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Cite this article: Voutsa V et al. 2021
Two classes of functional connectivity
in dynamical processes in networks. J. R. Soc.
Interface 18: 20210486.
https://doi.Org/10.1098/rsif.2021.0486
Received: 12 June 2021
Accepted: 13 September 2021
Subject Category:
Reviews
Subject Areas:
biocomplexity
Keywords:
scale-free graphs, modular graphs,
random graphs, synchronisation,
excitable dynamics, chaotic oscillators
Author for correspondence:
Marc-Thorsten Hütt
e-mail: m.huett@jacobs-university.de
Electronic supplementary material is available
online at https://doi.org/10.6084/m9.figshare.
c.5648010.
THE ROYAL SOCIETY
PUBLISHING
Two classes of functional connectivity
in dynamical processes in networks
Venetia Voutsa1
, Demian Battaglia2,3
, Louise J. Bracken5
, Andrea Brovelli4
,
Julia Costescu5
, Mario Diaz Muhoz6
, Brian D. Fath7
'8
'9
, Andrea Funk10
'11
,
Mel Guirro5
, Thomas Hein1 0 , 1 1
, Christian Kerschner6,9
, Christian Kimmich9,12
,
Vinicius Lima2,4
, Arnaud Messe13
, Anthony J. Parsons5
, John Perez5
,
Ronald Poppl15
, Christina Prell14
, Sonia Recinos10
, Yanhua Shi9
,
Shubham Tiwari5
, Laura Turnbull5
, John Wainwright5
, Harald Waxenecker9
and Marc-Thorsten Hutt1
department of Life Sciences and Chemistry, Jacobs University Bremen, 28759 Bremen, Germany
2
Aix-Marseille Universite, Inserm, Institut de Neurosciences des Systemes (UMR 1106), Marseille, France
3
University of Strasbourg Institute for Advanced Studies (USIAS), Strasbourg 67083, France
4
Aix-Marseille Universite, CNRS, Institut de Neurosciences de la Timone (UMR 7289), Marseille, France
department of Geography, Durham University, Durham DH1 3LE, UK
department of Sustainability, Governance and Methods, Modul University Vienna, 1190 Vienna, Austria
department of Biological Sciences, Towson University, Towson, Maryland 21252, USA
advancing Systems Analysis Program, International Institute for Applied Systems Analysis, Laxenburg 2361,
Austria
department of Environmental Studies, Masaryk University, 60200 Brno, Czech Republic
10
lnstitute of Hydrobiology and Aguatic Ecosystem Management (IHG), University of Natural Resources and
Life Sciences Vienna (BOKU), 1180 Vienna, Austria
"wasserCluster Lunz - Biologische Station GmbH, Dr. Carl Kupelwieser Promenade 5,3293 Lunz am See,
Austria
12
Regional Science and Environmental Research, Institute for Advanced Studies, 1080 Vienna, Austria
"Department of Computational Neuroscience, University Medical Center Eppendorf, Hamburg University,
Germany
"Department of Cultural Geography, University of Groningen, 9747 AD, Groningen, The Netherlands
^Department of Geography and Regional Research, University of Vienna, Universitatsstr. 7,1010 Vienna, Austria
VV, 0000-0001-6079-0292; MG, 0000-0003-1565-5686; TH, 0000-0002-7767-4607;
AM, 0000-0001-9081-3088; YS, 0000-0001-7575-6177; LT, 0000-0002-3307-1214;
M-TH, 0000-0003-2221-423X
The relationship between n e t w o r k structure a n d d y n a m i c s is one of the most
extensively investigated problems i n the theory of c o m p l e x systems of recent
years. U n d e r s t a n d i n g this relationship is o f relevance to a range
of d i s c i p l i n e s — f r o m neuroscience to g e o m o r p h o l o g y . A major strategy of
investigating this relationship is the quantitative c o m p a r i s o n of a representation
of n e t w o r k architecture (structural connectivity S C ) w i t h a
(network) representation of the d y n a m i c s (functional connectivity F C ) .
Here, w e s h o w that one c a n d i s t i n g u i s h t w o classes of functional connect
i v i t y — o n e based o n simultaneous activity (co-activity) of nodes, the other
based o n sequential activity o f nodes. W e delineate these t w o classes i n
different categories of d y n a m i c a l processes—excitations, regular a n d chaotic
oscillators—and p r o v i d e examples for S C / F C correlations of b o t h classes i n
each of these models. W e e x p a n d the theoretical v i e w of the S C / F C relationships,
w i t h conceptual instances o f the S C a n d the t w o classes of F C f o r
v a r i o u s application scenarios i n g e o m o r p h o l o g y ecology systems b i o l o g y
neuroscience a n d socio-ecological systems. Seeing the organisation of
d y n a m i c a l processes i n a n e t w o r k either as governed b y co-activity o r b y
sequential activity a l l o w s u s to b r i n g some order i n the m y r i a d of
observations relating structure a n d function of c o m p l e x networks.
© 2021 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution
License http://creativecommons.0rg/licenses/by/4.O/, which permits unrestricted use, provided the original
author and source are credited.
1. Introduction
The relationship between network structure a n d dynamics
has been at the forefront of investigation i n the field of complex
systems d u r i n g the past decades, w i t h networks serving as
p o w e r f u l abstract representations of real-world systems. H o w ever,
a solid theoretical understanding of the generic features
relating network structure a n d d y n a m i c s is still missing. Here,
our strategy of investigating these features is v i a the quantitative
comparison of network architecture (structural connectivity,
SC) w i t h a network (or matrix) representation of the dynamics
(functional connectivity, F C ) . W e establish k e y relationships
using simple m o d e l representations of dynamics: excitable
dynamics represented b y a stochastic cellular automaton,
coupled phase oscillators, chaotic oscillators represented b y
coupled logistic maps. W e validate these relationships i n
coupled F i t z H u g h - N a g u m o oscillators, i n the excitable a n d
the oscillatory regimes. Furthermore, w e give examples of
h o w the t w o classes of F C can be applied to various application
domains, i n w h i c h networks play a prominent role.
The simplest w a y of representing time series of d y n a m i c a l
elements as a network is to compute pairwise correlations.
Often, one also k n o w s about the 'true' or 'static' connectivity
of these d y n a m i c a l elements beforehand. The statistical question
then arises i n a natural w a y , whether the k n o w n
network (SC) a n d the network d e r i v e d f r o m the d y n a m i c a l
observations (FC) are similar. A s w e w i l l see i n the applications,
functional connectivity c a n either be thought of as
d y n a m i c a l similarities of nodes o r flows (of material, activity,
information, etc.) connecting t w o nodes.
The simplicity of the d y n a m i c s i n c l u d e d i n our investigation
allows us to w o r k w i t h this correlation-based approach. In case
of a large heterogeneity of d y n a m i c a l elements, v e r y noisy
dynamics, poor statistics (temporal sampling) or incomplete
information, more sophisticated representations of d y n a m i c a l
relationships a m o n g nodes are required [1-5].
Originating i n neuroscience [6], research into S C / F C
correlations has become a p r o m i s i n g marker for changes i n systemic
function a n d a means for exploring the principles
u n d e r l y i n g the relationship between network architecture
a n d d y n a m i c s — i n systems b i o l o g y [7,8], social sciences
[9-11], g e o m o r p h o l o g y [12-14] a n d technology [15-17], just
to name a few of the application areas.
S u c h S C / F C relationships are at the same time markers
for certain forms of systemic behaviour (e.g. a loss of S C / F C
correlation m a y indicate pathological brain activity patterns
[18]) a n d h i g h l y informative starting points for a mechanistic
understanding of the system (e.g. revealing h i g h l y connected
elements—hubs—as centres of self-organized excitation
waves i n scale-free graphs [19,20]).
W h i l e the systemic implications a n d the key results have
been reviewed elsewhere [21], here w e w o u l d like to s h o w
that across a range of d y n a m i c a l processes a n d n e t w o r k
architectures some f u n d a m e n t a l c o m m o n principles exist.
W e argue that one needs to distinguish between t w o types
of functional connectivity, one related to synchronous activity
(or co-activation), the other related to chains of events
(or sequential activation). A system, like phase oscillators
[22-24], f a v o u r i n g o n e type of functional connectivity (for
this example, synchronization) c a n also d i s p l a y the other
type of S C / F C correlations under certain conditions.
A c o n d i t i o n here is characterized b y the n e t w o r k type, the
strength of the c o u p l i n g of the d y n a m i c a l elements a n d the
choice of further (intrinsic) parameters of each of the d y n a m i cal
elements. Here, w e s h o w m a n y examples of transitions
f r o m one type of S C / F C correlations to another type u n d e r
changes of these conditions.
Stylized models of d y n a m i c s often offer a deep mechanistic
understanding of the d y n a m i c a l processes a n d phenomena
and, i n particular, help discern h o w network architecture
shapes the d y n a m i c a l behaviour. This point is illustrated b y
the intense research over the past decades o n networks of
coupled phase oscillators as a stylized m o d e l of oscillatory
dynamics. T w o prominent examples of this line of investigation
are the topological determinants of synchronizability
[23,25], the lifetimes of intermediate synchronization patterns
i n a time course towards full synchronization a n d their
relationships to the network's m o d u l a r organization [22].
Remarkably, it is precisely this f o r m a l distinction between
functional connectivity based o n co-activation a n d sequential
activation that is often h a r d to discriminate i n more detailed
(e.g. continuous) m o d e l s [26] a n d experimental data [27].
In the case of S C / F C correlations, the best-investigated
stylized m o d e l is the—three-state cellular automaton—SER
m o d e l of excitable d y n a m i c s [19,28,29]. K e y results include
that the topological overlap [30] is h i g h l y associated w i t h functional
connectivity based o n simultaneous activity, F C s m v a n d
that v i a this m e c h a n i s m — a clustering of high topological overlap
values w i t h i n m o d u l e s — m o d u l a r graphs display h i g h S C /
F C correlations, w h i l e scale-free graphs tend to display low, or
even systematically negative S C / F C correlations w i t h this definition
of F C [28,30]. Furthermore, a large asymmetry of the
sequential activation matrix (which is the foundation of functional
connectivity based o n sequential activation, FCseq) can
be associated w i t h self-organized waves a r o u n d hubs [20].
A d d i t i o n a l l y the role of cycles for organizing S C / F C correlations
has been investigated [31] a n d i n the deterministic
limit of the m o d e l , a theoretical framework for predicting S C /
F C correlations has been established [30].
A s a first illustration of the tremendous p o w e r of p r o b i n g
networks w i t h v a r i o u s types of d y n a m i c s , i n order to understand
h o w n e t w o r k architecture determines some of the
d y n a m i c a l features, i n figure 1, w e s h o w snapshots of d y n a m
i c a l states for three real-life n e t w o r k s c o m i n g f r o m different
d o m a i n s — N e u r o s c i e n c e (the macaque cortical area n e t w o r k
f r o m [32]), systems b i o l o g y (the core metabolic system of
the g u t bacterium Escherichia coli f r o m [33]) a n d social
sciences (intra-organizational n e t w o r k of skills awareness i n
a c o m p a n y f r o m [34])—under the action of three types of
dynamics—excitable d y n a m i c s , phase oscillators, the logistic
m a p as a n example of a chaotic oscillator.
The real-life networks s h o w n i n figure 1 c a n all be considered
examples of structural connectivity. T h e detailed
description of the structure of these networks is g i v e n i n
the electronic supplementary material.
A n important question a r o u n d figure 1 is whether the
three types of d y n a m i c s are plausible for the networks at
h a n d . First, w e w o u l d like to emphasize that the strategy of
our investigation is to probe n e t w o r k architectures b y
simple prototypes (or v e r y stylized forms) of d y n a m i c s ,
rather than d e v i s i n g realistic m o d e l s of the most plausible
f o r m of d y n a m i c s for each of these networks.
In the case of the cortical area network, the excitable
d y n a m i c s as w e l l as the oscillatory d y n a m i c s can be seen as
stylized b u t plausible d y n a m i c a l probes a n d , i n fact, those
have been p r e v i o u s l y e m p l o y e d to explore such n e t w o r k
neural network metabolic network social network
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Figure 1. An illustrative example of applying different categories of dynamical processes to real networks with different structures. Neural network: macaque cortical
area network from [32]. Metabolic network: core metabolic system of the gut bacterium Escherichia coli from [33]. Social network: skills awareness network from
[34]. SER: The mean activity of each node after 1000 timesteps, with a rate of spontaneous activity f= 0.001 and a recovery probability p = 0.1. Phase oscillators:
The average effective frequency of each node for ten simulations of length T= 200 initialized with a uniform distribution of eigenfrequencies. Logistic map: The
average standard deviation of the time series of each node for 10 simulations of 500 timesteps with the parameter R for each node randomly selected from a
uniform distribution with ffmin = 3.7 and /?m a x = 3.9.
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structures [20,35-37]. B u t also chaotic d y n a m i c s , as i n the
third r o w of figure 1, have been used to s t u d y neuronal connectivity
patterns [38,39].
In the case of metabolic networks, synchronous activity
patterns, a n d hence c o u p l e d phase oscillators, are a plausible
f o r m of d y n a m i c s (see, for example, the arguments i n [40],
where enzymes are described as cyclically operating devices,
as w e l l as the prominent usage of correlation networks i n
metabolomics [41^13]). A more pathway-oriented v i e w of
m e t a b o l i s m m i g h t emphasize the propagation o f activity
a n d , hence, w o u l d b e closer to t h e excitable d y n a m i c s
s h o w n i n the first r o w of figure 1. Chaotic oscillators are
clearly less relevant for this application d o m a i n .
Interaction d y n a m i c s , contact d y n a m i c s a n d information
f l o w i n a corporate setting unite aspects of excitable
d y n a m i c s (as i n t h e case of r u m o u r spreading, [44]) o r
synchronization [45,46]. B u t also chaotic d y n a m i c s have
been e m p l o y e d to m o d e l decision d y n a m i c s a n d activity i n
corporate settings [47-49].
