We aim at identifying factors that may affect the characteristics of evolving weighted networks derived from empirical observations. To this end, we employ various chains of analysis that are often used in field studies for a data-driven derivation and characterization of such networks. As an example, we consider fully connected, weighted functional brain networks before, during, and after epileptic seizures that we derive from multichannel electroencephalographic data recorded from epilepsy patients. For these evolving networks, we estimate clustering coefficient and average shortest path length in a time-resolved manner. Lastly, we make use of surrogate concepts that we apply at various levels of the chain of analysis to assess to what extent network characteristics are dominated by properties of the electroencephalographic recordings and/or the evolving weighted networks, which may be accessible more easily. We observe that characteristics are differently affected by the unavoidable referencing of the electroencephalographic recording, by the time-series-analysis technique used to derive the properties of network links, and whether or not networks were normalized. Importantly, for the majority of analysis settings, we observe temporal evolutions of network characteristics to merely reflect the temporal evolutions of mean interaction strengths. Such a property of the data may be accessible more easily, which would render the weighted network approach—as used here—as an overly complicated description of simple aspects of the data.

1.
S.
Boccaletti
,
V.
Latora
,
Y.
Moreno
,
M.
Chavez
, and
D.-U.
Hwang
, “
Complex networks: Structure and dynamics
,”
Phys. Rep.
424
,
175
308
(
2006
).
2.
A.
Arenas
,
A.
Díaz-Guilera
,
J.
Kurths
,
Y.
Moreno
, and
C.
Zhou
, “
Synchronization in complex networks
,”
Phys. Rep.
469
,
93
153
(
2008
).
3.
E.
Bullmore
and
O.
Sporns
, “
Complex brain networks: Graph theoretical analysis of structural and functional systems
,”
Nat. Rev. Neurosci.
10
,
186
198
(
2009
).
4.
J. F.
Donges
,
Y.
Zou
,
N.
Marwan
, and
J.
Kurths
, “
The backbone of the climate network
,”
Europhys. Lett.
87
,
48007
(
2009
).
5.
K.
Lehnertz
,
G.
Ansmann
,
S.
Bialonski
,
H.
Dickten
,
C.
Geier
, and
S.
Porz
, “
Evolving networks in the human epileptic brain
,”
Physica D
267
,
7
15
(
2014
).
6.
D. S.
Bassett
and
O.
Sporns
, “
Network neuroscience
,”
Nat. Neurosci.
20
,
353
364
(
2017
).
7.
A.
Barrat
,
M.
Barthélemy
, and
A.
Vespignani
, “
Weighted evolving networks: Coupling topology and weight dynamics
,”
Phys. Rev. Lett.
92
,
228701
(
2004
).
8.
M. E. J.
Newman
, “
Analysis of weighted networks
,”
Phys. Rev. E
70
,
056131
(
2004
).
9.
Z.
Wu
,
L. A.
Braunstein
,
V.
Colizza
,
R.
Cohen
,
S.
Havlin
, and
H. E.
Stanley
, “
Optimal paths in complex networks with correlated weights: The worldwide airport network
,”
Phys. Rev. E
74
,
056104
(
2006
).
10.
W.
Li
,
Y.
Lin
, and
Y.
Liu
, “
The structure of weighted small-world networks
,”
Physica A
376
,
708
718
(
2007
).
11.
J.-P.
Onnela
,
J.
Saramäki
,
J.
Hyvönen
,
G.
Szabo
,
M. A.
de Menezes
,
K.
Kaski
,
A.-L.
Barabasi
, and
J.
Kertesz
, “
Analysis of a large-scale weighted network of one-to-one human communication
,”
New J. Phys.
9
,
179
(
2007
).
12.
C.
Zhou
,
L.
Zemanova
,
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
).
13.
M.
Jalili
,
A. A.
Rad
, and
M.
Hasler
, “
Enhancing synchronizability of weighted dynamical networks using betweenness centrality
,”
Phys. Rev. E
78
,
016105
(
2008
).
14.
L.
and
T.
