Hidden early-warning signals in scale-free networks
Hidden early-warning signals in scale-free networks
Georg JaÈ ger 0 1
Christian Hofer 0 1
Marie Kapeller 0 1
Manfred FuÈ llsack 0 1
0 Institute of Systems Sciences, Innovation and Sustainability Research, University of Graz , Graz , Austria
1 Editor: Sergio GoÂmez, Universitat Rovira i Virgili , SPAIN
Critical transitions of complex systems can often be predicted by so-called early-warning signals (EWS). In some cases, however, such signals cannot be detected although a critical transition is imminent. Observing a relation of EWS-detectability and the network topology in which the system is implemented, we simulate and investigate scale-free networks and identify which networks show, and which do not show EWS in the framework of a two state system that exhibits critical transitions. Additionally, we adapt our approach by examining the effective state of the system, rather than its natural state, and conclude that this transformation can reveal hidden EWS in networks where those signals are otherwise obscured by a complex topology.
Data Availability Statement: All relevant data are
within the paper.
Funding: We acknowledge the financial support by
the University of Graz regarding the publication
fees. The funders had no role in study design, data
collection and analysis, decision to publish, or
preparation of the manuscript.
Competing interests: The authors have declared
that no competing interests exist.
Many systems in various scientific fields are known to exhibit so-called critical transitions, in
which a system changes abruptly from one state into another. Prominent examples include
climate change [
], asthma attacks , epileptic seizures [
], systemic market crashes 
and catastrophic shifts in ecosystems [
For each of these systems, in particular in cases of transitions from beneficial to
disadvantageous system states, it is of vital importance to accurately predict such critical transitions or at
least find signals whether a critical transition is imminent or not [
]. One approach to making
such predictions is looking for early-warning signals (EWS), i.e. various statistical indicators
related to certain properties of the system [
However, not all systems that exhibit critical transitions show these EWS [
investigation of EWS and the question in which systems they can and cannot be observed is a very
active field, particularly in the context of networks. Especially in complex networks, scanning
for EWS is very challenging. For example, the critical transition in a complex network might
not correspond to a bifurcation [
]. However, there are network techniques that can be used
to make detecting EWS easier, like considering spatial correlation in contrast to pure time
]. In order to monitor changes in this spatial correlation, an interaction network
approach can be used [
]. Thus, the idea of including topological information in an EWS
analysis is well established. However, most research regarding the combination of EWS and
networks focuses on one specific network and it is difficult to generalize insights from these
investigations to different networks. We are therefore interested in obtaining results that can
be generalized to a whole category of networks, in our case the important class of scale-free
In this paper we look at a simulated two-level system that is known to exhibit critical
transitions and investigate it in implementation in different scale-free networks. The big advantage
of such an approach, in comparison to other research in this field, is that we do not focus on
one specific example for a network, like a bank network [
] or an ecosystem [
], but we
investigate randomly generated scale-free networks and simulate many such networks. This
means that our findings are easier to generalize to all scale-free networks. Additionally we
propose a simple way to include topology in EWS investigations. Normally this is a difficult
process, but in this paper we condense all topological information, combined with the state of
each node in the network, to one single scalar that can then be analyzed for EWS in the same
way an ordinary time series is analyzed. A further advantage of our approach is that we are
able to investigate scale-free networks with different power laws and compare the results. All
differences in behavior we observe, can then be attributed to the different power law, since
other parameters are fixed and details of the topology average out.
We find that in some topologies EWS can be detected; in others however, they remain
hidden. Furthermore we explore a method to make these hidden EWS visible by transforming the
system from its natural state to a so-called effective state, by a method which was recently
suggested by Gao et al. [
This paper is organized as follows: Section 2 gives a short introduction to EWS. Section 3
introduces in the kind of scale-free networks examined. The example system we use for our
simulations is explained in Section 4 and mathematical details on how to calculate the effective
state of a system are given in Section 5. Numerical details of the simulation can be found in
Section 6. Results are presented in Section 7 and discussed in Section 8.
Systems theory describes dynamical systems as having an `eigen-behavior' [
] which makes
them evolve towards stable states. Internal interaction dynamics can cause systems to have
several such stable states, spanning basins of attraction in the reach of which dynamics tend to
return to after (small) perturbations [
]. The edges of these attractor basins (or potential
wells) mark tipping points at which transitions to alternative stable states may occur.
Depending on the strength of positive feedback effects, these transitions can be abrupt, showing
sudden shifts of regime [
], so called `critical transitions'.