The three m a i n messages of the illustration of d y n a m i c s o n
real-life networks s h o w n i n figure 1 are: (1) T h e representation
of complex systems as networks enables the p r o b i n g of such
complex structures w i t h dynamics. (2) Different networks
react differently to o n e type of dynamics. This general point
can be seen for example i n figure 1 b y f o l l o w i n g one type of
d y n a m i c s (e.g.excitable dynamics; first r o w i n figure 1) across
the three networks a n d observing that groups of nodes
acting together (similar colour, representing similar d y n a m i c a l
states) c a n be either i n the periphery or i n the centre of these
network representations. (3) A given network reacts differently
to different d y n a m i c a l probes. This general feature c a n be seen
b y f o l l o w i n g a single network across different types of
d y n a m i c s (columns i n figure 1). Regions i n the g r a p h w i t h a
similar d y n a m i c a l state (same colours) for one d y n a m i c s look
heterogeneous (different colours) for another d y n a m i c s . A l s o ,
similarities occur. T h e periphery a n d the centre of the networks
tend to behave differently i n a l l the examples o f d y n a m i c s
s h o w n i n figure 1.
It is obvious that s u c h a n illustration c a n o n l y p r o v i d e a
single snapshot of the diverse d y n a m i c s possible o n such
networks, even for a single type of dynamics, as the internal
parameters at each node, as w e l l as the c o u p l i n g type a n d
strength a m o n g t h e m c a n have different values. I n the following,
w e w a n t to further explore the systematic changes of these
d y n a m i c a l patterns as a function o f network architecture,
coupling a n d internal d y n a m i c a l parameters a n d h o w this
theoretical framework c a n be a p p l i e d to various disciplines.
2. Results and discussion
W e create different instances that indicate the behaviour of the
two classes of F C u s i n g various numerical schemes. T h e
means of enhancing or destroying S C / F C correlations c a n be
structural (i.e. d r i v e n b y network architecture) or d y n a m i c a l
(induced b y changing the parameters o f the d y n a m i c a l
model). T h e investigation i s organized a r o u n d the f o r m of
change: §2.1 topological changes, §2.2 changes i n the c o u p l i n g
strength, §2.3 changes of the intrinsic parameters of the i n d i v i d ual
elements. In §2.4, w e illustrate these principles i n a case
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Figure 2. (a) SC/FCsjm and SC/FCseq correlations across the randomization of a modular network, (b) Illustration of the SC and the FCsjm, FCseq matrices for three
network cases, pointed out by the dashed vertical black lines on the left figure (original modular network, 30% randomized network and completely randomized
network). The dynamical model used for the FC is the SER model (parameters: fmax = 10, NR = 10000, p = 0.1, f= 0.001.)
study o n a network of coupled F i t z H u g h - N a g u m o oscillators
i n the excitable a n d oscillatory regimes. U s i n g the three
examples f r o m figure 1, i n §2.5, w e s h o w the behaviour of
S C / F C correlations o n these real-world networks.
2.1. Topological changes
The first part of our investigation is related to the effect of topology
i n the S C / F C correlations. W e started w i t h networks w i t h
a distinct structure (modular graph, hierarchical graph, regular
graph), w h i c h w e gradually destroyed either b y r a n d o m i z i n g or
b y rewiring the initial network (see M e t h o d s , §5).
Figure 2 introduces the c o m p a r i s o n of structural connectivity
a n d functional connectivity o n the matrix level, b y
depicting the adjacency matrices o f t w o networks, together
w i t h examples of the corresponding functional connectivity
matrices d e r i v e d f r o m d y n a m i c s (here: the co-activation a n d
sequential activation matrices obtained f r o m simulations of
the S E R m o d e l ; see M e t h o d s , §5). This matrix v i e w o n S C /
F C relationships is similar to fig. 1 i n [26] a n d fig. 1 i n [28]
a n d a l l o w s u s to v i s u a l l y discern the strong positive correlation
between the adjacency matrix a n d the co-activation
matrix i n the case o f the m o d u l a r g r a p h (first row) a n d the
apparent lack thereof i n the more r a n d o m g r a p h (second
row), for w h i c h w e , however, can v i s u a l l y perceive a n agreement
between the adjacency matrix a n d the sequential
activation matrix. S o , here a change i n n e t w o r k t o p o l o g y
goes a l o n g w i t h a change f r o m o n e type of S C / F C correlations
(co-activation to sequential activation). This is the
p h e n o m e n o n w e set out to explore further i n the following.
In the electronic s u p p l e m e n t a r y material, figures S2 a n d
S3 s h o w the same matrix view, but for c o u p l e d phase oscillators
a n d logistic m a p s , respectively. I n figure S2 (phase
oscillators) i n the electronic supplementary material, a
v i s u a l inspection clearly s h o w s that the S C / F C correlations
based o n sequential activation are m u c h weaker than the
ones based o n co-activation. A l s o , S C / F C s j m remains visibly
h i g h d u r i n g r a n d o m i z a t i o n . In figure S3 i n the electronic supplementary
material (logistic maps), the lack o f correlation
between co-activation a n d the m o d u l a r structure is clearly
seen, as is the (faint, but discernible) agreement of this m o d ular
structure w i t h sequential activation. C a r e f u l v i s u a l
inspection also reveals the persisting positive S C / F C s e q correlations,
as w e l l as the negative S C / F C g j m correlations,
under r a n d o m i z a t i o n of the m o d u l a r network. I n figure S4
i n the electronic supplementary material examples of
space—time plots for single runs o f the chaotic d y n a m i c s are
s h o w n a n d this thus provides a microscopic v i e w o f the
results s u m m a r i z e d i n figure 2.
In figure 3, w e g o f r o m rather structured n e t w o r k topologies
to rather unstructured r a n d o m n e t w o r k topologies.
Figure 3 supports the v i s u a l impression f r o m the matrix
examples s h o w n i n figure 2 b y s h o w i n g the t w o types of
S C / F C correlations as a function of n e t w o r k r a n d o m i z a t i o n
procedures, f o r the S E R m o d e l ( w h i c h w a s also used i n
figure 2), as w e l l as t w o other types of d y n a m i c s , n a m e l y
c o u p l e d phase oscillators a n d c o u p l e d logistic m a p s i n the
chaotic regime (see M e t h o d s , §5). It s h o u l d b e noted that
each o f these d y n a m i c a l m o d e l s has been instrumental i n
the past i n a d v a n c i n g o u r u n d e r s t a n d i n g of f u n d a m e n t a l
relationships between n e t w o r k architecture a n d d y n a m i c s
(see, e.g. [19,30,50] for the S E R m o d e l , [22,24] f o r c o u p l e d
phase oscillators, a n d [51,52] for the logistic maps).
For the S E R m o d e l , w e see a trend that structured topologies
favour h i g h S C / F C correlations o f b o t h types, whereas
unstructured r a n d o m networks favour h i g h S C / F C s e q correlations.
W e c a n also see that S C / F C s j m is v e r y sensitive to
topological changes, i n contrast to S C / F C s e q , w h i c h , i n this
case, shows a more stable behaviour. The networks of coupled
phase oscillators behave i n almost the opposite way, where
co-activation (rather than sequential activation) is favoured
b y r a n d o m network structures a n d shows a more stable
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0.2 0.4 0.6 0.8 1.0
SC/FCsim
SC/FQ„
0.2 0.4 0.6 0.8 1.0
5C/FCsim5C/FCsim
0.2 0.4 0.6 0.8
randomization
1.0
Figure 3. SC/FCsjm and SC/FCseq correlations across the range of randomization/rewiring processes. First column: randomization of a modular graph. Second
column: rewiring of a regular graph. Third column: randomization of a hierarchical graph in the three models. First row: SER model; parameters: fmax = 10,
NR (over different initial conditions) = 10000, Wff (over different initial graphs) = 10, p = 0.1, f= 0.001. Second row: coupled phase oscillators; parameters:
fmax = 50, Wff (over different initial conditions) = 100, NR (over different initial graphs) = 10, co e (0,1), k = 10, a = 0.25, u e (0,1). Third row: logistic map
(chaotic oscillators); parameters: fmax = 500, Wff (over different initial conditions) = 50, R e (3.7, 3.9), k = 2.
behaviour under topological changes. I n the case o f chaotic
oscillators, the details about the n e t w o r k architecture a n d the
selection o f the type of c o u p l i n g matter. For this case, the
transition f r o m structured to unstructured networks does not
affect the S C / F C s e q , but leads to strong negative correlations
of the S C / F C s m v The hierarchical n e t w o r k is the o n l y one,
though, i n w h i c h the destruction of the m o d u l a r i t y is n o t
revealed f r o m the dynamics. For this graph, all the d y n a m i c a l
models s h o w that the r a n d o m i z a t i o n does not essentially affect
the value of S C / F C correlations, instead constant, l o w positive
correlations o f S C / F C s e q a n d constant, l o w negative correlations
o f S C / F C s j m a r e m a i n t a i n e d d u r i n g t h e
r a n d o m i z a t i o n process.
2.2. Changes in coupling strength
The second set o f o u r n u m e r i c a l experiments pertains to
changes i n the c o u p l i n g strength a m o n g nodes. For this type
of change, o n l y the models of the phase a n d chaotic oscillators
can be used, as the S E R m o d e l — i n the f o r m used here—has no
c o u p l i n g parameter ( w h i c h c o u l d , however, b e introduced v i a
a relative excitation threshold, as i n [29,53]).
Figure 4 shows that i n the case of c o u p l e d phase oscillators,
all the n e t w o r k architectures stabilize S C / F C g ^ against
changes o f c o u p l i n g strength. Large values o f c o u p l i n g
strength lead t o r a p i d synchronization (co-activity o f the
nodes) a n d therefore to inadequate a m o u n t o f information
for the sequential activation. A s a result, seeing the structure
of the n e t w o r k through the d y n a m i c s u s i n g the sequential activation
is, i n this case, not possible. For the chaotic oscillators,
w e observe general trends o f increasing S C / F C s e q w i t h
increasing c o u p l i n g , reaching a m a x i m u m , a n d gradually
decreasing for further increase of the c o u p l i n g , essentially
across all n e t w o r k architectures (figure 4).
2.3. Changes in intrinsic parameters
Each d y n a m i c a l m o d e l i s characterized b y specific intrinsic
parameters that determine the b e h a v i o u r o f the i n d i v i d u a l
elements a n d of the system, too. Changes i n the values of
the intrinsic parameters m a y result i n drastic changes to the
functional connectivity. I n this part o f the investigation, the
t w o types of functional connectivity are studied as a function
of such intrinsic parameters. W e are here attempting t o
address the f o l l o w i n g question: i s there at least one class of
the functional connectivity that c a n s u r v i v e u n d e r the
changes of a d y n a m i c a l parameter of the m o d e l ? O r relatedly
is it possible to observe the structure of the n e t w o r k t h r o u g h
the d y n a m i c s even i f w e are consistently c h a n g i n g a n
intrinsic parameter?
The stochastic S E R m o d e l is characterized b y the recovery
probability p, that determines if a node i n the refractory state
w i l l return to the susceptible state. For the phase oscillators,
w e use the range of natural frequencies as the intrinsic parameter.
The logistic m a p has o n l y one intrinsic parameter, R,
w h i c h defines the d y n a m i c a l behaviour o f the u n c o u p l e d
modul argraph Erdos-Rényi graph Barabási-Albert graph hierarchical graph
LQ
b
i3
Figure 4. SC/FCsjm and SC/FCseq correlations as a function of the coupling strength among the nodes in the two types of oscillators applied to different network
architectures. First column: modular graph. Second column: Erdos-Renyi graph. Third column: Barabasi-Albert graph. Fourth column: hierarchical graph. First row:
coupled phase oscillators; parameters: fmax = 50, Wp = 100 (over different initial conditions), Wp = 10 (over different initial graphs), w e (0,1), k = 10, a= 0.25,
u e (0,1). Second row: logistic map (chaotic oscillators); parameters: fmax = 500, Wp = 50, /?e (3.7, 3.9), k=2.
modular graph Erdos-Rényi graph Barabási-Albert graph hierarchical graph o
CO
(-0.1,0.1) (-0.1,0.1) (-0.6,0.6)
0.4
0.3
0.2
0.1
0
-0.1
-0.2
- SC/FC,h
- S C / F C
0.4
0.3
0.2
0.1
0
-0.1
-0.2
- SC/FC,i (
- SC/FC,
0.4
0.3
0.2
0.1
0
-0.1
-0.2
- SC/FC ,i (
- SC/FC.
0.4
0.3
0.2
0.1
0
-0.1
-0.2
- SC/FC,i (
- S C / F C , , ,
Figure 5. SC/FCsjm and SC/FCseq correlations under changes of dynamical parameters in the three models. First row: SER model with increasing recovery probability
(parameters: fmax =10, Wff = 10 000 (over initial conditions), Ng = 10 (over different initial graphs), f= 0.001). Second row: coupled phase oscillators under a
widening of the distribution of the natural frequencies (parameters: fmax = 50, Wff = 100 (over initial conditions), Ng = 10 (over different initial graphs), k = 10,
rj= 0.25, u e (0,1)). Third row: logistic map under a shift of /?average of the distribution of R within the interval (3.7, 3.9), keeping the width equal to 0.2
(parameters: fmax = 500, Wff = 50 (over initial conditions), k=2). Four network architectures were used for each model: First column: modular graph. Second
column: Erdos-Renyi graph. Third column: Barabasi-Albert graph. Fourth column: hierarchical graph.
oscillator. W e here v a r y the average R such that the u n c o u p l e d
oscillator w o u l d reside i n the chaotic regime (3.7, 3.9).
Figure 5 s h o w s the results of this part of the investigation.
For the S E R m o d e l , w e see that n e t w o r k effects are consistent
across the w h o l e parameter range. W e c a n see that S C / F C s m l
is consistently h i g h f o r the m o d u l a r g r a p h a n d v e r y close
to zero f o r a l l the other graphs, where, i n contrast, t h e
S C / F C s e q h a s positive correlation values. F o r the phase
oscillators, t h e w i d t h o f the frequency distribution
matters: increasing w i d t h leads to a consistent decrease of
S C / F C s m v b u t leaves S C / F C s e q intact i n a l l graphs, except
for the m o d u l a r , i n w h i c h the behaviour of S C / F C s e q is similar
t o S C / F C s m v T h e logistic m a p does n o t s h o w a n y
parameter sensitivity of S C / F C correlations i n the different
n e t w o r k architectures.