Zhou
, “
Link prediction in weighted networks: The role of weak ties
,”
EPL
89
,
18001
(
2010
).
15.
S.
Horvath
,
Weighted Network Analysis: Applications in Genomics and Systems Biology
(
Springer Science & Business Media
,
New York, Dordrecht, Heidelberg, London
,
2011
).
16.
R. K.
Pan
and
J.
Saramäki
, “
The strength of strong ties in scientific collaboration networks
,”
EPL
97
,
18007
(
2012
).
17.
M.
Wiedermann
,
J. F.
Donges
,
J.
Heitzig
, and
J.
Kurths
, “
Node-weighted interacting network measures improve the representation of real-world complex systems
,”
EPL
102
,
28007
(
2013
).
18.
G.
Menichetti
,
D.
Remondini
,
P.
Panzarasa
,
R. J.
Mondragon
, and
G.
Bianconi
, “
Weighted multiplex networks
,”
PLoS ONE
9
,
e97857
(
2014
).
19.
A.
Allard
,
M. Á.
Serrano
,
G.
García-Pérez
, and
M.
Boguñá
, “
The geometric nature of weights in real complex networks
,”
Nat. Commun.
8
,
14103
(
2017
).
20.
M.
Rubinov
,
S. A.
Knock
,
C. J.
Stam
,
S.
Micheloyannis
,
A. W. F.
Harris
,
L. M.
Williams
, and
M.
Breakspear
, “
Small-world properties of nonlinear brain activity in schizophrenia
,”
Hum. Brain Mapp.
30
,
403
416
(
2009
).
21.
S. C.
Ponten
,
L.
Douw
,
F.
Bartolomei
,
J. C.
Reijneveld
, and
C. J.
Stam
, “
Indications for network regularization during absence seizures: Weighted and unweighted graph theoretical analysis
,”
Exp. Neurol.
217
,
197
204
(
2009
).
22.
M.-T.
Horstmann
,
S.
Bialonski
,
N.
Noennig
,
H.
Mai
,
J.
Prusseit
,
J.
Wellmer
,
H.
Hinrichs
, and
K.
Lehnertz
, “
State dependent properties of epileptic brain networks: Comparative graph-theoretical analyses of simultaneously recorded EEG and MEG
,”
Clin. Neurophysiol.
121
,
172
185
(
2010
).
23.
D. S.
Bassett
,
N. F.
Wymbs
,
M. A.
Porter
,
P. J.
Mucha
,
J. M.
Carlson
, and
S. T.
Grafton
, “
Dynamic reconfiguration of human brain networks during learning
,”
Proc. Natl. Acad. Sci. U. S. A.
108
,
7641
7646
(
2011
).
24.
M.-T.
Kuhnert
,
C.
Geier
,
C. E.
Elger
, and
K.
Lehnertz
, “
Identifying important nodes in weighted functional brain networks: A comparison of different centrality approaches
,”
Chaos
22
,
023142
(
2012
).
25.
M.
Boersma
,
D. J.
Smit
,
D. I.
Boomsma
,
E. J. D.
Geus
,
H. A.
Delemarre-van de Waal
, and
C. J.
Stam
, “
Growing trees in child brains: Graph theoretical analysis of electroencephalography-derived minimum spanning tree in 5-and 7-year-old children reflects brain maturation
,”
Brain Connect.
3
,
50
60
(
2013
).
26.
F. D.
Vico Fallani
,
J.
Richiardi
,
M.
Chavez
, and
S.
Achard
, “
Graph analysis of functional brain networks: Practical issues in translational neuroscience
,”
Philos. Trans. R. Soc. B
369
,
20130521
(
2014
).
27.
C.
Geier
,
S.
Bialonski
,
C. E.
Elger
, and
K.
Lehnertz
, “
How important is the seizure onset zone for seizure dynamics?
,”
Seizure
25
,
160
166
(
2015
).
28.
H.
Dickten
,
S.
Porz
,
C. E.
Elger
, and
K.
