.
Simmons
,
U.
Andrae
,
V. D. C.
Bechtold
,
M.
Fiorino
,
J. K.
Gibson
,
J.
Haseler
,
A.
Hernandez
,
G. A.
Kelly
,
X.
Li
,
K.
Onogi
,
S.
Saarinen
,
N.
Sokka
,
R. P.
Allan
,
E.
Andersson
,
K.
Arpe
,
M. A.
Balmaseda
,
A. C. M.
Beljaars
,
L. V. D.
Berg
,
J.
Bidlot
,
N.
Bormann
,
S.
Caires
,
F.
Chevallier
,
A.
Dethof
,
M.
Dragosavac
,
M.
Fisher
,
M.
Fuentes
,
S.
Hagemann
,
E.
Hólm
,
B. J.
Hoskins
,
L.
Isaksen
,
P. A. E. M.
Janssen
,
R.
Jenne
,
A. P.
Mcnally
,
J.-F.
Mahfouf
,
J.-J.
Morcrette
,
N. A.
Rayner
,
R. W.
Saunders
,
P.
Simon
,
A.
Sterl
,
K. E.
Trenberth
,
A.
Untch
,
D.
Vasiljevic
,
P.
Viterbo
, and
J.
Woollen
, “
The ERA-40 re-analysis
,”
Q. J. R. Meteorol. Soc.
131
,
2961
3012
(
2005
).
118.
E.
Ravasz
and
A.-L.
Barabási
, “
Hierarchical organization in complex networks
,”
Phys. Rev. E
67
,
026112
(
2003
).
119.
J.
Dall
and
M.
Christensen
, “
Random geometric graphs
,”
Phys. Rev. E
66
,
016121
(
2002
).
120.
R. V.
Donner
,
J.
Heitzig
,
J. F.
Donges
,
Y.
Zou
,
N.
Marwan
, and
J.
Kurths
, “
The geometry of chaotic dynamics – a complex network perspective
,”
Eur. Phys. J. B
84
,
653
672
(
2011
).
121.
L.
Lacasa
,
B.
Luque
,
J.
Luque
, and
J. C.
Nuno
, “
The visibility graph: A new method for estimating the Hurst exponent of fractional Brownian motion
,”
Europhys. Lett.
86
,
30001
(
2009
).
122.
N.
Marwan
,
M. C.
Romano
,
M.
Thiel
, and
J.
Kurths
, “
Recurrence plots for the analysis of complex systems
,”
Phys. Rep.
438
,
237
329
(
2007
).
123.
N.
Marwan
and
J.
Kurths
, “
Complex network based techniques to identify extreme events and (sudden) transitions in spatio-temporal systems
,”
Chaos
25
,
097609
(
2015
).
124.
M. C.
Romano
,
M.
Thiel
,
J.
Kurths
, and
C.
Grebogi
, “
Estimation of the direction of the coupling by conditional probabilities of recurrence
,”
Phys. Rev. E
76
,
036211
(
2007
).
125.
Y.
Zou
,
M. C.
Romano
,
M.
Thiel
,
N.
Marwan
, and
J.
Kurths
, “
Inferring indirect coupling by means of recurrences
,”
Int. J. Bifurcation Chaos
21
,
1099
1111
(
2011
).
126.
N.
Marwan
, “
A historical review of recurrence plots
,”
Eur. Phys. J. ST
164
,
3
12
(
2008
).
127.
G. M.
Ramírez Ávila
,
A.
Gapelyuk
,
N.
Marwan
,
H.
Stepan
,
J.
Kurths
,
T.
Walther
, and
N.
Wessel
, “
Classifying healthy women and preeclamptic patients from cardiovascular data using recurrence and complex network methods
,”
Auton. Neusci.
178
,
103
110
(
2013
).
128.
J. F.
Donges
,
R. V.
Donner
,
N.
Marwan
,
S. F.
Breitenbach
,
K.
Rehfeld
, and
J.
Kurths
, “
Non-linear regime shifts in Holocene Asian monsoon variability: Potential impacts on cultural change and migratory patterns
,”
Clim. Past
11
,
709
741
(
2015
).
129.
N. H.
Packard
,
J. P.
Crutchfield
,
J. D.
Farmer
, and
R. S.
Shaw
, “
Geometry from a time series
,”
Phys. Rev. Lett.
45
,
712
716
(
1980
).