2.4. Additional case study
The F i t z H u g h - N a g u m o m o d e l c a n b e u s e d as a case s t u d y
verifying whether o u r previous results translate t o this
(a)
0.7 I
0.6-
0.5 u
0.4-
&
u
0.3 -
0.2-
0.1 -
0 FitzHugh-Nagumo:
excitable FitzHugh-Nagumo: oscillatory
randomization
SC/FC.r a
S C 7 F C e [ ]
- —
() 0.2 0.4 0.6 0 8 1.0 0.2 0.4 0.6 0.8
randomization
1.0
Figure 6. SC/FC correlations from the FitzHugh-Nagumo model in excitable (a) and oscillatory (b) regime while randomizing random modular networks. The blue
curves represent co-activation (i.e. a time window of 1 ms), while the red curves represent sequential activation (i.e. using a time window of 12 ms).
more detailed, more realistic m o d e l . T o this end, w e study the
behaviour of S C / F C correlations as a function of r a n d o m i z i
n g a m o d u l a r g r a p h i n the oscillatory regime (a = 0) a n d i n
the excitable regime (a = 0.8). T h e results of this more
complicated m o d e l s h o w n i n figure 6 c o n f i r m the general
observations d e r i v e d f r o m the t w o corresponding m i n i m a l
models: the excitable d y n a m i c s enhance the S C / F C s e q
across the transition of a m o d u l a r to a n E r d o s - R e n y i graph,
whereas oscillations favour the S C / F C s m l across the
r a n d o m i z a t i o n process.
2.5. SC/FC correlations in real networks
W e c a n n o w return to the real-life n e t w o r k s f r o m figure 1 a n d
study the t w o types of S C / F C correlations i n these networks
as a function of the intrinsic parameters of the d y n a m i c a l
models, as d o n e i n figure 5 f o r the abstract n e t w o r k architectures.
T h e results are s u m m a r i z e d i n figure 7. R e g a r d i n g
the S E R m o d e l , w e see h i g h S C / F C s e q correlations for the
neural system a n d , i n contrast, h i g h S C / F C s j m correlations
for the social system, u n d e r the increase of the recovery
probability, w h i l e i n the case of the metabolic system, the
type of S C / F C correlation that is higher depends strongly
on the parameter value. F o r the phase oscillators, w e see
initially h i g h correlations that approach zero v a l u e as w e
increase the w i d t h of the eigenfrequencies distribution,
w i t h the S C / F C g ^ to have constantly higher values. I n the
metabolic network, d o m i n a n t a n d relatively strong a n d
stable S C / F C s j m appears u n d e r the same changes of the
o) distribution, whereas the zero values of S C / F C g ^ for the
n a r r o w distributions give place to strong negative correlations
as w e m o v e to w i d e r distributions. T h e behaviour of
the social n e t w o r k is similar to the neural o n e , b u t w i t h
lower S C / F C correlation values. T h e results for the logistic
m a p are d o m i n a t e d b y S C / F C s e q correlations, independent
of n e t w o r k architecture a n d parameter value.
connectivity exist i n this d o m a i n a n d (3) h o w the t w o types
of functional connectivity appear i n this setting.
Throughout this investigation, w e have the f o l l o w i n g
scenario i n m i n d : given a network (structural connectivity)
a n d d y n a m i c a l processes for each of the nodes, w e analyse
the time series observed at each node a n d derive relationships
a m o n g the nodes (functional connectivity) i n order to understand
h o w network architecture determines o r shapes the
d y n a m i c a l relationships a m o n g nodes. This interplay of structure
a n d d y n a m i c s is then illustrated b y a n d quantified i n
terms of S C / F C correlations. T h e topic of d y n a m i c s o n
graphs is, of course, m u c h broader than w e describe it
here. T h e clear distinction between (static or s l o w l y changing)
structural c o n n e c t i v i t y — w h i c h serves as 'infrastructure' for
d y n a m i c a l processes—and (often r a p i d l y changing) functional
connectivity is not plausible for a l l applications. A s a consequence,
a debate about S C / F C correlations is not possible i n
important areas of research. Often, i n those disciplines, the
evolution of the network itself under the action of its agents
(nodes) is investigated, therefore w e c a n o n l y conceptualize
the F C i n the context of the evolution of the structure of the netw
o r k . Social network analysis ( S N A ) is the m e t h o d o l o g y of
choice for such situations [54] (see electronic supplementary
material for more details).
W h e n multiple relationships must b e taken into account to
provide a more realistic a n d precise description of a complex
system o r w h e n interactions g o b e y o n d the pairwise level
(with examples f r o m systems biology being protein complexes
or biochemical reactions), hypergraphs [55,56] can serve as a
useful framework for representing these systems. Furthermore,
if the structural network changes o n a similar timescale to functional
connectivity or even under the influence of the functional
dynamics, w e enter the rich field of adaptive networks [57-59].
In this case, inevitably, the topology influences the character of
the collective dynamics of the system, b u t d y n a m i c s affect topology,
too, leading to a continuing interplay between them. This
is of particular relevance i n social-ecological systems (see §3.5).
3. Applications
In this section, w e briefly review some areas of application to
illustrate, (1) h o w structural connectivity c a n b e defined i n
these contexts, (2) w h i c h approaches for d e f i n i n g functional
3.1. Application to geomorphology
W i t h i n h y d r o l o g y a n d geomorphology, the examples of structurally
connected pathways that w e w i l l discuss here are
those that direct the f l o w of water a n d sediment over the
S C / F C s t a
S C / F C s e q
3 u
3 ft.
o U
1 ft
° 5
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•
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network metabolic network social network
0 0.2 0.4 0.6 C
P
.8 1.0
S C / F C s i m
SC/FC.._
seq
(-0.6, 0.6)
0)
— S C / F C s i m
S C / F C s e q
S C / F C s i m
S C / F C s e q
0.2
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SC/FCs e q
•
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SC/FCs i r
SC/FCs e
n
\
SC/FCs i r
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n
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(-0.1,0.1)
0.4"
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S C / F C s i m
SC/FC
seq
(-0.6, 0.6)
3.65 3.70 3.75 3.80 3.85 3.65 3.70 3.75 3.80 3.85
" S C / F C s i m
3.65 3.70 3.75 3.80 3.85
Figure 7. SC/FCsjm and SC/FCseq correlations for the three real-life networks under changes of dynamical parameters in the three models, as in §2.3. First column:
neural network. Second column: metabolic network. Third column: social network. First row: SER model; parameters: fmax = 10, Wff (over different initial conditions)
= 10000, f= 0.001. Second row: coupled phase oscillators; parameters: fmax = 50, Wff=100 (over different initial conditions), /r = 10, <r=0.25, o e
(0,1). Third row: logistic map (chaotic oscillators); parameters: fmax = 500, Wff = 50 (over initial conditions), k=2.
surface a n d w i t h i n the near-surface zone. O n steeper slopes,
these structurally connected pathways are p r e d o m i n a n t l y
controlled b y t h e topography a n d vegetation, whereas
on slopes (ca < 5°) other surface characteristics such as microtopography
a n d soil h y d r a u l i c properties c a n become
relatively important. T h e d y n a m i c a l processes occurring
over this structural template a n d subsequent functional connectivity
are then a n emergent property of these structural
controls i n combination w i t h d y n a m i c a l inputs (e.g. precipitation).
T h e presence o f vegetation also (i) modifies soil
properties, (ii) often has a n associated m i c r o t o p o g r a p h y a n d
(iii) can impede/reduce flows d u e to friction a n d d a m m i n g
effects, s o that there are d y n a m i c feedbacks between t h e
structural a n d functional connectivity [12,13,60].
There are various approaches to assessing structural
connectivity i n h y d r o l o g y a n d g e o m o r p h o l o g y . If w e take a
river network, t h e structural connectivity o f the n e t w o r k
can b e defined based o n the pathways connecting all links
through w h i c h water c a n potentially flow, resulting i n a
graph structure most often i n the f o r m o f a tree [61,62],
w i t h the exception of b r a i d e d streams [63], deltas [64,65] or,
for a more b r o a d example, coastal sediment pathways [66].
Thus, the structural connectivity o f river networks c a n b e
quantified u s i n g , for example, the pairwise connectivity of
its u n d e r l y i n g tree structure—an approach that has been
used b o t h f o r natural a n d synthetic river networks [67].
These synthetic networks are useful as they i n f o r m o u r
mechanistic understanding o f these complex systems. O n e
such example i s o p t i m a l channel networks ( O C N s ) , w h i c h
can be generated for v a r y i n g values of the energy exponent
(a parameter that characterizes the mechanics of erosion processes
i n channel formation) i n order to reveal h o w such
topological factors lead to emergent n e t w o r k properties
[68]. O C N s replicate the major scaling features associated
w i t h river networks a r o u n d the w o r l d [69,70], a n d thus
bridge the g a p between r a n d o m graphs s h o w n i n figure 3
that go f r o m structured to unstructured n e t w o r k topologies
a n d river networks observed i n nature. Furthermore, river
networks have a directional structural template, w i t h links
connecting high-elevation nodes to low-elevation nodes. O n
hillslopes, structural connectivity has been measured based
on the upslope contributing area t o a particular node (e.g.
[71]), a n d o n the combination o f topographically connected
f l o w paths (using f l o w routing algorithms) a n d the presence
of vegetation (measured u s i n g remote sensing techniques)
that intersects ( a n d i n certain environments disconnects)
these f l o w paths (e.g. [72]). S i m i l a r l y i n h y d r o l o g i c a l analysis
of sub-surface flow, structural connectivity of a n e t w o r k of
wells m a y b e determined f r o m the d o w n s l o p e direction of
surface t o p o g r a p h y f r o m a n y w e l l (e.g. [73]).
In these examples f r o m h y d r o l o g y a n d g e o m o r p h o l o g y
w e are concerned w i t h (1) areas that have a similar response
to a d y n a m i c a l probe (e.g. rainfall event) a n d (2) connectivity
of fluxes, i.e. flows o f water a n d / o r sediment that are
transported t h r o u g h the n e t w o r k to a d o w n s l o p e / s t r e a m
location. These t w o types of functional connectivity m a p
onto F C s j m a n d F C s e q , respectively. A p p r o a c h e s used to
measure F C i n h y d r o l o g y a n d g e o m o r p h o l o g y are v a r i e d .
In relation to (1), F C s j m geostatistical analysis is often
used to assess h o w the scales of co-activation change i n
response to a d y n a m i c a l probe. F o r example, one c a n
measure the autocorrelation of soil-moisture content a n d
h o w this changes over time, b o t h i n response to a rainfall
event a n d then d u r i n g a refractory p e r i o d (see for example
[74]). I n the case of sub-surface flows, F C between t w o
wells (nodes) a a n d b is d e e m e d to exist if w e l l b is d o w n slope
of w e l l a a n d the wells are co-activated (i.e. water is
present i n both) (e.g. [75,76]).
In relation to (2), F C s e q is often assessed /inferred based
on gauges w i t h i n a n e t w o r k being activated at a range of
lag times, thus indicating the f l o w of water or sediment
between the t w o locations [77]. Geostatistical analysis has
also been u s e d to quantify h o w F C s e q changes throughout
a f l o o d event [67]. The F C s e q of fluxes t h r o u g h a real or synthetic
river n e t w o r k has also been s i m u l a t e d a n d
incorporated into a d y n a m i c tree approach b y analysing
dye propagation m o d e l s at successive snapshots [78]. S u c h
approaches c o u l d be particularly valuable f o r s t u d y i n g the
little u n d e r s t o o d impact of pulses of sediment [79], nutrients
[80] or other diffuse chemicals [81] transported b y surface
waters. I n the case of sub-surface flows, sequential activity
of the t w o wells m a y be inferred f r o m time-series analysis
of water levels at a range of time lags (e.g. [73]).
Suitable empirical data for measuring these examples of
functional connectivity are relatively scarce, a n d therefore
researchers often t u r n to process-based m o d e l l i n g as a w a y to
quantify b o t h types of functional connectivity. For example,
h i g h spatio-temporal resolution m o d e l l i n g c a n be used to
measure times d u r i n g a storm event w h e n infiltration w i l l be
locally satisfied a n d thus the onset of runoff generation
(excited) or not (susceptible/refractory) d u e to spatial variability
i n infiltration capacity rainfall intensity a n d antecedent
soil-moisture content. F r o m this h i g h spatio-temporal m o d e l ling,
the degree of synchronized functional connectivity of all
locations w i t h i n the m o d e l spatial d o m a i n c a n be derived.
The spatial pattern of these synchronized points i n t u r n determines
the sequential connectivity of runoff a n d sediment flux
[82,83]. F o r example, u s i n g high-resolution process-based
m o d e l l i n g [21] measured the length of connected f l o w paths
on grass a n d shrub hillslopes that h a d v a r y i n g lengths of structurally
connected pathways. In this example, the longer the S C ,
the higher the F C s e q of discharge a n d sediment flux, w h i c h is
similar to that observed i n the case of c o u p l e d phase oscillators
where F C s e q is destroyed w i t h a n increase i n the randomness
of the network.
Whereas some evidence exists for the impact of S C o n
F C s e q at a particular timescale [21], there remains scope to
examine patterns of F C s e q both at increasing l a g times a n d
i n response to d y n a m i c a l probes of different magnitudes for
their impact o n landscape change (topological changes). F u r thermore,
geomorphological assessment of the importance of
c o u p l i n g strength a m o n g nodes remains unexplored. A n
important point to highlight is that timescales of synchronized
versus sequential functional connectivity i n h y d r o l o g y a n d
g e o m o r p h o l o g y are often m a r k e d l y different. The w i d e s p r e a d
synchronization of activity over a spatial range is v a l i d for
a s m a l l time p e r i o d (mins/hours), w h i l e the sequential
propagation of fluxes t h r o u g h the n e t w o r k occurs over
longer time p e r i o d s — h o u r s to days to decades—depending
on the size/configuration/connectivity of the n e t w o r k /
system. S i m i l a r l y earthquake/storm-driven landslides tend
to be synchronized over timescales of h o u r s - d a y s , whereas
the resulting cascade of material t h r o u g h the n e t w o r k is
sequential over significantly longer timescales (e.g. [84-86]).
Likewise, the spatial scales associated w i t h s y n c h r o n i z e d
a n d sequential connectivity tend to differ. F o r example,
nearby nodes often exhibit synchronization, whereas sequential
flux propagation is observed at a larger spatial scale [87].
R o o d events highlight the potential f o r sequential p r o p a gation
of processes over large spatial scales over time
periods of hours to days. I n catastrophic f l o o d i n g i n the L o c k yer
valley i n Q u e e n s l a n d , A u s t r a l i a i n 2011, the h y d r o l o g i c a l
a n d sedimentological connectivity between the channel a n d
the f l o o d p l a i n w a s spatially variable d e p e n d i n g o n the m o r p
h o l o g y of the reach a n d whether it w a s e x p a n d i n g o r
contracting [88]. Hence, i n this example, the organization of
d y n a m i c a l processes i n the n e t w o r k w a s crucial to the
change i n channel m o r p h o l o g y despite assumptions that i n
such a large f l o o d i n g event thresholds f o r connectivity
w o u l d have been exceeded.