Lehnertz
, “
Weighted and directed interactions in evolving large-scale epileptic brain networks
,”
Sci. Rep.
6
,
34824
(
2016
).
29.
S. F.
Muldoon
,
E. W.
Bridgeford
, and
D. S.
Bassett
, “
Small-world propensity and weighted brain networks
,”
Sci. Rep.
6
,
22057
(
2016
).
30.
M.
Barthélemy
,
A.
Barrat
,
R.
Pastor-Santorras
, and
A.
Vespignani
, “
Characterization and modeling of weighted networks
,”
Physica A
346
,
34
43
(
2005
).
31.
J. P.
Onnela
,
J.
Saramäki
,
J.
Kertész
, and
K.
Kaski
, “
Intensity and coherence of motifs in weighted complex networks
,”
Phys. Rev. E
71
,
065103
(
2005
).
32.
M. A.
Serrano
,
M.
Boguñá
, and
R.
Pastor-Satorras
, “
Correlations in weighted networks
,”
Phys. Rev. E
74
,
055101
(
2006
).
33.
I.
Farkas
,
D.
Abel
,
G.
Palla
, and
T.
Vicsek
, “
Weighted network modules
,”
New J. Phys.
9
,
180
(
2007
).
34.
J.
Saramäki
,
M.
Kivelä
,
J. P.
Onnela
,
K.
Kaski
, and
J.
Kertész
, “
Generalizations of the clustering coefficient to weighted complex networks
,”
Phys. Rev. E
75
,
027105
(
2007
).
35.
T.
Opsahl
,
F.
Agneessens
, and
J.
Skvoretz
, “
Node centrality in weighted networks: Generalizing degree and shortest paths
,”
Soc. Networks
32
,
245
251
(
2010
).
36.
F.
Radicchi
,
J. J.
Ramasco
, and
S.
Fortunato
, “
Information filtering in complex weighted networks
,”
Phys. Rev. E
83
,
046101
(
2011
).
37.
M.
Eidsaa
and
E.
Almaas
, “
s-core network decomposition: A generalization of k-core analysis to weighted networks
,”
Phys. Rev. E
88
,
062819
(
2013
).
38.
A.
Garas
,
F.
Schweitzer
, and
S.
Havlin
, “
A k-shell decomposition method for weighted networks
,”
New J. Phys.
14
,
083030
(
2012
).
39.
J.
Alstott
,
P.
Panzarasa
,
M.
Rubinov
,
E.
Bullmore
, and
P.
Vertes
, “
A unifying framework for measuring weighted rich clubs
,”
Sci. Rep.
4
,
7258
(
2014
).
40.
A. S.
Pikovsky
,
M. G.
Rosenblum
, and
J.
Kurths
,
Synchronization: A Universal Concept in Nonlinear Sciences
(
Cambridge University Press
,
Cambridge, UK
,
2001
).
41.
S.
Boccaletti
,
J.
Kurths
,
G.
Osipov
,
D. L.
Valladares
, and
C. S.
Zhou
, “
The synchronization of chaotic systems
,”
Phys. Rep.
366
,
1
101
(
2002
).
42.
E.
Pereda
,
R.
Quian Quiroga
, and
J.
Bhattacharya
, “
Nonlinear multivariate analysis of neurophysiological signals
,”
Prog. Neurobiol.
77
,
1
37
(
2005
).
43.
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
).
44.
N.
Marwan
,
M. C.
Romano
,
M.
Thiel
, and
J.
Kurths
, “
Recurrence plots for the analysis of complex systems
,”
Phys. Rep.
438
,
237
329
(
2007
).
45.
K.
Lehnertz
,
S.
Bialonski
,
M.-T.
Horstmann
,
D.
Krug
,
A.
Rothkegel
,
M.
Staniek
, and
T.
Wagner
, “
Synchronization phenomena in human epileptic brain networks
,”
J. Neurosci. Methods
183
,
42
48
(
2009
).
46.
T.
Stankovski
,
T.
Pereira
,
P. V. E.
McClintock
, and
A.