130.
E. N.
Lorenz
, “
Deterministic nonperiodic flow
,”
J. Atmos. Sci.
20
,
130
141
(
1963
).
131.
S.
Schinkel
,
N.
Marwan
,
O.
Dimigen
, and
J.
Kurths
, “
Confidence bounds of recurrence-based complexity measures
,”
Phys. Lett. A
373
,
2245
2250
(
2009
).
132.
Y.
Zou
,
R. V.
Donner
,
J. F.
Donges
,
N.
Marwan
, and
J.
Kurths
, “
Identifying complex periodic windows in continuous-time dynamical systems using recurrence-based methods
,”
Chaos
20
,
043130
(
2010
).
133.
J. H.
Feldhoff
,
R. V.
Donner
,
J. F.
Donges
,
N.
Marwan
, and
J.
Kurths
, “
Geometric signature of complex synchronisation scenarios
,”
Europhys. Lett.
102
,
30007
(
2013
).
134.
J. F.
Donges
,
R. V.
Donner
,
K.
Rehfeld
,
N.
Marwan
,
M. H.
Trauth
, and
J.
Kurths
, “
Identification of dynamical transitions in marine palaeoclimate records by recurrence network analysis
,”
Nonlinear Processes Geophys.
18
,
545
562
(
2011
).
135.
J.
Rockström
,
W.
Steffen
,
K.
Noone
,
A.
Persson
,
F. S.
Chapin
 III
,
E. F.
Lambin
,
T. M.
Lenton
,
M.
Scheffer
,
C.
Folke
,
H. J.
Schellnhuber
,
B.
Nykvist
,
C. A.
de Wit
,
T.
Hughes
,
S.
van der Leeuw
,
H.
Rodhe
,
S.
Sorlin
,
P. K.
Snyder
,
R.
Costanza
,
U.
Svedin
,
M.
Falkenmark
,
L.
Karlberg
,
R. W.
Corell
,
V. J.
Fabry
,
J.
Hansen
,
B.
Walker
,
D.
Liverman
,
K.
Richardson
,
P.
Crutzen
, and
J. A.
Foley
, “
A safe operating space for humanity
,”
Nature
461
,
472
475
(
2009
).
136.
J. F.
Donges
,
R. V.
Donner
,
M. H.
Trauth
,
N.
Marwan
,
H.-J.
Schellnhuber
, and
J.
Kurths
, “
Nonlinear detection of paleoclimate-variability transitions possibly related to human evolution
,”
Proc. Natl. Acad. Sci. U.S.A.
108
,
20422
20427
(
2011
).
137.
N.
Marwan
,
M. H.
Trauth
,
M.
Vuille
, and
J.
Kurths
, “
Comparing modern and Pleistocene ENSO-like influences in NW Argentina using nonlinear time series analysis methods
,”
Clim. Dyn.
21
,
317
326
(
2003
).
138.
T. D.
Herbert
,
L. C.
Peterson
,
K. T.
Lawrence
, and
Z.
Liu
, “
Tropical ocean temperatures over the past 3.5 million years
,”
Science
328
,
1530
1534
(
2010
).
139.
D. J.
Kennett
,
S. F. M.
Breitenbach
,
V. V.
Aquino
,
Y.
Asmerom
,
J.
Awe
,
J. U. L.
Baldini
,
P.
Bartlein
,
B. J.
Culleton
,
C.
Ebert
,
C.
Jazwa
,
M. J.
Macri
,
N.
Marwan
,
V.
Polyak
,
K. M.
Prufer
,
H. E.
Ridley
,
H.
Sodemann
,
B.
Winterhalder
, and
G. H.
Haug
, “
Development and disintegration of Maya political systems in response to climate change
,”
Science
338
,
788
791
(
2012
).
140.
T. D.
Herbert
, “
Review of alkenone calibrations (culture, water column, and sediments)
,”
Geochem. Geophys. Geosyst.
2
,
2000GC000055
, doi: (
2001
).
141.
L.
Li
,
Q.
Li
,
J.
Tian
,
P.
Wang
,
H.
Wang
, and
Z.
Liu
, “
A 4-Ma record of thermal evolution in the tropical western Pacific and its implications on climate change
,”
Earth Planet. Sci. Lett.
309
,
10
20
(
2011
).
142.
M. B.
Kennel
,
R.