3.2. Application to freshwater ecology
O v e r the decades, ecosystem ecology has d e v e l o p e d a considerable
a m o u n t of methodologies f o r n e t w o r k analysis,
w h i c h contributed to the characterization of the e v o l u t i o n
a n d status of ecosystems [89]. T h e structural connectivity
(SC) is represented i n these models b y depicting standing
stocks (e.g. biomass, local c o m m u n i t i e s o r populations,
species, i n d i v i d u a l s , o r habitat patches) as nodes, a n d the
interactions between t h e m (e.g.feeding, the m o v e m e n t of animals
or diseases) as links [89]. W i t h i n landscape connectivity
the spatial structure of river n e t w o r k s (SC) plays a key role i n
structuring ecological patterns [90]. G r a p h representations of
river n e t w o r k s are often m o d e l l e d to resemble the hierarchical
structuring of habitat patches (nodes) a n d the potential
dispersal corridors (links) [91-95].
D y n a m i c a l approaches have not explicitly u s e d the terms
co-activation or sequential activation for describing functional
connectivity. H o w e v e r , some of the notions i n this
paper c a n also be d e d u c e d f r o m d y n a m i c a l approaches of
habitat connectivity already a p p l i e d i n aquatic ecology. T h e
focus o n a n i m a l m o v e m e n t a n d dispersal has been d r i v i n g
the theoretical a n d e m p i r i c a l w o r k i n the past few decades
[62,93,96-100], especially i n the light of fragmented l a n d scapes.
I n m o d e l s of o r g a n i s m a l - e n v i r o n m e n t interactions
based o n landscape's resistance to dispersal [101,102] a n d
i n m o d e l s that i n c l u d e the intrinsic dispersal abilities a n d
limitations of organisms (i.e. i n d i v i d u a l - b a s e d p o p u l a t i o n o r
m e t a c o m m u n i t y m o d e l s [90,103,104]), the m o v e m e n t of
animals is represented as the dispersal of i n d i v i d u a l s f r o m
node to n o d e (analogous to the f l o w of vehicles i n a
transport n e t w o r k [97].
C o m m u n i t y ecologists have l o n g seen i n d i v i d u a l p o p u lations
a n d c o m m u n i t i e s as oscillators [105]. T h e y focused
on the d y n a m i c s of a m o d u l a r network, inferred f r o m the
synchrony between the rate of change of the p o p u l a t i o n
density w i t h i n nodes [105]. A n o t h e r example is the ecohydrological
study of [103], where they p r o p o s e d the concept of
'fluvial synchrograms' to explain patterns of the geography
of metapopulations' synchrony w i t h i n a river network, u s i n g
the case of freshwater fishes of Europe. I n their empirically
d r i v e n approach based o n the geography of synchrony, they
d e v e l o p e d theoretical synchrograms u s i n g simulated timeseries
of species abundance f r o m the spatially explicit
d y n a m i c m e t a c o m m u n i t y m o d e l [103]. These fluvial synchrograms
depicted the decay i n synchrony over E u c l i d e a n ,
watercourse, a n d flow-connected distances [103]. S y n c h r o n y
w a s higher i n populations connected b y direct water f l o w
and decreased faster w i t h the E u c l i d e a n a n d watercourse
distances, h i g h l i g h t i n g the extent of spatial patterns of synchrony
that emerge f r o m dispersal [103]. Other approaches
like the ones of [103,105-108] are examples of investigations
focused o n the effect of the n e t w o r k topology (SC) o n the
synchronous d y n a m i c s of nodes (FC) ( S C / F C relationships).
Representations of synchronous functional connectivity g o
b e y o n d the movement-based approaches a n d c a n include
m o d e l s u s i n g the i n p u t - o u t p u t analysis described i n the ecological
n e t w o r k analysis ( E N A ) section (§3.5.2) (i.e. species
interactions m o d e l s q u a n t i f y i n g predator-prey, mutualistic
or competitive relations [109]; species-resource interactions
m o d e l s q u a n t i f y i n g consumer-resources relations [110-112];
f o o d webs m o d e l s that trace energy m o v e m e n t [113-115];
and nutrient c y c l i n g m o d e l s [116]). I n competitive consumer-resource
systems, consumers can overlap their diet a n d
resources interact w i t h o n e another, w h i c h makes it possible
to visualize t h e m as c o u p l e d oscillators [110,117]. T o explain
the d y n a m i c s of communities i n these systems, Hajian-Forooshani
& V a n d e r m e e r [110] a p p l i e d the e n d u r i n g L o t k a a n d
Volterra equations [111,112] a n d the K u r a m o t o m o d e l [118] i n
a s i m p l i f i e d three-oscillator system. I n this system c o m p o s e d
of three consumers a n d three resources, they measured t w o
distinct types of c o u p l i n g : trophic-coupling (the strength of
cross-feeding) a n d resources-coupling (strength of competition
between resources) [110]. G i v e n a persistent oscillator i n the
L o t k a - V o l t e r r a formulations, trophic-coupling i m p l i e d eventual
synchrony (all oscillators are i n the same point i n circle
space) a n d resource-coupling i m p l i e d asynchrony [110]. T h e
simulations i n b o t h of the t w o models h a d similar results,
suggesting that c o u p l e d oscillators a n d the application of the
notions of the K u r a m o t o m o d e l can provide theoretical contributions
o n ecosystem a n d c o m m u n i t y organization [110].
To illustrate the idea of sequential activation i n freshwater
systems, w e consider examples u s i n g r a n d o m w a l k s .
R a n d o m w a l k is the most c o m m o n approach to simulate anim
a l s ' m o v e m e n t a n d c a n be considered as sequential F C ,
since the sequential steps of their dispersal can p r o v i d e v a l u able
information about the n e t w o r k structure. For theoretical
studies that m o d e l the distribution of local species' persistence
i n time, r a n d o m w a l k w i t h o u t drift is the simplest
baseline d e m o g r a p h i c m o d e l [119]. In originations a n d extinctions
models w o r k i n g w i t h macroevolutionary timescales,
the abundance of a species i n a node has the same probability
of increasing o r decreasing b y o n e i n d i v i d u a l i n each time
step [119]. Then, the increase of one i n d i v i d u a l w i l l represent
the colonization of a free site b y a n i n d i v i d u a l of a n e w
species i n the system, or a r a n d o m l y s a m p l e d i n d i v i d u a l
w i t h i n the c o m m u n i t y [119]. A n a s s u m p t i o n of this m o d e l
to account for l i m i t e d dispersal effects is that o n l y offspring
of the nearest neighbours of the d y i n g i n d i v i d u a l are a l l o w e d
to colonize the e m p t y space [119]. A d d i t i o n a l l y , the local
extinction corresponds to a first passage of a r a n d o m
walker equal to zero, leaving a persistence time distribution
f o l l o w i n g a p o w e r - l a w decay w i t h exponent 3/2 [119]. A s i m pler
alternative m o d e l w i l l be the stochastic patch occupancy
m o d e l ( S P O M ) that describes the presence/absence of a
focal species i n a node (simulates o n l y colonization a n d
not colonization-extinction dynamics) [120,121]. Here, i n a
given structural river n e t w o r k w i t h discrete habitat patches
as nodes, each node has a probability to be c o l o n i z e d b y
species b e l o n g i n g to the regional species p o o l [120]. A t the
starting point of each simulation, a sequential colonization
process starts. F r o m initially o c c u p i e d nodes, o r initially
introduction sites of the n e w species, the e m p t y patches
can become o c c u p i e d i n a sequential manner (successive
snapshots). The potential occupancy of a node w i l l be dependent
o n a chain of colonization events a n d the presence of
u n o c c u p i e d nodes w i t h i n a certain range (only e m p t y
nodes c o u l d become colonized). T h e S C / F C relationships
i m p l i e d b y the aforementioned studies are different f r o m
the ones described i n this paper. T h e dispersal of animals
(which serves as sequential F C ) finally determines the structure
of the n e t w o r k a n d i n the approaches of [115] a n d [114]
this o n e is built u s i n g time-ordered graphs (i.e. continuoustime
M a r k o v chain f o r [115]). T h e m a i n difference is that
this provides a time-resolved v i e w of the d y n a m i c s , whereas
in the approach of the current paper, the time is eliminated b y
suggesting the time-average v i e w o n d y n a m i c s . B r i d g i n g
over these t w o types of time v i e w s o n d y n a m i c s requires
further investigation o n h o w the F C that d e r i v e d f r o m the
temporal graphs contributes to the time average information
of d y n a m i c s . Ecological applications of d y n a m i c a l
approaches, like the ones m e n t i o n e d above, a n d the classification
of the t w o types of F C addressed i n this paper b r i n g
n e w perspectives to assess functional connectivity i n freshwater
ecology. A d d i t i o n a l l y , evaluating the S C / F C
relationships c a n highlight the importance of specific nodes
in facilitating the overall colonization processes, w h i c h c a n
help to estimate a n u m b e r of effective reserves necessary to
achieve a particular conservation goal [103,122-126].
3.3. Application to systems biology
In systems biology, w e f i n d m a n y instances w h e r e a
distinction between simultaneous a n d sequential events i n
networks is made, even t h o u g h the t e r m i n o l o g y of
'functional connectivity' is rarely used. O n the level of gene
regulation, f o r example, temporal p r o g r a m m e s structurally
i m p l e m e n t e d v i a single i n p u t m o d u l e s [127], l e a d i n g to a
'just-in-time' p r o d u c t i o n of proteins f o r specific biological
functions i n a bacterial cell [128] are a n example of a contrib
u t i o n to functional connectivity based o n sequential
activation. N o t e that here the u n w e i g h t e d g r a p h w o u l d
lead to a m i s l e a d i n g relationship between structural a n d
functional connectivity, as structure i n the unweighted g r a p h
w o u l d suggest a co-activation, rather than a sequential
activation. T h e latter, i n fact, is i m p l e m e n t e d v i a distinct
weights f r o m the regulator to the target genes or operons.
A s w e see i n a l l the case studies presented here, such details
matter, w h e n b r i n g i n g these abstract concepts to a specific
d o m a i n of application.
A n o t h e r example is the sequence of events d u r i n g the
yeast cell cycle, w h i c h is h a r d - w i r e d into the corresponding
gene regulatory n e t w o r k [129] a n d can be understood u s i n g
simple, discrete cellular automata-type m o d e l s [130,131],
n a m e l y Boolean n e t w o r k m o d e l s [132].
A more c o m m o n approach i n systems b i o l o g y addressi
n g the relationship of gene expression (or transcriptome)
data—simultaneous measurements of gene activity v i a h i g h throughput
technologies—and the u n d e r l y i n g regulatory netw
o r k is n e t w o r k inference. This approach s u m m a r i z e s a
range of statistical methods to infer the regulatory n e t w o r k
f r o m expression profiles [3,133]. It s h o u l d be noted that this
approach already starts w i t h assumptions about S C / F C
relationships, i n particular, that i n d e e d structural connectivity
can be reconstructed f r o m observations of the d y n a m i c a l
states [134].
In the case of metabolic networks—bipartite graphs of metabolites
a n d biochemical reactions available i n a cell, where
reactions c a n be represented either b y enzymes or b y the
genes encoding these enzymes [33,135,136]—the activity
levels of genes encoding enzymes can be thought of as a representation
of the metabolic state of a cell. These activity
levels are given b y transcriptome data. Statistical analysis
of transcriptome data (either b y repeated measurements—
replicates—or b y contrasting the cellular condition w i t h a
baseline or reference conditions (e.g. mutant gene expression
levels w i t h w i l d - t y p e gene expression levels)) leads to sets of
genes characterizing a cellular state, for example, the sets
of 'upregulated' a n d 'downregulated' genes or the sets of
genes w i t h 'high' or ' l o w ' expression levels w i t h i n a large set
of samples. A possible definition of functional connectivity
then is the i n d u c e d s u b g r a p h of a gene-centric projection of
the metabolic n e t w o r k spanned b y such a gene set derived
f r o m transcriptome data [137].
The connectivity of such a s u b g r a p h c o m p a r e d to rand
o m l y d r a w n gene sets is a convenient a n d frequently
e m p l o y e d measure f o r S C / F C correlations i n this context
[7,138-140], as it addresses the statistical question of h o w
clustered such a gene set (representing functional connectivity)
is w i t h i n a g i v e n (metabolic) n e t w o r k (representing
structural connectivity).
The conceptual m o d e l of functional connectivity b e h i n d
such a n investigation is that of synchronous activity. Distinguishing
between the t w o types of functional connectivity
is challenging, given the current state of 'omics' data i n systems
biology, d u e to the lack of suitable time-resolved data.
This observation is further u n d e r l i n e d b y the fact that predictive
theories of genome-scale metabolic activity (e.g. fluxbalance
analysis, [141,142]) are also based o n a steady-state
assumption. I n order to discriminate between co-activity
a n d sequential activity, o n e needs to resort to time-resolved
models, typically based o n o r d i n a r y differential equations
(ODEs), w h i c h , however, d u e to their often huge n u m b e r of
required parameters are restricted to single p a t h w a y s or
other suitably defined cellular subsystems, rather than the
scale of a w h o l e cell.
B e y o n d these t w o basic situations characterized b y a gene
regulatory n e t w o r k a n d a metabolic n e t w o r k as structural
connectivity, respectively, there are m a n y other examples of
sequential a n d synchronous usages of a g i v e n 'hardware' i n
systems biology. M e t a b o l i c control analysis [143,144] relates
the distribution of control i n biochemical n e t w o r k s to their
structure. Protein interaction networks [145] s u m m a r i z e
h o w selective b i n d i n g patterns (structural) a n d protein
complexes (the d y n a m i c assemblies to execute biological
function) are interlinked. Fermentation processes (e.g. i n
cocoa fermentation) often rely o n a sequential activation of
microbial populations [146].
S u m m a r i z i n g the S C / F C situation i n systems biology i n a
qualitative form, one can conclude that gene regulatory networks
lean towards sequential activation, w h i l e protein interaction
networks functionally lean towards co-activation (protein complexes)
a n d metabolic networks m a y display aspects of both
(steady-state activity versus metabolic pathways).