Stefanovska
, “
Coupling functions: Universal insights into dynamical interaction mechanisms
,”
Rev. Mod. Phys.
89
,
045001
(
2017
).
47.
L.
Antiqueira
,
F. A.
Rodrigues
,
B. C. M.
van Wijk
,
L.
da F. Costa
, and
A.
Daffertshofer
, “
Estimating complex cortical networks via surface recordings–a critical note
,”
NeuroImage
53
,
439
449
(
2010
).
48.
S.
Bialonski
,
M.-T.
Horstmann
, and
K.
Lehnertz
, “
From brain to earth and climate systems: Small-world interaction networks or not?
,”
Chaos
20
,
013134
(
2010
).
49.
B. C. M.
van Wijk
,
C. J.
Stam
, and
A.
Daffertshofer
, “
Comparing brain networks of different size and connectivity density using graph theory
,”
PLoS ONE
5
,
e13701
(
2010
).
50.
N.
Langer
,
A.
Pedroni
, and
L.
Jäncke
, “
The problem of thresholding in small-world network analysis
,”
PLoS ONE
8
,
e53199
(
2013
).
51.
D.
Papo
,
M.
Zanin
, and
J. M.
Buldú
, “
Reconstructing functional brain networks: Have we got the basics right?
,”
Front. Hum. Neurosci.
8
,
107
(
2014
).
52.
S.
Porz
,
M.
Kiel
, and
K.
Lehnertz
, “
Can spurious indications for phase synchronization due to superimposed signals be avoided?
,”
Chaos
24
,
033112
(
2014
).
53.
M.
Fraschini
,
M.
Demuru
,
A.
Crobe
,
F.
Marrosu
,
C. J.
Stam
, and
A.
Hillebrand
, “
The effect of epoch length on estimated EEG functional connectivity and brain network organisation
,”
J. Neural Eng.
13
,
036015
(
2016
).
54.
D.
Papo
,
M.
Zanin
,
J. H.
Martínez
, and
J. M.
Buldú
, “
Beware of the small-world neuroscientist!
,”
Front. Hum. Neurosci.
10
,
96
(
2016
).
55.
A.
Barrat
,
M.
Barthélemy
,
R.
Pastor-Satorras
, and
A.
Vespignani
, “
The architecture of complex weighted networks
,”
Proc. Natl. Acad. Sci. U. S. A.
101
,
3747
3752
(
2004
).
56.
J. G.
Foster
,
D. V.
Foster
,
P.
Grassberger
, and
M.
Paczuski
, “
Link and subgraph likelihoods in random undirected networks with fixed and partially fixed degree sequences
,”
Phys. Rev. E
76
,
046112
(
2007
).
57.
T.
Opsahl
,
V.
Colizza
,
P.
Panzarasa
, and
J. J.
Ramasco
, “
Prominence and control: The weighted rich-club effect
,”
Phys. Rev. Lett.
101
,
168702
(
2008
).
58.
D.
Garlaschelli
, “
The weighted random graph model
,”
New J. Phys.
11
,
073005
(
2009
).
59.
G.
Ansmann
and
K.
Lehnertz
, “
Constrained randomization of weighted networks
,”
Phys. Rev. E
84
,
026103
(
2011
).
60.
S.
Bialonski
,
M.
Wendler
, and
K.
Lehnertz
, “
Unraveling spurious properties of interaction networks with tailored random networks
,”
PLoS ONE
6
,
e22826
(
2011
).
61.
T.
Schreiber
and
A.
Schmitz
, “
Surrogate time series
,”
Physica D
142
,
346
382
(
2000
).
62.
G.
Ansmann
and
K.
Lehnertz
, “
Surrogate-assisted analysis of weighted functional brain networks
,”
J. Neurosci. Methods
208
,
165
172
(
2012
).
63.
K.
Schindler
,
H.
Leung
,
C. E.
Elger
, and
K.
Lehnertz
, “
Assessing seizure dynamics by analysing the correlation structure of multichannel intracranial EEG
,”
Brain
130
,
65
77
(
2007
).