In respect to the suddenness of these shifts, attempts have been made to derive signals for
the approach of a tipping point at which critical transitions occur. One such indicator is the
time a system needs to return to equilibrium after perturbation. Slowing down of recovery
time, so called `critical slowing down' (CSD), has been found to indicate the loss of a system's
resilience and thus the approach of a transition [
CSD shows in the form of various statistical signals. One of them concerns changes in the
correlation structure of a time series, caused by an increase in the `short-term memory' (i.e.
the correlation at low lags) of a system prior to a transition and is measured with
autocorrelation at-lag-1 of consecutive observations [
]. Another indication for CSD is provided by the
tendency of a system to drift more widely around its stable state when approaching a tipping
point. This causes the standard deviation in the time series to increase. And finally, variance
can show asymmetries and more extreme values when a system's state gets close to the
attraction of an alternative stable state. This causes skewness and kurtosis to change.
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There are several case studies, where EWS are used for a variety of problems, such as
predicting regime shifts in ecosystems [
], climate change [
] or land surface change [
In a scale-free network the degree distribution follows a power law: The fraction of nodes with
degree k is proportional to k−γ:
Scale-free networks are relatively common, for example social networks, the World Wide
Web [24±26], the internet [27±29], citation networks [
], scientific collaboration networks
[31±33] and metabolic networks [
] are scale-free. These networks have many intriguing
properties: They contain so-called hubs, i.e. nodes with a degree much higher than the average
degree of the network, their clustering coefficient distribution follows a power law, and they
are very fault tolerant. [35±37]
In order to generate such a scale-free network we use a mechanism proposed by Goh et al.
]. We start by generating N nodes and numbering them with i = 1, 2, . . ., N. Then we
arbitrarily choose a control parameter α in the interval [0, 1). Each node is then assigned a
connection probability pi = i−α. Next we select two nodes a and b with probabilities equal to pa/∑i pi
and pb/∑i pi respectively. If those two nodes are not already connected, a link is established.
This process is repeated, until a previously determined number of links is reached. Since the
links are distributed according to this probability, it can be calculated that the degree of the
nodes obeys the power law [
where γ is closely related to the control parameter α via
That way it is possible to generate a scale-free network with well-defined γ
2 by choosing α
A two state system
In order to gain universal results, not depending on the specifics of a certain system, we
investigate a simple two state system, consisting of a network of N nodes and L links, connecting
these nodes. Each node can be in one of two different states, denoted as `state 1' and `state 2'.
The meaning of these states depends on the investigated system. For a cooperative system of
human beings the states could represent `cooperating' and `not cooperating', for an ecosystem
comprised of different species that live in symbiosis the states could represent `not endangered'
and `endangered' and many more interpretations are possible.
We can generalize our results to all systems that have the following properties: All nodes
initially start in state 1 and have a gradually increasing chance to change to state 2.
Neighboring nodes can, however, transfer a node in state 2 back to state 1. This mixture of positive and
negative feedback-loops guarantees a critical transition and is a valid approximation to many
real-life systems. Details on the system are given in section 6.
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The effective state of a system
The natural state of a system (in our system this corresponds to the number of nodes in state
1) can be used to scan for EWS and for many systems this approach is well-established
[39±41]. However, it is equally possible to calculate the so-called effective state of a system and
use this property to scan for EWS. Calculating effective states was proposed by Gao et al. [
in order to derive a one-dimensional dynamic from a complex multi-dimensional system.
There, effective parameters are calculated as
IT A x
IT A I
with the adjacency matrix A, the unit vector I, and x, the investigated property of the system
(in our case x = 1 for nodes in state 1 and x = 0 for nodes in state 2). This operation
corresponds to taking an average over the whole network, where each node is weighted with its
degree, i.e. the number of links attached to it. For networks where all nodes have the same
degree, the effective state and the natural state are equivalent. The difference becomes more
important for networks where some nodes are very well connected, while others are not. Take
a very simple network of 4 nodes, where one node serves as a hub and is connected to all other
nodes, while there are no other links in the system. Suppose the hub is in state 1, all other
nodes are in state 2. The natural state of the system, scaled to the number of nodes can then be
1 0 0 0
The interpretation of this property is that 25% of all nodes are in state 1, i.e. if you select one
node of the system at random, the chance that it is in state 1 is 25%. However, the fact that the
central hub of this network, the node with the most influence on the system, is the one in state
1, is neglected in this simple average. As long as exactly one node is in state 1, the natural state
is 0.25. This is different for the effective state. Using the same example, we can calculate
1 0 1 1 1 1
BBB 1 CCC BBB 1 0 0 0 CCCBBB 0 CCC
B@ 1 AC B@ 1 0 0 0 ACB@ 0 AC
1 1 0 0 0 0
0 1T 0 1 00 1T 0
BB C B CC
BB 1 C B 1 CC
BB C B CC
B@B@ 1 AC B@ 1 ACCA
The interpretation of this number is not straightforward, because it does not mean that 50% of
the nodes are in state 1. However, if you select a random link from the system and then select
one of the two connected nodes at random, the chance that this selected node is in state 1 is
f1f is thus related to the chance to select a node in state 1 with a selection process that
favors very well connected nodes, to be precise, all selection processes where the selection
chance of each node is directly proportional to the degree of the node.