3.4. Application to Neuroscience
Neuroscience is one of the first disciplinary fields i n w h i c h
the need to formalize notions of S C a n d F C w a s felt. Perhaps
this w a s d u e to the fact that n e t w o r k descriptions i n neuroscience
g o b e y o n d a mathematical representation b u t
correspond to a n actual, concrete reality: neural circuits are
n e t w o r k e d systems, w i t h their 'reticular' ( f r o m L a t i n f o r
'little net') nature debated since at least the t u r n of past century
[147]. N e t w o r k nodes can be, d e p e n d i n g o n the scale of
observation, i n d i v i d u a l neuronal cells (at the micro-scale),
populations i n v o l v i n g thousands of neurons (at the mesoscale),
u p to entire brain regions (at the macro-scale). T h e
relations d e f i n i n g links are different d e p e n d i n g o n the considered
type of connectivity a n d are defined b o t h i n terms
of a n a t o m y of w i r i n g a n d of i n f o r m a t i o n exchange.
It is natural to consider S C i n Neuroscience as the description
of anatomical connections p h y s i c a l l y existing between
n e t w o r k nodes: i n d i v i d u a l synaptic connections f o r m i n g
electrochemical junctions between the o u t w a r d axons a n d the
i n w a r d dendrites of different neurons ( w i t h i n v o l u m e s <
1 m m 3
, already containing approx. 1 0 4
— 1 0 5
neurons); or b u n dles
of long-range connection axons c o u p l i n g together smaller
or larger groups of neurons, separated b y v a r y i n g distances
(approx. 1 - 1 0 m m for mesoscale circuits u p to approx. 10 -
100 m m for macroscale, b r a i n - w i d e networks). A t a l l these
scales, one usually refers to the c o m p i l a t i o n of all structural
connections between probed n e t w o r k nodes as to a connectome
[148,149]. Different techniques must be used to extract
S C information at different scales, even if a systematic review
of t h e m is not possible here. W e can briefly mention, t h o u g h ,
that w e dispose of w h o l e matrices of S C for rodent, n o n h u m a n
a n d h u m a n primate brains [32,146-151], as w e l l as detailed
microcircuit reconstructions [152-154].
Studies of S C i n Neuroscience often revolve a r o u n d : the
search for general architectural [159] or w i r i n g o p t i m i z a t i o n
[160,161] principles i n connectivity, o r the identification of
characteristic motifs of connectivity that are over-represented
w i t h respect to chance-level [162,163] a n d special structures
such as dense clusters at the micro-scale level [164], or cores
a n d 'rich-clubs' at the macro-scale level [151,165]. Recently,
attempts have also been m a d e to use topological data analyses
techniques [166,167] to characterize the 'shape' of
networks b e y o n d the limitations of g r a p h theoretical descriptions,
w h i c h greatly emphasize strictly local or strictly global
aspects but are deficient i n c a p t u r i n g intermediate structures
at arbitrary meso-scales. Other lines of research a i m at l i n k i n g
specific structural motifs to specific functions, as i n the case of
specific arrangements of positively w e i g h e d excitatory connections
a n d negatively w e i g h e d inhibitory connections
a l l o w i n g m o d u l a t i o n s of perception [168] or of specific patterns
of interconnection between cortical layers at different
depths i n the tissue thickness, a l l o w i n g the regulation of sensory
a n d predictive i n f o r m a t i o n f l o w s [169]. Finally, m a n y
efforts have been devoted to identify S C alterations that
m a y be indicative of d e v e l o p i n g a n d progressing
neurological or psychiatric pathologies, a n d thus serve as
diagnostic or predictive biomarkers [165,170]. H o w e v e r , a
comprehensive survey of all these applications largely transcends
the scope of this review.
N o t o n l y are neural network nodes (neurons or p o p u lations)
w i r e d together b y l i v i n g cables, but messages are
continually exchanged along these cables. A l l information processing
related to our perception, o u r cognition a n d our
behaviour is generally believed to arise f r o m the exchange of
'spikes'—propagating pulses of electric depolarization of the
cell membrane, able to elicit neurotransmitter release at synaptic
terminals—between synaptically connected neurons. S u c h
spikes, i n d i v i d u a l l y or g r o u p e d i n m o r e complex spatiotemporal
patterns, represent 'codewords' encoding inform
a t i o n about external a n d internal w o r l d s i n still largely
u n k n o w n languages. Streams of spike-encoded information
are thus copied, transferred a n d m e r g e d between system's
components l i n k e d b y S C , assembling into emergent neural
algorithms that ultimately underlie functions a n d behaviours
[171]. These computations are h i g h l y distributed a n d the c o m m
u n i c a t i o n between system's units that they involve can also
be seen as g i v i n g rise to networks, but this time of functional
rather than structural nature. T w o units are thus defined as
functionally connected if they 'interact'. The p r o b l e m is therefore
to operationally define h o w a n 'interaction' can be
pragmatically measured f r o m observing h o w coordinated
neural activity unrolls through time.
Some measures of F C define 'interaction' as 'synchrony7
between activity fluctuations a n d modulations. This is the
case for instances of so-called resting state F C [172], describi
n g linear covariance between the fluctuations of different
b r a i n regions d u r i n g unconstrained m i n d - w a n d e r i n g , as
revealed b y functional M R I (fMRI). S u c h metrics of connectivity
are a k i n to the first f o r m of F C p r e v i o u s l y described.
G i v e n the remarkable oscillatory components present i n a n
neural activity a n d simultaneously at different frequency
b a n d s [173,174], analyses of synchronization-type F C i n
neuroscience are often conducted i n the spectral d o m a i n ,
tracking coherence a n d phase-locking [175]. I n d i v i d u a l neurons
are not necessarily oscillating a n d can keep firing i n
irregular manner, nevertheless neuronal populations can collectively
oscillate because of the interplay between excitatory
a n d inhibitory currents w i t h i n local recurrent microcircuits
[176]. S u c h collective oscillations produce periodic m o d u lations
of the excitability of neurons w i t h i n large
populations, so that efficient transmission between synaptically
c o u p l e d populations can occur o n l y if their respective
local oscillations are suitably aligned i n phase ('communication-through-coherence'
hypothesis [177]). Thus, t w o
neuronal populations that are structurally connected m a y
become functionally disconnected if their oscillations are,
for example, i n antiphase a n d spikes emitted i n p r o x i m i t y
of the sender population's oscillation peaks reach postsynaptic
neurons at the throughs of the target population's
oscillations a n d hit thus against a w a l l of locally generated
inhibitory blockade, preventing information carried b y
i n p u t spikes f r o m b e i n g transduced into output activity.
U n d e r this hypothesis, it is thus possible to flexibly 'switch
on a n d o f f F C o n top of a static S C link, just b y adjusting
the relative phase of the sender's a n d target's oscillations. A
natural generalization of linear correlation f r o m the time to
the spectral d o m a i n w h e n d e a l i n g w i t h oscillatory neural
activity is inter-regional coherence or phase synchronization
[175]. Coherence i n the g a m m a b a n d (40-100 H z ) between
frontal/prefrontal a n d sensory regions is k n o w n , for instance,
to be boosted i n sensorimotor coordination or attention
[178,179]. Furthermore, F C can also be established between
populations oscillating at different frequencies v i a nonlinear
cross-frequency c o u p l i n g [180]. N o t o n l y cognition but also
pathology can perturb coordinated neural oscillations a n d
the associated F C [181], but once again a detailed coverage
of the use of oscillation analyses for b i o m a r k i n g goes
b e y o n d the limits of the present w o r k .
Other measures of F C go b e y o n d mere synchrony or
correlation—beyond the first f o r m of F C — a n d attempt
reflecting actual causal influence. U n l i k e correlation, w h i c h
is symmetric a n d thus gives rise to undirected graphs of
F C , measures of causal interdependence between time
series of neural activity give rise to directed networks. In
Neuroscience, 'functional' a n d 'effective connectivity' are
sometimes u s e d as distinct terms, w i t h the latter serving as
a F C measure that tracks causality. H o w e v e r , here, w e keep
n a m i n g connectivity of functional type all connectivity
relations that d o not express anatomical interconnection. A
very s i m p l e w a y to account for the direction of interaction
can be to assess the temporal precedence of the 'causing' o n
the 'caused' fluctuation. This can be achieved for instance
b y u s i n g lagged cross-correlation or m u t u a l information
rather than zero-lag correlation (e.g. [183]), since the effect
cannot precede the cause. In this sense, these metrics are
related to the second f o r m of F C , reflecting sequential activation,
as p r e v i o u s l y discussed. H o w e v e r , the
correspondence is o n l y partial, i n this case a n d u n l i k e for
neural F C s of the first f o r m . If exact sequences of neural activation
can be p r o d u c e d b y neural architectures k n o w n as
'synfire chains' [184], they have been o n l y rarely sought for
i n actual recording a n d n e u r o i m a g i n g data [185,186] a n d
not been p u t i n relation w i t h notions of functional c o u p l i n g .
Directed F C measures i n neuroscience are defined operationally
i n terms of time-series-based statistical metrics, rather
than i n terms of explicitly d y n a m i c considerations. Thus,
w h i l e i n m a n y cases the statistically-inferred directed F C
goes, for example, f r o m the phase-leading to the phase-lagging
neuronal p o p u l a t i o n [187,188], i.e. respects a sequentiality criterion,
i n other cases the relation can be inverted, reflecting
nonlinear interactions between populations, as anticipatory
synchronization [189] or heterogeneities i n internal synchrony
levels [190]. M o r e explicitly, causality c o u l d be captured: b y
the detection of remote effects o n distant regions triggered
b y interventions i n local regions (as i n d y n a m i c causal m o d e l ling
[182]); or b y s h o w i n g that consideration of the past activity
of a putative causal source region improves the prediction of
the future activity of a target region, as i n Granger causality
analyses of neural time series [191-193]. Importantly, Granger
causality can also be spectrally d e c o m p o s e d [194], a l l o w i n g the
detection of the contribution of different oscillatory c o m ponents
of neural activity to inter-regional causal influences.
It has thus become possible to observe that causal influences
i n different directions can be mediated b y oscillations i n different
frequency bands, e.g. i n the g a m m a - b a n d (approx. 40 H z )
for b o t t o m - u p a n d i n the beta-band (approx. 20 H z ) for t o p d
o w n information exchange between prefrontal a n d v i s u a l
regions [195,196].
M o r e recently, emphasis has been put o n the fact that F C
networks are not static but change i n very flexible ways through
time, i.e. they are better described as temporal networks [197].
The n e w term of 'chronnectome' has been introduced to stress
how, b e y o n d analyses of the static functional connectome, explicit
consideration of the spontaneous reconfiguration dynamics
of F C over time m a y help to better discriminate cohorts of subjects
a n d patients, b y disentangling temporal f r o m inter-subject
variability [198]. A t the macro-scale, different F C networks are
sequentially recruited along the u n f o l d i n g of cognitive tasks,
potentially signaling different neurocomputational steps
[199,200]. Spontaneous resting state F C networks w a x a n d
wane i n a seemingly stochastic f l o w that is not r a n d o m l y structured
b u t displays characteristic long-term m e m o r y [59] a n d
whose rate of reconfiguration a n d degree of temporal structuring
predict cognitive performance at the single subject level
[201,202]. A t the micro-scale as well, the emergence a n d dissolution
of transient synchronous assemblies of cells w i t h i n
h i p p o c a m p u s a n d enthorinal cortex c a n be m o d e l l e d w i t h
temporal network descriptions [203]. Future studies w i l l be
needed to understand whether this complex F C network
dynamics can be seen as a measurable fingerprint of o n g o i n g
neural computations l i n k e d to functional behaviour [204].
The definitions of S C a n d F C g i v e n i n the previous subsections
are i n principle completely independent: o n e c o u l d
i n d e e d assess the existence of F C based o n the analyses of
multivariate neuronal activity time series w i t h o u t k n o w i n g
anything about the u n d e r l y i n g a n a t o m y a n d S C . This
scenario of a 'perfect separation' between S C a n d F C is
o b v i o u s l y unlikely. H o w e v e r , a n equally naive scenario d o m inates
the discussion of m a n y articles i n the literature i n w h i c h
a structural cause is necessarily sought for to explain a n y
change arising at the level of F C . I n reality the flexibility of
F C o n very fast behavioural time-scales w i t h respect to physiological
processes reshaping S C (at least at the meso- a n d macroscales)
already suggests that F C cannot just be a passive m i r r o r
of the u n d e r l y i n g S C . W e have previously proposed that F C is
the measurable by-product of u n d e r l y i n g collective d y n a m i c s
[187,205]. I n this proposed theoretical v i e w [206], alternative
m o d e s of a system's dynamics, or states w i t h i n the ' d y n o m e '
[207]—or d y n a m i c a l repertoire [208]—of a system w o u l d
give rise to alternative F C configurations o n t o p of a same
u n d e r l y i n g S C (functional multiplicity).
A n a l o g o u s l y circuits w i t h very different S C but that give
rise nevertheless to equivalent d y n a m i c a l m o d e s — a property
k n o w n i n systems neuroscience as functional homeostasis
[209]—would give rise to similar F C (structural degeneracy)
[210]. A n example of degeneracy can be f o u n d i n simulations
of neuronal cultures in vitro, i n w h i c h h i g h clustering of F C is
invariantly f o u n d because of collective n e t w o r k activity
bursting, independently f r o m the simulated culture's S C
being w e a k l y or strongly clustered [211].
