Predicting Crashes in a Model of Evolving Networks
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Transcript of Predicting Crashes in a Model of Evolving Networks
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Predicting Crashes in a Model of Evolving
Networks
ANDREAS KRAUSE
University of Bath, School of Management, Bath BA2 7AY, United Kingdom
Received April 3, 2003; revised January 5, 2004; accepted January 5, 2004
We consider an evolving network of interacting species which exhibits self-organization. The system is charac-
terized by repeated crashes in which a large number of species are extinct and subsequent recoveries. We
investigate the macroscopic properties of this system before such crashes, concentrating on the variance of the
relative population sizes of species and its evolution over time. A simple score function is constructed to
determine the probability of a crash within a certain time interval to be used as a predictor for crashes. 2004
Wiley Periodicals, Inc. Complexity 9: 2430, 2004
Key Words: crash; prediction; self-organization; evolving networks
Many time series are characterized by rare but re-
peated large and sudden changes (crashes) with
subsequent slow reversals (recoveries). Examples of
such behavior include earthquakes, sand piles, the extinc-
tion of species in biological evolution, and chemical pro-
cesses as well as social systems, where stock markets have
been of particular interest. Empirical as well as theoretical
contributions in most cases focus on the distribution of the
size of such crashes and the waiting time between them.
However, for practical applications it would be of great
importance to find properties of these systems that allow to
predict the occurrence of crashes. Recently it has been
proposed that earthquakes are preceded by log-periodic
oscillations [1, 2]. A similar result has been reported for
crashes in stock markets [3], and the literature mentioned
therein, but the empirical evidence for these precursors in
stock market crashes is thus far not conclusive. Further-
more, the origin of these oscillations remains undetected
such that these precursors up to now lack a sound theoret-
ical foundation for the mechanism causing the crash.
In this article we consider a model of evolving networks
of interacting species as has been found to be useful for a
wide range of applications, e.g., in social systems [4 6],
evolutionary models [7, 8], or chemical processes [9]. Inves-
tigating the network structure of such a model [10], finds
that large crashes are the consequence of a structural
change in the network itself, called a core-shift. A further
result is that the network structure also affects the robust-
ness of the network to minor exogenous changes causing
crashes. In deriving these results it was necessary to know
the complete structure of the network at any point of time;
this we will call the microscopic properties of a network.
Correspondence to: Andreas Krause, E-mail: mnsak@
bath.ac.uk
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When investigating real networks such precise knowledge is
usually not obtainable. We therefore focus on aggregate
outcomes of the network structure, i.e., its macroscopic
properties, which are much more easily observed.
After briefly outlining the model, we establish that there
exists a close relationship between microscopic and macro-scopic properties of the network. We then use these mac-
roscopic properties to determine the probability of a crash
within a given time interval and then, finally, develop a
simple score function to predict the probability of a crash
within a certain time period.
1 . THE MOD EL
We describe a network of species by a directed graph withn
nodes representing the species, which is characterized by its
adjacency matrix Ai (atij) 0. The edges of the graph
represent any links between species. If there is a link orig-
inating from nodejand directly terminating at nodeiwe set
atij
1 andatij
0 otherwise. In a chemical process we caninterpret this situation as jbeing a catalyst for the produc-
tion ofior for biological species thatjincreases the chances
of i surviving. We will exclude self-catalytic processes by
settingatii 0.
A fast dynamical variable is given by the relative popu-
lation of the species, xti, where 0 xt
i 1 and thext
i sum up
to unity. This variable evolves according to the following
differential equation until it reaches its fixed point:
x ti
i1
n
atijxt
j xt
i k1
n
j1
n
atkjxt
j.
We only consider these fixed points in the further analysis.
Rather than investigating the dynamics of the relative pop-
ulation, we are here concerned with the dynamics of the
graph itself, serving as the slow dynamic variable.