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In a way the effective state of the system takes the different importance of each node into
account, in contrast to the natural state, where this information is completely lost. It is
therefore intuitive that the effective state of the system offers more insight into the particularities of
the system and should hence be used to scan for EWS, especially for very complex network
The numerical details of our simulations are as follows: We use a population of N = 500,
connected via L = 2500 links. The chance of a node switching from state 1 to state 2 begins at s12 =
0 percent and is increased by ds12 = 0.0002 percent each time step. The chance to switch back
to state 1, s21, is dependent on the link-neighbors: Each link-neighbor in state 1 gives a node in
state 2 a 1 percent chance to switch back to state 1. Note that the actual switch-chances have no
influence on the qualitative behavior of the system, they would only translate the time of the
critical transition. Different values of ds12 and s21 were tested. The only qualitatively different
behavior is observed, if s21 is too small to give the system any resilience, in which case state 1 is
not stable and all nodes reach state 2 nearly instantly.
In our simulations we analyze the system detailed above by counting the nodes in state 1 in
each time step to find N(1), the number of nodes in state 1. Additionally we calculate Ne
effective number of nodes in state 1 according to Eq (4).
A typical time development of the natural state of such a system is shown in Fig 1: N(1)
decreases until it reaches 0 after a critical transition. The constant increase of standard
deviation is a property of the system; it is, however, also clearly visible, that the standard deviation
of N(1) increases sharply right before the critical transition. In order to verify that this can
indeed be seen as an EWS we also need to investigate the autocorrelation, depicted in the
lower panel of Fig 1. Here we can see that the autocorrelation increases significantly before the
critical transition as well.
The difference between the effective and the natural state is illustrated in Fig 2. Both serve
as a measurement of what fraction of the system is in state 1, yet the particularities concerning
the individual importance of each node, detailed above, lead to a slight difference, most
nently visible in a higher variance of Neff
In order to scan for EWS we analyze the standard deviation and the autocorrelation of both
N(1) and Ne
f1f in a rolling window of 100 time steps. In order to constitute an EWS, both
standard deviation and autocorrelation must increase before the critical transition. We simulate
scale-free networks with α values of 0.125, 0.25, 0.5, and 1 corresponding to γ values of 9, 5, 3,
and 2 respectively.
The results of our simulations are reported in Figs 3±6. The left side of each figure (blue)
shows standard deviation (upper panel) and autocorrelation (lower panel) for the natural state
N(1). The right side (green) shows the same properties for the effective state Ne
f1f. For each
value of γ the results were obtained by averaging over simulations of 10 different networks in
order to rule out that the observed effects are random features of a specific network, but rather
features of all scale-free networks with this γ value. Different networks of course lead to
different times for the critical transition Tc, so in order to average over them it is necessary to
measure time relative to Tc. As indicated on the x-axes, we plot all values against the difference
between the time Tc and the time t.
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Fig 1. Typical time development of the number of nodes in state 1. N(1) decreases over time, until there is a critical transition at t = Tc (red
vertical line). When looking at the standard deviation and the autocorrelation of N(1), one can see that it increases significantly prior to the critical
transition, which constitutes an EWS.
Fig 3 shows the resulting EWS for scale-free networks with γ = 9. In these networks we can
observe an increase in standard deviation as well as autocorrelation for both the natural state
(blue) as well as for the effective state (green). Here, effective and natural state are very similar,
since a large fraction of the nodes has a degree very close to the average degree. Therefore, also
standard deviation and autocorrelation of N(1) and Ne
f1f do not differ much.
Fig 4 presents our simulation results for scale-free networks with γ = 5. Again, standard
deviation as well as autocorrelation increase before the critical transition at t = Tc. This effect
can be seen in the natural state (blue) as well as in the effective state (green).
Fig 5 displays results for scale-free networks with γ = 3. Here, there is a visible difference
between investigating the natural state of the system (blue) and the effective state of the system
(green). While both properties lead to an EWS, i.e. an increase in both standard deviation and
autocorrelation, the signal is much clearer for Ne
f1f. In these networks, not all nodes are equally
6 / 14
Fig 2. Natural and effective state of the system. N(1) and N
e1ff behave similarly, however, the effective state has a higher variance.
well connected. Some nodes have very few links, while others have a very high degree, which
gives them more influence on the total state of the system. Neglecting these differences by
investigating the natural state leads to a less distinct, but still visible EWS.