It is important to stress that, i n o u r view, F C goes b e y o n d
the level of the node-specific d y n a m i c s a n d includes collective
d y n a m i c a l m o d e s of the entire neural system. This is
m a d e evident i n studies that attempt to predict large-scale
F C i n spontaneous resting-state activity conditions starting
f r o m simulations of SC-connectome-based simulations. I n
these cases, a g o o d fit between simulated a n d e m p i r i c a l F C
is obtained o n l y w h e n the m o d e l is t u n e d to operate close
to a critical point of d y n a m i c operation [212,213], indicating
that F C manifests a peculiar d y n a m i c a l regime, certainly
shaped b u t n o t fully constrained b y structure. For instance,
waves [19,20] or 'connectome harmonics' [214] shape largescale
coordinated activity a n d , hence, F C . T h e importance
of being t u n e d into specific d y n a m i c a l regimes to account
for the qualitative features of large-scale coordinated activity
has also been c o n f i r m e d b y m i n i m a l models, w i t h reduced
realism b u t enhanced possibility to rigorously understand
mechanisms [27,28,215]. A predicted consequence of this
hypothesis is that local perturbation to i n d i v i d u a l nodes
w i t h i n a neural n e t w o r k m a y induce a n e t w o r k - w i d e reconfiguration
of F C , i n c l u d i n g of remote nodes n o t directly
connected to the perturbed n o d e [205]. I n a nonlinear
system the exact same perturbation c a n lead to different
effects i n different d y n a m i c a l states a n d since the F C clearly
depends o n the d y n a m i c a l state of the system, the effects of
a local perturbation can be dependent o n F C rather than o n
S C . O n c e again, computational simulations of v i r t u a l b r a i n
models i n f o r m e d b y e m p i r i c a l S C information b u t a u g m e n ted
w i t h nonlinear b r a i n d y n a m i c s c o n f i r m the v a l i d i t y of
our prediction [216]. V i r t u a l brain m o d e l s t u n e d to regimes
m a x i m i z i n g the degeneracy of their ' d y n o m e ' , s a m p l e d v i a
a noise-driven exploration, also succeed i n qualitatively
reproducing the s w i t c h i n g 'chronnectome' observed i n resti
n g state f M R I [217]. C o m p u t a t i o n a l m o d e l l i n g thus
provides strong evidence i n favour of our hypothesis of flexible
F C being the b y - p r o d u c t of a complex d y n a m i c a l system,
w h o s e behaviour is constrained but not f u l l y determined b y
the u n d e r l y i n g S C . I n other w o r d s , function follows
dynamics, not structure.
3.5. Application to social-ecological systems
Social-ecological systems (SES) are complex adaptive a n d m u l tilevel
(poly centric) systems attributed w i t h interplays
between h u m a n a n d n o n - h u m a n entities (nodes) at spatial
a n d temporal scales [218], through the metabolic flows of
material a n d energy (links). The concept of 'social metabolism',
taken f r o m cellular metabolism, is central i n the study of SES
[219]. N e t w o r k analysis has increasingly been used to study
coupled, o r social-ecological systems [220-224]. Here, the SES
is often depicted as a multilevel social-ecological network
(SEN), where s o c i a l / h u m a n actors comprise o n e network,
natural entities a separate network, a n d flows are captured
between a n d w i t h i n each network level. S u c h multilevel netw
o r k s are m o d e l l e d v i a a stochastic environment, such as a
M u l t i l e v e l E R G M [225]. Here, micro-configurations are specified,
consisting of actors a n d / o r entities f r o m either or b o t h
the t w o networks, such as the tendency for t w o social nodes
to share a coordination tie w h e n both nodes are likewise
l i n k e d to the same natural resource. These microconfigurations
are then m o d e l l e d , alongside other competing tendencies (such
as the general tendency of a network to exhibit transitive closure),
to test hypotheses l i n k i n g S E N patterns to sustainable
(or unsustainable) management practices.
3.5.1. Social networks
The standard S C / F C approaches are u s u a l l y u n c o m m o n i n
social networks because the distinction between the 'hardware'
(structural connectivity) a n d the d y n a m i c s (functional connectivity)
is less clear than i n other fields. H o w e v e r , measuring the
(often rapid) flows a l o n g the edges of a more stable (slowly
changing) network, as the example of the competence perceptions
a n d the everyday information exchanges i n a c o m p a n y
shows, c o u l d be a n interesting perspective for future research
approaches, incorporating t w o theoretical frameworks: social
systems theory [226,227] a n d S N A [54,228,229], e.g. i n the
context of relational events models [230].
b
o
C O
B o t h approaches are debating h o w the internal function (of
networks or systems) c a n produce emergent properties that
transform the structure a n d vice versa. H o w e v e r , i n social
science, these debates are not w i t h o u t tensions (i.e. between
structure a n d agency) a n d criticisms (i.e. b y more conflictoriented
approaches). W e argue that the overlap between
'system' a n d 'network' c o u l d be h e l p f u l for S C / F C i n social
science, specifically to understand the connections between
actors e m b e d d e d i n different social subsystems a n d h o w
u n d e r l y i n g n e t w o r k topologies a m o n g those actors impact
the subsystems (for example, e c o n o m y a n d democracy), a n d
vice versa. Signed contracts between p u b l i c institutions (PI)
a n d private companies (PC) c a n serve as a n illustrative
example. T h e outcome of a relational analysis of p u b l i c procurement
is a multilevel, multi-relational, t w o - m o d e n e t w o r k
of business-government connections, w h o s e nodes a n d
relations are e m b e d d e d i n different social subsystems: the
economic system, the political system a n d the State. Here,
the m a i n focus is the S C / F C relationships that emerge
f r o m the analysis of the procurement network.
F C s e q is the l o n g i t u d i n a l a n d d y n a m i c procurement process,
analysing the sequential configurations to understand
w h y a P I is issuing a contract, whether to o n e a n d not to
another P C . Some of these companies c o u l d be important
market leaders o r potential corrupters. F C s e q is especially
significant i n the case of private actors, as w e assume that
very outstanding degree-peaks of a f e w companies c o u l d
be evidence of a corrupted n e t w o r k d y n a m i c . C o m p a n i e s
w i t h a n extraordinarily h i g h n u m b e r of contracts i n a short
p e r i o d m a y have extraordinary political influences (i.e. interlocks
o r bribes). B o t h the relational positions a n d the
d y n a m i c a l peaks c o u l d correlate directly to structural
n e t w o r k transitions (collapse or even fragmentation) a n d also
to system-related implications, such as the resource distrib
u t i o n i n the economic system o r the decision-making
process i n the State. F C g ^ is the specific linkage-configuration
at each time step: nodes that have active links to the same
node(s) are co-active. T h e co-activation through shared links
is changing i n each time step w h e n a n e w link is created, a n d
an existing link is decaying. F C g ^ applies to P I a n d also to
P C , a n d c a n be seen as a n indicator for strong relational
positions of other nodes (in the o p p o s i n g type) i n the network.
For example, m a n y co-active institutions are a 'pointer' to
influential companies, whether important market leaders or
potential corrupters.
3.5.2. Ecological networks
Ecological n e t w o r k analysis ( E N A ) is a systems-oriented metho
d o l o g y developed b y ecologists to understand whole-system
d y n a m i c s a n d properties [231,232]. This m e t h o d o l o g y is based
on n e t w o r k a n d information theory a n d derives itself f r o m
i n p u t - o u t p u t analysis, m o d e l l i n g ecosystems as a set of
nodes a n d ties (vertices, edges) [233,234]. U n d e r this framew
o r k , species, aggregation of species into functional groups,
or n o n - l i v i n g resource pools are taken as nodes w h i l e the
exchange of material or energy between species is taken as
edges. I n a d d i t i o n , E N A m e t h o d o l o g y has also been w i d e l y
applied to analyse direct a n d indirect exchange of energy
a n d carbon emissions between economic sectors at u r b a n /
country level f r o m a system perspective [235]. This m e t h o d ology
is useful to evaluate system properties such as cycling
index, total t h r o u g h f l o w a n d relational interactions b y pairwise
components i n the system through t h e r m o d y n a m i c a l l y
conserved transactions of a chosen currency [236].
A l t h o u g h not explicitly u s i n g the language of structural
connectivity a n d functional connectivity E N A internal
logics resemble those used i n connectivity science [21].
U n d e r the E N A f r a m e w o r k , S C is defined b y the n u m b e r
a n d position of functional groups—species, aggregation of
species or economic sectors—forming the nodes a n d the
flows of material a n d energy between t h e m (edges) [231].
This set of arrangements defines the n e t w o r k architecture
or n e t w o r k t o p o g r a p h y a n d , therefore, the 'hardware' o n
w h i c h d y n a m i c processes take place, n o r m a l l y represented
w i t h a n adjacency matrix [114,231]. Ecosystems are open,
t h e r m o d y n a m i c , far-from e q u i l i b r i u m systems, w h i c h implies
that they require continual i n p u t f l o w of h i g h - q u a l i t y l o w entropy
energy [237]. O n c e energy enters the system, it is
the structural connectivity that defines the system's overall
d y n a m i c flow-storage patterns [236]. E N A is a p p l i e d to
steady-state systems, therefore capturing, i n a snapshot,
b o t h the structural connectivity a n d the cumulative behaviour
of a g i v e n h i g h l y d y n a m i c network.
N o w , w e t u r n to the functional connectivity u n d e r the
E N A f r a m e w o r k , e m p l o y i n g a basic input-state-output
m o d e l frame. A s o p e n systems require continual i n p u t , a n
ecosystem is sustained b y the d y n a m i c co-activation pulses
entering across the b o u n d a r y . In nature, these pulses c o u l d
be seen as the solar energy received b y the p r i m a r y producers
(multiple i n d i v i d u a l s or m u l t i p l e species, d e p e n d i n g o n
scale). I n this manner, w e interpret co-activation as nodes
sharing a functional similarity, such as trophic level, a n d
thus being charged simultaneously. This is different f r o m
v i e w i n g co-activation as t w o or more attributes to a l i g n for
activation to occur. T h e latter m a y n o t have a direct analogy
i n E N A . In this case, the i n p u t of energy is simultaneous to
several nodes d u e to their inner characteristics (e.g. they a l l
belong to the same trophic level). O n c e co-activation
occurs, the energy/material flows sequentially f r o m n o d e to
node. A l t h o u g h ecological networks have complex connection
patterns i n c l u d i n g cycling, each i n d i v i d u a l sequential
p a t h w a y c a n be 'decomposed' a n d identified as a u n i q u e
carrier of energy matter f r o m initial activation to final dissipation
b e y o n d the system border. These energy flows are the base
of a l l exchanges a n d f o r m the m o d e l structure encompassing a
diversity of nodes a n d trophic levels. The sequential activation
is captured a l o n g these cascading indirect pathways f r o m the
initial co-activation pulse. Therefore, the most straightforward
w a y to visualize functional connectivity based o n the sequential
activation of nodes is w i t h a linear f o o d chain. T h e initial
input of energy triggers the sequential activation of nodes
d o w n the f o o d chain, whereas each component is dependent
on the previous for its f l o w source [237]. E v e n t u a l l y as the
initial pulse travels throughout the m a n y n e t w o r k e d pathways,
it is dissipated, its useful energy spent, c o m i n g to rest outside
the system b o u n d a r y (as higher entropy) a n d completing the
input-state-output triumvirate. Ecological n e t w o r k analysis
can expose some of the interesting properties that emerge i n
the state based o n those i n p u t - o u t p u t relations.
W h a t is particular about ecological, a n d therefore also SES,
systems a n d networks is that o n e major element conditions
both their architecture a n d their functional connectivity over
the l o n g r u n : net energy (as a n indicator of l o w entropy)
[238-241]. A s l o n g as there are h i g h levels of net energy connectivity
( a n d therefore complexity i n terms of nodes a n d
b
o
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functions) c a n increase, as n e w 'agents' o r 'elements' are
attracted to the system or d r a w n into it b y existing agents.
N o d e s that h a p p e n to (or m a n a g e d to) control large
amounts of net-energy flows c a n leverage their relative position
i n the n e t w o r k a n d exert p o w e r over other n e t w o r k
members. This means that s u c h agents then have substantial
p o w e r to adapt the n e t w o r k architecture to their o w n preferences,
e.g. to increase their relative p o w e r [242]. It acts
collectively o n major sets of nodes, thus it contributes to synchronous
F C , b u t it c a n also trigger cascading effects w i t h i n
the network, contributing this w a y to sequential F C . P o w e r
enables agents to exert a certain level of control over other
agents a n d even allows t h e m to eliminate or a d d other
agents or nodes. In particular, they m a y control the distrib
u t i o n of f l o w s as they m o v e t h r o u g h the system.
3.5.3. Social-ecological networks
A rich b o d y of literature o n social-ecological system analysis
focuses o n the structures a n d patterns of interdependent
social a n d ecological interactions (SC), w h i c h are further
associated w i t h p h e n o m e n a of interests like cooperation
and conflict [223,243-246]. M o r e specifically it investigates
the actor-to-actor relationship i n the social system, the ecological
component-to-component interdependencies i n the
ecological system, as w e l l as the actor-to-component relationship
across the social a n d ecological system [247]. Altogether
it forms a m u l t i l e v e l n e t w o r k configuration m a d e of nodes
and links between different system entities. I n terms of S C /
F C relationships, one line of research is exploring h o w certain
social-ecological system configurations c a n facilitate successful
adaptation a n d transformation i n S E S to address
resource management challenges [243,248,249]. B o t h adaptivity
a n d transformability are critical elements of resilience
s t u d y describing the capacity of the interdependent s o c i a l ecological
systems dealing w i t h u n k n o w n o r unforeseen
shocks [250,251].
A l t h o u g h u s i n g different terms, other lines of research
have identified t w o types of cascading effects that connect
various regime shifts, the directional a n d bidirectional links
[252,253]. O n e is called the d o m i n o effect, w h i c h reveals a
one-way directional dependence [254]. W e argue that it fits
more w i t h sequential functional connectivity because the
feedback f r o m one regime shift affects the drivers a n d outcomes
of another regime shift, w h i l e the other one is
termed h i d d e n feedback, s h o w i n g a s e l f - a m p l i f y i n g / d a m p ing
bidirectional cycle [246,255], w h i c h w e argue is more of
a synchronous connectivity nature.
V a r i o u s analytical f r a m e w o r k s have been a p p l i e d to capture
the process of co-evolution, such as the M u S I A S E M
(multi-scale-integrated assessment of societal a n d ecosystems
metabolism) f r a m e w o r k . F r o m a M u S I A S E M perspective,
flows of material a n d energy m o v e t h r o u g h a system (or netw
o r k ) i n order to fulfil certain societal functions. W e argue
that it departs f r o m a set of k n o w n structural connections
(e.g. the m i x of p r i m a r y energy sources for a society a n d its
end uses) a n d then tries to describe functional connectivity
of a central element of a n e t w o r k structure b y u s i n g ratios
that are c o m p o s e d of b o t h a f l o w a n d a f u n d element. R o w
is the element that either disappears over the duration, such
as p r i m a r y energy or appears b y the e n d of the d u r a t i o n
like the product, w h i l e f u n d c a n be seen as a converter that
transforms i n p u t f l o w s into o u t p u t flows d u r i n g the enterexit
duration, e.g. labour, l a n d or machinery. Moreover,
funds are i m p e r m a n e n t structures w h o s e existence depends
on the availability of flows [239]. These ratios give (among
other things) i n f o r m a t i o n about the relative p o w e r of
nodes/agents i n multi-level networks. H i g h rates of metabolized
energy p r o v i d e increased p o w e r to (1) control a n d
b o t h create a n d synchronously co-activate m a n y nodes/
agents i n a n e t w o r k (hierarchy-dependency effect) a n d (2)
influence sequential activation b y controlling flows
(controlling-the-tap-effect). M u S I A S E M tries to p r o v i d e
measures a n d indicators for such relations, i n order to
guide the transformation towards a P o s t - C a r b o n society.