Brown
, and
H. D. I.
Abarbanel
, “
Determining embedding dimension for phase-space reconstruction using a geometrical construction
,”
Phys. Rev. A
45
,
3403
3411
(
1992
).
143.
G. H.
Haug
and
R.
Tiedemann
, “
Effect of the formation of the Isthmus of Panama on Atlantic Ocean thermohaline circulation
,”
Nature
393
,
673
676
(
1998
).
144.
M.
Medina-Elizalde
and
D. W.
Lea
, “
The mid-Pleistocene transition in the tropical Pacific
,”
Science
310
,
1009
1012
(
2005
).
145.
Z.
An
,
Y.
Sun
,
W.
Zhou
,
W.
Liu
,
X.
Qiang
,
X.
Wang
,
F.
Xian
,
P.
Cheng
, and
G. S.
Burr
, “
Chinese loess and the East Asian Monsoon
,” in
Late Cenozoic Climate Change in Asia
, Developments in Paleoenvironmental Research Vol.
16
, edited by
Z.
An
(
Springer Netherlands
,
Dordrecht
,
2014
), pp.
23
143
.
146.
C.
Karas
,
D.
Nürnberg
,
A. K.
Gupta
,
R.
Tiedemann
,
K.
Mohan
, and
T.
Bickert
, “
Mid-Pliocene climate change amplified by a switch in Indonesian subsurface throughflow
,”
Nat. Geosci.
2
,
434
438
(
2009
).
147.
Y.
Sun
,
Z.
An
,
S. C.
Clemens
,
J.
Bloemendal
, and
J.
Vandenberghe
, “
Seven million years of wind and precipitation variability on the Chinese Loess plateau
,”
Earth Planet. Sci. Lett.
297
,
525
535
(
2010
).
148.
B.
Luque
,
L.
Lacasa
,
F.
Ballesteros
, and
J.
Luque
, “
Horizontal visibility graphs: Exact results for random time series
,”
Phys. Rev. E
80
,
046103
(
2009
).
149.
Y.
Zou
,
J.
Heitzig
,
R. V.
Donner
,
J. F.
Donges
,
J. D.
Farmer
,
R.
Meucci
,
S.
Euzzor
,
N.
Marwan
, and
J.
Kurths
, “
Power-laws in recurrence networks from dynamical systems
,”
Europhys. Lett.
98
,
48001
(
2012
).
150.
X.-H.
Ni
,
Z.-Q.
Jiang
, and
W.-X.
Zhou
, “
Degree distributions of the visibility graphs mapped from fractional Brownian motions and multifractal random walks
,”
Phys. Lett. A
373
,
3822
3826
(
2009
).
151.
L.
Lacasa
,
A.
Nuñez
,
E.
Roldán
,
J. M. R.
Parrondo
, and
B.
Luque
, “
Time series irreversibility: A visibility graph approach
,”
Eur. Phys. J. B
85
,
217
(
2012
).
152.
J.
Theiler
,
S.
Eubank
,
A.
Longtin
,
B.
Galdrikian
, and
J. D.
Farmer
, “
Testing for nonlinearity in time series: the method of surrogate data
,”
Physica D
58
,
77
94
(
1992
).
153.
P. M.
Grootes
and
M.
Stuiver
, “
Oxygen 18/16 variability in Greenland snow and ice with 10–3- to 105-year time resolution
,”
J. Geophys. Res.
102
,
26455
26470
, doi: (
1997
).
154.
C.-F.
Schleussner
,
D.
Divine
,
J. F.
Donges
,
A.
Miettinen
, and
R.
Donner
, “
Indications for a North Atlantic ocean circulation regime shift at the onset of the Little Ice Age
,”
Clim. Dyn.
(published online 2015).
155.
T.
Schreiber
and
A.
Schmitz
, “
Surrogate time series
,”
Physica D
142
,
346
382
(
2000
).
156.
M.
Thiel
,
M. C.
Romano
,
J.
Kurths
,
M.
Rolfs
, and
R.
Kliegl
, “
Twin surrogates to test for complex synchronisation
,”
Europhys. Lett.
75
,
535
(
2006
).
157.
M.
Paluš
,
D.
Hartman
,
J.
Hlinka
, and
M.
Vejmelka
, “
Discerning connectivity from dynamics in climate networks
,”
Nonlinear Processes Geophys.
18
,
751
763
(
2011
).

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