64.
D.
Yao
,
L.
Wang
,
R.
Oostenveld
,
K.
Dremstrup Nielsen
,
L.
Arendt-Nielsen
, and
A. C. N.
Chen
, “
A comparative study of different references for EEG spectral mapping: The issue of the neutral reference and the use of the infinity reference
,”
Physiol. Meas.
26
,
173
184
(
2005
).
65.
C.
Rummel
,
G.
Baier
, and
M.
Müller
, “
The influence of static correlations on multivariate correlation analysis of the EEG
,”
J. Neurosci. Methods
166
,
138
157
(
2007
).
66.
S.
Hu
,
M.
Stead
,
Q.
Dai
, and
G. A.
Worrell
, “
On the recording reference contribution to EEG correlation, phase synchorony, and coherence
,”
IEEE Trans. Syst. Man Cybern.
40
,
1294
1304
(
2010
).
67.
K.
Schindler
,
H.
Leung
,
K.
Lehnertz
, and
C. E.
Elger
, “
How generalised are secondarily ‘generalised’ tonicclonic seizures?
,”
J. Neurol., Neurosurg. Psychiatry
78
,
993
996
(
2007
).
68.
K.
Schindler
,
S.
Bialonski
,
M.-T.
Horstmann
,
C. E.
Elger
, and
K.
Lehnertz
, “
Evolving functional network properties and synchronizability during human epileptic seizures
,”
Chaos
18
,
033119
(
2008
).
69.
A.
Joudaki
,
N.
Salehi
,
M.
Jalili
, and
M. G.
Knyazeva
, “
EEG-based functional brain networks: Does the network size matter?
,”
PLoS ONE
7
,
e35673
(
2012
).
70.
F.
Mormann
,
K.
Lehnertz
,
P.
David
, and
C. E.
Elger
, “
Mean phase coherence as a measure for phase synchronization and its application to the EEG of epilepsy patients
,”
Physica D
144
,
358
369
(
2000
).
71.
J.
Gotman
, “
Interhemispheric interactions in seizures of focal onset: Data from human intracranial recordings
,”
Electroencephalogr. Clin. Neurophysiol.
67
,
120
(
1987
).
72.
M. E. J.
Newman
, “
The structure and function of complex networks
,”
SIAM Rev.
45
,
167
256
(
2003
).
73.
D. J.
Watts
and
S. H.
Strogatz
, “
Collective dynamics of ‘small-world’ networks
,”
Nature
393
,
440
442
(
1998
).
74.
J. C.
Reijneveld
,
S. C.
Ponten
,
H. W.
Berendse
, and
C. J.
Stam
, “
The application of graph theoretical analysis to complex networks in the brain
,”
Clin. Neurophysiol.
118
,
2317
2331
(
2007
).
75.
M.
Rubinov
and
O.
Sporns
, “
Complex network measures of brain connectivity: Uses and interpretations
,”
NeuroImage
52
,
1059
1069
(
2010
).
76.
D. S.
Bassett
and
E. T.
Bullmore
, “
Small-world brain networks revisited
,”
Neuroscientist
23
,
499
(
2017
).
77.
M. E. J.
Newman
, “
Scientific collaboration networks. II. Shortest paths, weighted networks, and centrality
,”
Phys. Rev. E
64
,
016132
(
2001
).
78.
S.
Maslov
,
K.
Sneppen
, and
A.
Zaliznyak
, “
Detection of topological patterns in complex networks: Correlation profile of the internet
,”
Physica A
333
,
529
540
(
2004
).
79.
S. J.
Schiff
,
D.
Colella
,
G. M.
Jacyna
,
E.
Hughes
,
J. W.
Creekmore
,
A.
Marshall
,
M.
Bozek-Kuzmicki
,
G.
Benke
,
W. D.
Gaillard
,
J.
Conry
, and
S. R.
Weinstein
, “
Brain chirps: Spectrographic signatures of epileptic seizures
,”
Clin. Neurophysiol.
111
,
953
958
(
2000
).
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