A n o t h e r concrete example of a n S E S here is the global
c o m m o d i t y trade system connecting resource extraction a n d
final demands. Here, nodes are the trading partners such
as cities/countries/regions (at various jurisdiction levels),
w h i c h c a n be l i n k e d t h r o u g h flows of products, material,
monetary value a n d environmental footprints. Altogether,
the established static trading structure w i t h complex
interactions constitutes the n e t w o r k architecture (SC). F o r
instance, i n the p a l m o i l trading market, Indonesia a n d M a l a y sia
have been the m a i n producers, exporting products to
countries like the E U , C h i n a a n d India. The identified relational
structure between the countries is the S C . O n the other h a n d ,
n e t w o r k d y n a m i c s (FC) describe the d y n a m i c s of the f l o w
(i.e. the quantity of trade; the environmental footprint)
e m b e d d e d i n the relational patterns. I n p u t - o u t p u t analysis
[234,256] has been w i d e l y used to capture the i n p u t flows
a m o n g each sector of trade partners i n the network. T h e f l o w
d y n a m i c i n the I O table is rather synchronous, i n the sense
that it is the market interaction where price co-activates both
s u p p l y a n d d e m a n d sides. F o r instance, w i t h the E U passing
a stricter sustainability regulation w h i l e i m p o r t i n g p a l m o i l ,
big producers like Indonesia tend to export more of their products
to less regulated markets like C h i n a . T h e network
structure remains the same, yet the f l o w d y n a m i c changes
synchronously as d r i v e n b y the market price (i.e. higher standards
w i l l increase the p r o d u c t i o n costs, thus the price w i l l
rise accordingly). Sequential activation cannot be m o d e l l e d
using i n p u t - o u t p u t analysis (IOA), as it is more like a snapshot
of a n economic system i n a given m o m e n t i n time. In fact, this
static nature of I O A is often criticized as one of its major shortcomings
that have o n l y partially been overcome b y the
development of d y n a m i c models.
A l t h o u g h connectivity terminology is n o t explicitly used
here, the p h e n o m e n o n of network evolution through actions
of its agents (nodes) is f o u n d quite evident. The theoretical fram
e w o r k developed i n the paper regarding the distinction
between synchronous a n d sequential events has a great potential
to p r o v i d e a different n e t w o r k perspective to understand
the u n d e r l y i n g mechanisms i n social-ecological networks.
4. Conclusion
Here, w e have attempted to u n i f y the b r o a d range of S C / F C
approaches w i t h i n a c o m m o n framework. W e have reproduced
key findings f r o m the literature a n d extended t h e m towards
additional variations of n e t w o r k topology a n d d y n a m i c a l
characteristics i n order to see c o m m o n properties a n d underl
y i n g principles a n d offer a deep mechanistic understanding
of the major contributors to S C / F C correlation.
b
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M i n i m a l models (small toy m o d e l representations of
certain classes of dynamics) are h e l p f u l to explore these
generic features. O u r challenge here w a s to describe h o w the
strengths of the t w o types of S C / F C correlations—based o n
co-activation a n d sequential activation—depend o n the class
of d y n a m i c s , the n e t w o r k architecture, the c o u p l i n g a n d
the internal d y n a m i c a l parameters. W e used n u m e r i c a l s i m u lations
to derive some universal behaviours of S C / F C
correlations u n d e r changes of these system properties a n d to
a p p l y this k n o w l e d g e to real-life systems or data.
The strength of S C / F C correlations can be shifted
between the t w o classes—functional connectivity based o n
co-activation a n d sequential a c t i v a t i o n — i n basically three
ways: (1) m o d i f i c a t i o n of n e t w o r k architecture (e.g. the grad
u a l r a n d o m i z a t i o n of a m o d u l a r graph), (2) change i n
parameters of the d y n a m i c s (e.g. increasing or decreasing
the noise or the coupling) a n d (3) a change i n the temporal
resolution i n w h i c h d y n a m i c a l data are observed (e.g. b y
temporally coarse-graining the observed time series).
The basic challenge of this type of investigation is that the
strength of each type of S C / F C correlation d e p e n d s not o n l y
on the class of d y n a m i c s , the n e t w o r k architecture, the c o u p l
i n g strength a n d the d y n a m i c a l parameters, but also o n the
type of statistics that are a p p l i e d . In some cases, the effect
of the different statistics is so strong that there is a noticeable
change i n the properties that are preserved or not.
A n i m p o r t a n t question is h o w to assess the reliability of
the results. In order to c o n f i r m that a n u m e r i c a l l y observed
b e h a v i o u r of S C / F C correlations (under systemic changes)
is reliable, w e p e r f o r m e d the f o l l o w i n g tests: (1) W e v a r y
the other system parameters s l i g h t l y i n order to study the
robustness of the result. (2) W e reproduce the b e h a v i o u r
observed i n a m i n i m a l m o d e l also i n a richer representation
of the same class of d y n a m i c s .
Of course, the question is more i n v o l v e d o n the technical
level than o u r brief introduction hints at. There are different
w a y s of assessing functional connectivity b e y o n d pairwise
correlations. A c r o s s a l l disciplines, the reliability a n d completeness
of structural data are a n important issue. In the case of
brain networks o n the level of cortical areas (or connectomes),
one issue is whether or not to regard these networks as
weighted or u n w e i g h t e d graphs [32,257]. Furthermore, most
systems for w h i c h such correlations are of interest w i l l have
some f o r m of multiscale organization [258]. Hence, any analysis
o n S C / F C relationships w i l l require selecting suitable
spatial a n d temporal scales. O n some level, w e can furthermore
expect that the structural n e t w o r k (often thought of as 'static' i n
the context of S C / F C correlations) w i l l also change w i t h time,
t h o u g h often o n a longer timescale than functional connectivity.
W e can furthermore envision a coevolution of
structural a n d functional connectivity towards jointly ensuring
a reliable functioning of the system [50].
E v e n the s m a l l discussion of the plausibility of these
stylized forms of d y n a m i c s i n the context of the application
d o m a i n s s h o w n i n figure 1 illustrates h o w real-life complex
systems contain a range of d y n a m i c a l usages (functional
activity patterns) of a g i v e n infrastructure (structural connectivity).
It is less clear, however, that even a f o r m of d y n a m i c s ,
w h i c h b y definition seems to favour one type of functional
connectivity (sequential activation for excitable d y n a m i c s ;
synchronous activity for c o u p l e d oscillators) can d i s p l a y
strong S C / F C signals for the other type of functional connectivity
if the constellations of n e t w o r k architecture, c o u p l i n g
and d y n a m i c a l parameters are right. This point is illustrated
w i t h the n u m e r i c a l simulations discussed here.
W e believe that subsequent investigations m i g h t e m p l o y
the pattern of S C / F C correlations as a means of i d e n t i f y i n g
f r o m a g i v e n n e t w o r k structure, w h i c h type of d y n a m i c s is
most plausible, i.e. w h i c h type of d y n a m i c s this n e t w o r k
w a s 'built for'. O u r current u n d e r s t a n d i n g of d y n a m i c s o n
networks does not yet a l l o w for such a detailed assessment.
A n o t h e r direction of extending our investigation is to have
continuous chaotic systems. Here, the L o r e n z system [259,260],
Rossler system [261,262] or Stuart-Landau system [263,264]
w o u l d be suitable candidates. The question is then, h o w the
results described above change w h e n g o i n g f r o m discrete to
continuous time a n d w h e n g o i n g f r o m a one-dimensional
system (at each node) to a higher-dimensional system. A s w e
have s h o w n above, i n the case of excitable d y n a m i c s , the step
f r o m discrete to continuous time leaves the results qualitatively
intact, as does the step f r o m a one-dimensional to a t w o dimensional
system i n the case of regular oscillations. For
chaotic systems, this needs to be investigated i n detail.
C o m p l e x behaviour (patterns w i t h long-range correlations,
in contrast to chaotic d y n a m i c s w i t h o u t order o n a larger scale)
can emerge near critical points. G i v e n our hypothesis that h i g h
S C / F C s e q is associated w i t h large-scale patterns (e.g. excitation
waves [20]), the distinction between chaotic a n d complex behaviour
m a y have a strong effect o n S C / F C correlations. This
aspect requires further investigation a n d c o u p l e d electronic
chaotic oscillators m a y be a n interesting test case for this, as
they p r o v i d e a h i g h level of realism, i n particular for brain
dynamics, together w i t h a detailed understanding of their collective
behaviours [265,266]. Electronic chaotic oscillators can
be used as a physical m o d e l of brain d y n a m i c s [265,266] as similar
d y n a m i c s to neuronal activity are observed i n such networks
of diffusively c o u p l e d single-transistor oscillators.
Seeing sequential activation as a p r o x y of large-scale patterns
certainly has its limitations. In particular, w e expect that
criticality—power-law distributions of activity a n d l o n g range
correlations that have been studied i n great detail i n
general n e t w o r k s [267,268] a n d i n particular i n b r a i n
d y n a m i c s [27,269-271]—cannot be identified i n this w a y .
O n the technical level, v a r i o u s definitions of co-activation
and sequential activation are plausible, e.g. different n o r m a l izations,
time delays a n d discretizations. W e d i d not explore
these aspects i n detail. A discussion of the impact of these
aspects can be f o u n d for example i n [30,50].
A s often w i t h n u m e r i c a l investigations, some seemingly
small 'design decisions' affect the results. In the S E R m o d e l ,
for example, near the deterministic limit, longer runs d o not
p r o v i d e more information, as the system r a p i d l y settles into a
(periodic) attractor. Then, o n l y a large n u m b e r of short runs
can reveal the u n d e r l y i n g n e t w o r k architecture. The same is
true for phase oscillators, w h i c h p r o v i d e d the c o u p l i n g is
h i g h e n o u g h g i v e n a certain spread of eigenfrequencies,
rapidly settle into a fully synchronized state no longer i n f o r m a tive
about the architecture of the network. H e r e also, transients
f r o m m a n y runs need to be collected.
In the case of c o u p l e d phase oscillators, delayed c o u p l i n g
[272,273] a n d phase shift c o u p l i n g [274] can alter the synchronization
properties a n d the d y n a m i c a l behaviour dramatically.
In the case of time delays, w e can expect that strength is shifted
between co-activation a n d sequential activation. T i m e delays
are p r e s u m e d to be a key ingredient i n reproducing realistic
neural d y n a m i c s [187,275,276]. Phase shift c o u p l i n g , o n the
other h a n d , can transform phase oscillators into excitable units
[274]. These points, together w i t h our observation that the exact
f o r m of the c o u p l i n g is relevant for the behaviour of S C / F C
correlations, underlines that even for the simple models discussed
here more investigations are necessary.
W i t h o u r investigations, w e set o u t to understand
w h i c h n e t w o r k features a n d w h i c h class of d y n a m i c s rather
enhance S C / F C g ^ or rather enhance S C / F C s e q . A s a rule,
w e f i n d that m o d u l a r i t y enhances S C / F C g ^ , w h i l e b r o a d
degree distribution or randomness enhances S C / F C s e q .
Increase i n c o u p l i n g favours h i g h S C / F C s m v w h i l e h i g h
parameter diversity tends to enhance S C / F C s e q .
F r o m the v i e w of d y n a m i c s , excitation models tend to
favour F C s e q , a trend w e observe both w i t h the m i n i m a l
(SER) m o d e l of the excitable d y n a m i c s a n d w i t h the more realistic
F i t z H u g h - N a g u m o m o d e l . B y contrast, regular oscillators
favour F C s m l , as w e see w i t h the stylized (coupled phase
oscillator) m o d e l a n d , at a higher level of realism, w i t h the
F i t z H u g h - N a g u m o m o d e l i n its oscillatory regime. I n the
case of the chaotic oscillators, the choice of the c o u p l i n g term
used here leads to a persistent dominance of h i g h positive
S C / F C s e q , but for a complete v i e w about the S C / F C strengths
further investigation w i t h other types of c o u p l i n g is needed.
The conceptualization of the synchronous a n d sequential
activity i n different application scenarios is more sophisticated
than i n the case of m i n i m a l models, w h i c h indicates
that broader definitions of the t w o notions are needed. H o w ever,
as w e s h o w i n the second half of our investigation, the
systematics extracted f r o m investigating m i n i m a l m o d e l s help
us better organize the diverse findings i n the application
d o m a i n s a n d thus provides a fresh perspective o n d y n a m i c a l
processes i n network-like systems i n these fields. Specifically,
w e argue that F C s j m is associated w i t h simultaneous
measurements either of the d y n a m i c a l activity of nodes or
of links, where the concept of 'simultaneous events' introduces
a timescale, at w h i c h events are considered to be
'synchronous'. Relatedly, i n terms of a more general v i e w
on FCseq/ w e argue that it can be seen as f l o w of information
or materials i n the system, s u m m a r i z i n g concepts s u c h as
influence a n d diffusion.
5. Methods
5.1. Network topologies
The simulations were performed o n a set of abstract graphs
(modular, Erdos-Renyi, Barabasi-Albert, Newman-Watts-Strogatz
and hierarchical [277]) and three real-life networks (neural
[32], social [34], metabolic [33]). The description of the network
architectures of the graphs is given below:
M o d u l a r graph: includes 60 nodes and has density 0.23. Each
graph is constructed by starting from four cliques, i n w h i c h every
node is linked w i t h all the other nodes i n the same clique. Then,
edges are randomly rewired w i t h probability p = 0.23 to link
different cliques.
Watts-Strogatz graph: includes 60 nodes and every node
of the graph is linked w i t h its 15 nearest neighbours i n a ring
topology [278].
Erdos-Renyi (ER) graph: includes 60 nodes and the probability
of edge creation for each node is 0.23 [279].
Barabasi-Albert (BA) graph: Each B A graph consists of 60
nodes a n d it is grown b y attaching n e w nodes, each with 8
edges that are preferentially attached to existing nodes with high
degree [280].
Hierarchical graph: includes 64 nodes, 174 edges a n d it
has a scale-free topology w i t h modular structure. The detailed
construction process is described i n [277].