Fig 6 reports the findings of our simulations for scale-free networks with γ = 2. Here, the
results are qualitatively different from those obtained for networks with γ 3. When
investigating the natural state of the system (blue), no EWS can be detected. Both standard deviation
and autocorrelation even decrease before Tc. However, when looking at the effective state of
the system (green) the EWS are still clearly visible. The huge difference between the behavior
of N(1) and Ne
f1f can be attributed to the network topology, specifically to the parameter γ, since
it is the only difference to the other simulations.
We conclude that analyzing a scale-free network for EWS is not straightforward, especially
when considering networks with γ 3. In our example system we observe the standard
deviation and the autocorrelation of both N(1), the natural state of the system, as well as Ne
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Fig 3. EWS for γ = 9. An increase in standard deviation as well as autocorrelation can be observed prior to the critical transition, both for the natural state N(1)
(blue) and the effective state N
e1ff (green) in networks with γ = 9.
effective state of the system. In scale-free networks with γ < 3 we are able to detect clear signals
both for the natural as well as for the effective state. For scale-free networks with γ = 3 the
signals are visible for both N(1) and Ne
f1f, however they were clearer for the effective state. For γ =
2 the situation is different. Scanning for EWS in the natural state of the system reveals no
EWS; both standard deviation and autocorrelation even decrease shortly before the critical
transition. Nevertheless, analyzing the effective state of the system does produce EWS. Since
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Fig 4. EWS for γ = 5. An increase in standard deviation as well as autocorrelation can be observed prior to the critical transition, both for the natural state N(1)
(blue) and the effective state N
e1ff (green) in networks with γ = 5.
the only difference between the investigated systems is the parameter γ, we deduce that the
exponent of the power law for the scale-free network is responsible for this qualitative
difference. The same system, implemented in different networks, can show EWS in one case, while
showing none in the other case, even though there is a critical transition in both topologies.
This finding substantiates that not only the interaction between the nodes of a network, but
also the topology itself influences, if EWS are detectable or not. Especially when investigating
9 / 14
Fig 5. EWS for γ = 3. An increase in standard deviation as well as autocorrelation can be observed prior to the critical transition, both for the natural state N(1)
(blue) and the effective state N
e1ff (green) in networks with γ = 3. Note that the effective state shows a much clearer signal with less fluctuations than the natural
networks with well connected hubs it is therefore paramount to include topological effects
when scanning for EWS, possibly by using the effective state of a system, rather than its natural
In scale-free networks with a small γ parameter the degree distribution is far from uniform.
There are very big hubs, i.e. nodes that have many links and are therefore very important for
the development of the system as a whole. Nodes that have only few connections have less
10 / 14
Fig 6. EWS for γ = 2. While the natural state of the system N(1) (blue) shows no EWS (both standard deviation and autocorrelation decrease), the EWS are
clearly visible for the effective state N
e1ff (green) in networks with γ = 2.
influence. When investigating the natural state of the system we completely ignore these
differences and scanning for EWS yields no results, since this implicitly assumes that the sum of
nodes in one state is a good measurement for the overall state of the system. This assumption
is valid for scale-free networks with high γ, but it is questionable for networks where the degree
distribution is far from uniform. If we want to include the difference in importance of the
individual nodes we have to investigate the effective state of a system, rather than its natural state.
11 / 14
In conclusion, our investigations show that there are systems in which one cannot find
EWS, although a critical transition is imminent, when only considering the natural state of the
system. Analyzing the effective state of the system can reveal these hidden EWS. Our research
suggests that the benefit of using the effective state of a system rather than its natural one is
greater, the less uniform the links are distributed. It is likely, that the possibility for hidden
EWS is not only a property of scale-free networks, but rather of all networks, where the degree
distribution is far away from uniform. To illuminate this, further research is required and
should focus on investigating different systems that show critical transitions and different
types of networks beyond the scope of scale-free ones in order to find universal insights into
the connection between topology and EWS detectability.
We are grateful to the anonymous reviewers whose comments greatly improved this
manuscript. We acknowledge the financial support by the University of Graz regarding the
Conceptualization: Georg JaÈger, Manfred FuÈllsack.
Formal analysis: Georg JaÈger, Christian Hofer, Manfred FuÈllsack.
Investigation: Georg JaÈger.
Methodology: Georg JaÈger, Christian Hofer, Marie Kapeller, Manfred FuÈllsack.
Software: Georg JaÈger, Christian Hofer, Marie Kapeller, Manfred FuÈllsack.
Writing ± original draft: Georg JaÈger, Manfred FuÈllsack.
Writing ± review & editing: Georg JaÈger, Christian Hofer, Marie Kapeller, Manfred FuÈllsack.
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