Neural graph: includes 89 nodes (cortical areas) a n d 676
edges derived v i a thresholding a n d symmetrization from the
29 x 91 connectivity matrix (inter-areal connection strength
measurements) described i n [32]. A n edge between two nodes
is accepted if the decimal logarithm of the corresponding connection
strength measurement is above the threshold value 10 (see
electronic supplementary material for more details).
Metabolic graph: includes 72 nodes (metabolites) and 486
edges [33]. W e use the systems biology markup language
(SBML) model 'e-coli-core' from the B I G G database (bigg.ucsd.edu)
and extract the stoichiometric matrix S. The adjacency
matrix of the metabolite-centric metabolic network shown i n
figure 1 is then obtained b y mapping a l l non-zero entries i n
SST
to 1, where ST
is the transpose of S.
Social graph: includes 77 nodes (people) and 875 edges [34]. It
is an undirected graph and an edge between two nodes is created,
if one's knowledge about the skills of others within the company
exceeds a threshold equal to 5.0 (see electronic supplementary
material for more details about the real-life networks).
5.2. Topological changes
In the first instance, three initial graphs that have distinct structure
were randomized or rewired i n different proportions, such as the
ratio N o Changes/No Edges ^ 0.11 corresponds to a percentage
of 10% of randomization/rewiring process. Thus, for the modular
and the regular graph every 10% of randomization/rewiring
process corresponds to 50 swaps/rewiring changes of edges. The
degree of the nodes is preserved and only the structure of the network
changes. For the hierarchical graph, 20 swaps of the edges for
every 10% of randomization are enough to end up w i t h a scale-free
graph, whose modularity is completely destroyed. A s a consequence,
the randomized network retains its degree distribution
and the presence of hubs, but without the embedded modularity
that it initially had, similar to a scale-free topology as the preferential
attachment model from [280]. The modular a n d the
hierarchical networks were randomized, according to the
Markov chain algorithm [281]: pairs of randomly selected edges
are swapped, providing no self-loop or multiple edges between
two nodes are created. The rewiring process was performed o n
the Watts-Strogatz model according to the scheme from [278]: a
randomly selected link was destroyed and a new one was created
between one of the two nodes and a randomly selected one; the
requirement of self-loops and multiple edges between two nodes
must be, also, satisfied. During the rewiring process and before
we e n d u p w i t h an Erdos-Renyi graph, the network passes
through a 'small w o r l d ' regime [278].
5.3. SER model
The models we used to highlight the two classes of functional connectivity
cover a range of different types of dynamical processes:
excitable dynamics, regular oscillations and chaotic oscillations.
The SER model, a simple cellular automaton model of excitable
dynamics, acts o n discrete time and the update rules are simultaneously
applied as follows to go from the state at time t to the
state at time t + 1: (1) A node i n the susceptible state (S) changes
into a node i n the state of the excited nodes (£) if one or more of
its neighbours are excited. Alternatively, a node can go from S to
E i n a stochastic way with a given (usually small) rate of spontaneous
excitation, /. (2) A node i n the excited state (E) changes
into a node i n the refractory state (R). (3) A node i n the refractory
state (R) changes into a node i n the susceptible state (S) i n a
stochastic way w i t h a given refractory probability p. This model
has been originally studied as a model of self-organized criticality
[282] a n d later been applied to address abstract questions of
excitable dynamics o n graphs [29,31,283], as well as topics i n
neuroscience [19,50]. In the deterministic limit, p = 1, /= 0, the contribution
of the three cycles significantly affects the collective
dynamics [28,30]. Due to its discreteness i n time and states, i n
the SER model co-activity and sequential activity of the nodes
can be defined i n a parameter-free way: each node can be found
in one of the three states x,(ř) e (S, E, R], however, i n the analysis
of S C / F C relationships we only distinguish two states:
Í 1 *,(*)= E
, {
> ~\0 Xi(t) = S o r R .
Separating the nodes into the two categories (active or inactive)
is a convenient way to define the t w o classes of functional
connectivity. The co­activation matrix is
Q = 5>(f)c,­(f),
t
and the sequential activation matrix is
S? = $ > ( f ) c , ­ ( f ­ l ) .
t
It should be noted that different normalizations of these quantities
can be envisioned (see [28] for a detailed discussion).
For all the cases where the SER model was used, we simulated
NR = 10 000 runs of i m a x = 10 (unit timestep) with randomly generated
initial conditions, with 6% of the nodes to be i n the E state
and the rest to be i n S or R state with an equiprobability. The information
for the F C matrices was accumulated by initially taking the
sum over the time of each matrix, and then by taking the s u m over
the multiple runs. The S C / F C correlations were computed with
the Pearson correlation between the flattened adjacency and the
co­activation/sequential activation matrix. The final average
value was computed as the mean of the correlations from the 10
different initial graphs, and the errors as the standard deviation
of these correlation values. W e obtain the main results using the
recovery probability p = 0.1 and transmission probability/= 0.001.
5.4. Phase oscillators
The second, also well studied, model studied here is the
Kuramoto model [118,284]. It describes the behaviour of a
large set of coupled phase oscillators and their transition to
synchronization. W e use it here i n a variant, where the oscillators
are coupled according to the architecture of a given network [24].
Each of the oscillators has a n intrinsic natural frequency (or
'eigenfrequency') &>,­ and all of them are equally coupled with
their neighbours with coupling k. The evolution of the phase
of a node i n a population of N oscillators is governed b y the
following dynamics:
- £ = W I +
Ň S A i j s i n { d i
~ 0 i )
' i = 1 N
This
model has been instrumental i n the past for understanding
h o w network topology determines synchronizability [23] and
h o w synchronization patterns emerge from architectural features
of networks [22].
Investigating the behaviour of the two classes of FC , i n this
model, requires oscillators that have not reached the total synchronization,
w h i c h indicates the absolute ' w i n ' of the co­activation.
Thus, Gaussian noise, scaled b y amplitude a, was added i n
order to delay the synchronization process.
d-0- k ^
-r1
= COÍ + — V AH siní Oj -6Í) + <TU, i = 1, . . . , N
at N
;=i
The matrix of functional connectivity, i n this case, is constructed
from the correlation coefficient between the time series
of the effective frequency:
Qy(Si) = c o r r t ( A ( i ) , + St)), (5.1)
where
m = (Ae! (i)>( = ^ ' +
f ' W + 1 ) ­ ei{f)
Z a l
t'=t-M
for some suitable choice of a time w i n d o w At.
For a continuous model, such as the coupled phase oscillators,
the definition of the two classes of functional connectivity is not
possible i n a parameter­free manner. In equation (5.1), for St = 0,
we have strict co­activation and with increasing time lag St a transition
from correlations dominated by co­activation to correlations
dominated b y sequential activation (before the two timeseries
of effective frequencies essentially de­couple). Particularly, the
decision of the appropriate selection of the time lag for the sequential
activation was based o n the results of S C / F C correlations as a
function of the coupling strength for different values of time lag. In
the electronic supplementary material, figure S5 shows the multiple
curves of the different time delay values for a modular and
an E R graph. While the effect of the increasing time delay i n a m o d ular
graph is the gradual decrease of the S C / F C correlation, i n the
ER graph three groups of curves emerge. The first one corresponds
to the co­activity of the nodes (includes the zero and time lag equal
to 1), the second group includes the curve that corresponds to the
time­delay 2 and, i n this case, is the appropriate selection for the
sequential activation, since larger values for the time delay,
which constitutes the third group of curves, have zero contribution
in the sequential activation.
For this case, we simulated NR = 100 runs over f m a x = 50 using
the Euler method, with randomly generated initial conditions from
the uniform distribution (—n, TI) o n different graphs with non­identical
oscillators. The integration timestep for the solution of the
system was equal to 0.1. The Gaussian noise was selected to
have zero mean, unit variance and it was scaled b y amplitude
a = 0.25. The eigenfrequencies were uniformly selected from the
interval (0,1). The size of the time w i n d o w we selected At for the
effective frequency was equal to 20 and the F C matrices were constructed
from the Pearson correlation of the effective frequencies
between each pair of nodes. The diagonal elements are zero, b y
default. A s i n the SER model, the S C / F C correlations were computed
with the Pearson correlation of the flattened adjacency and
F C matrices. F or the latter one, the sum, over multiple runs, was
taken and the average correlation values derived from the S C /
F C correlations of 10 different initial networks; the corresponding
errors derived from the standard deviation of these 10 values. F or
the main results, we selected a coupling strength equal to 10.
5.5. Logistic map
The third model that was used as a dynamical probe of network
architectures is the logistic map. Such dimensional maps (also
termed finite­difference equations or recursion relations) are used
to describe the evolution of one variable over discrete steps i n
time, following a template of the form xM =f(xf). The logistic map
xi+i = Rx,(l - xt)
is the most well­known example of this class of dynamical models
[285]. Starting from a stable fixed point at low R, the system undergoes
a sequence of period­doubling bifurcations with increasing R
leading to a large regime of deterministic chaos, occasionally interrupted
b y small periodic windows. Systems of coupled logistic
maps have been studied extensively as a model for spatio­temporal
pattern formation [286] and o n networks [51,52,287].
The coupled system has the form
k N
Xi(t + l)=RjXi(t)(l-Xi(t)) + t ;
1 = 1,
where k is the coupling strength a n d Ay is the network's adjacency
matrix (structural connectivity). Note that w e impose
additional constraints o n the system to force each xt(t) to be i n
the interval x e [0, 1]. W e define F C as the correlation between
the timeseries of the nodes for zero time lag (co-activation) and
a time lag of 1 (sequential activation):
Cjj = corr,(x;(f), Xj{t)), Si, = corr,(x,-(f), Xj(t + 1)).
We simulated NR = 50 runs over f m a x = 500 (unit timestep)
w i t h randomly generated initial conditions from the uniform
distribution (0,1). The parameter R was randomly selected b y
each oscillator from the interval (3.7, 3.9). For the m a i n results,
the coupling strength that was used was equal to 2. The F C
matrices were constructed from the Pearson correlation between
the time series of the x variable (diagonal elements are zero b y
default). The S C / F C correlations derived from the comparison
of the flattened adjacency a n d F C matrices, b y taking the
Pearson correlation, after each run. The average correlation
value derived from the mean correlation values over the multiple
runs a n d the errors from the standard deviation of the correlation
values over the multiple runs.
5.6. FitzHugh-Nagumo model
A s a more sophisticated model of excitable dynamics a n d
regular oscillations, w e use the F i t z H u g h - N a g u m o model
[288,289], a two-dimensional model of ordinary differential
equations (ODEs).
The F i t z H u g h - N a g u m o model is composed of two coupled
variables, where x represents the membrane potential a n d y is
the recovery variable:
x?(f)
•Vi{t)
(d)
the randomization process of a modular graph i n the excitable
and i n the oscillatory regime.
A s w i t h the logistic map, coupled F i t z H u g h - N a g u m o oscillators
have been employed i n a range of investigations focusing
on spatio-temporal pattern formation [290] a n d collective
dynamics i n networks [26].
We simulated 10 runs, using the Euler method to solve the
system. The total time of each simulated r u n was 180 s a n d the
integration step 0.1 ms. W e downsampled the output at 1 ms
and we used this to calculate the F C . The F C s m l matrix derived
from the s u m of the co-activation matrices over the time of each
run. The co-activation matrices were constructed as i n the SER
model, after discretizing the time series (spike detection) w i t h a
threshold equal to one a n d using a time w i n d o w equal to
1 ms. For the FCseq matrices, various widths of time w i n d o w s
were selected i n order to discretize the time series a n d detect
the spikes. Larger time w i n d o w s include both spikes that occur
simultaneously a n d sequentially, thus, from the whole activity
within the window, the co-activity (time w i n d o w 1 ms) was subtracted.
The calculation of S C / F C correlations derived from the
flattened adjacency a n d functional connectivity matrices, after
excluding the diagonal elements. The final correlation values
came from the mean value of the 10 correlation values from
the different runs a n d the errors from the corresponding standard
deviation. The co-activity of the nodes, as well as the
sequential activity of the nodes, using different w i n d o w sizes,
were tested under different values for the coupling strength
and the noise amplitude for both the excitable (a = 0.8) a n d oscillatory
regime (a = 0) (see electronic supplementary material, S6).
The selected parameter values for the system are /? = 0.6, y=l,
rx = 0.001, Ty = 0.1. The random numbers for the noise ux were
selected from a normal distribution w i t h zero mean a n d unit
variance, whose amplitudes were scaled b y a a n d w i t h a n
additional scaling parameter y/dt/rx (At is the size of the integration
step). The scaling term for the uy is equal to zero. For
the main results, we selected the coupling strength (divided b y
the average degree i n the network) equal to 0.044, the amplitude
of the noise equal to a = 0.15 a n d the time w i n d o w of 12 ms for
the sequential activation.
b
ho
o
ho
o
C O
and
ßyit)+a,
where (d) is the average degree i n the network, xx, Ty are the timescale
parameters for each variable, again k the coupling strength
among the connected nodes, vx, vy are random variables d r a w n
from a Gaussian distribution of zero mean a n d unit variance
and a the amplitude of the noise. In the xy plane, w e can distinguish
three regions a n d the intersection of the nullclines of
the system (see electronic supplementary material, figure SI),
dxj(t)/dt = 0 A dyi(t)/dt = 0 defines the fixed point. Hence,
depending o n the region that the fixed point is placed, the
system can be found either i n the oscillatory or i n the excitable
regime. By shifting the linear nullcline (changing the parameter
a), w e can move from region 1 (excitable regime) to region 2
(oscillatory regime). Here, we plot the correlation values during
Data accessibility. This article has no additional data. The Python codes
used for the numerical simulations will be available from the authors
upon request.
Authors' contributions. V.V. and M.-T.H. designed research. V.V. and A . M .
wrote computational code and performed simulations. V.V. and
M.-T.H. analysed results and wrote the framework of the paper.
L.J.B., J.C., M.G., A.J.P., J.P., R.P., ST., L.T. and J.W. wrote the geomorphology
section. A.F., T . H . and S.R. wrote the freshwater
ecology section. M.T.H. wrote the systems biology section. D.B.,
A.B. and V L . wrote the neuroscience section. M . D . M . , B.F., C.Ker.,
C.Kim., Y.S. and H.W. wrote the social-ecological systems section.
C P . wrote §1.2 of the Electronic Supplementary Material and contributed
to the application section. A l l authors read and approved the
final version of the paper.
Competing interests. We declare we have no competing interests.
Funding. This project has received funding from the European Union's
Horizon 2020 research and innovation programme under the Marie
Sklodowska-Curie grant agreement N o . 859937.
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