After the fixed point is reached, the node with the least
populated species is extinct and replaced with a new species
by reassigningatik andat
ki for alli kto unity with proba-
bilityqand zero with probability 1 qand giving a small
random population 0 xtk 1 to the new species; the
relative populations are reweighed to ensure they sum up to
unity. It has to be noted that the fixed point of the differ-
ential equation does not depend on the initial conditions
but only on the adjacency matrixAt. After this change of the
graph (graph update), the relative population evolves again
according to the above equation. We define time as the
number of graph updates. The initial graph is chosen such
that for all i j atij1 with probabilityqand at
ij0 with
probability 1 q. This model has been introduced in Jain
and Krishna [17], based on the well-documented Bak-Snep-
pen model of evolution [11], and its properties have exten-
sively been investigated in the literature [1216], besides
others.
The number of populated species, i.e., those with posi-
tive relative populations, show repeated sudden crashes
followed by subsequent slow recoveries. The properties of
these crashes and recoveries are very well documented in a
large number of the before-mentioned articles and are the
topic of investigation here.In this article we use a simulation with n 100 species
and q 0.0025, which has become a standard parameter-
ization of this model in the literature. A crash is here defined
as a situation where in a single time period 60 or more
species are depopulated. This choice of 60 depopulated
species is arbitrary and aims only to capture the need for a
substantial fraction of species to become depopulated in
order to be classified as a crash. Simulations with other
thresholds gave qualitatively similar results to those pre-
sented below. In all we simulate 4,000,000 time periods,
which in our realization include 1450 crashes, or 0.03625%
of all observations.
2. MACROSCOPIC VS. MICROSCOPIC PROPERTIES
In the simple model outlined above, the only observable
variable that does not directly relate to the network struc-
ture and can be used to predict crashes is the distribution of
the species relative population. Obviously, the normaliza-
tion to unity prevents the mean to be informative, so that an
obvious choice would be to use the variance of the relative
population, which we define as
t2 Va rxt
i.
Apparently this can be observed without reference to the
details of the network structure; hence, it is a macroscopic
variable.
The network structure is represented by its adjacency
matrix, as mentioned above. The dynamic properties of the
network depend at least in part on its largest eigenvalue, t1,
as shown in Jain and Krishna [17]. Ift1 0, there are only
a few crashes and any changes in the number of populated
species are random. For t1 1 the number of populated
species tends either to grow, with the possibility of a few
smaller setbacks in this process, or all species are populated.
An eigenvalue 0 t1 1 cannot be observed in this model.
A crash is often, but not always, associated with a significant
change in the network structure, a core-shift, which mani-
fests itself in a significant change of the largest eigenvalue of
the adjacency matrix. In order to determine the eigenvalues
of the adjacency matrix, we need to know the entire network
structure; hence, it is a microscopic variable.
There is a strong relationship between the largest eigen-
value of the adjacency matrix and the variance of the rela-
tive population. Conducting a regression of this variance,
t2, and the eigenvalue we obtain with Dtbeing a dummy
variable that equals 1 for t01 and zero otherwise:
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t2 2.643 103 7.357 103Dt
1.795 1031 Dt t1,
R2 0.940.
Other specifications are consistent with this result and
show a similar goodness of fit, R2.
From the above result we can deduct that the variance of
the relative population depends on the largest eigenvalue of
the adjacency matrix. Hence macroscopic and microscopic
properties are very close substitutes, and we can use the
variance of the relative population as a good approximation
to reflect the network structure.
3. VARIANCE AND THE OCCURRENCE OF CRASHES
When investigating the behavior of the variance before a
crash, we see from Figure 1 that it decreases until approxi-
mately 150 time periods before a crash and then increases at
an accelerating rate until the crash. Consequently we ob-
serve that the growth rate of the variance, relative to 100
time periods earlier in order to eliminate any short term
fluctuations and defined as
FIGURE 1
Behavior of the median variance of the relative population and its growth rate (as defined in the text) before a crash.
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tt
2 t100
2
t1002 ,
becomes positive shortly before the crash and then in-
creases ever faster. These observations indicate that we can
use the variance and its growth rate as a predictor for
FIGURE 2
Probability of a crash as a function of the variance of the relative population (horizontal axis) and its growth rate (vertical axis). Note: The plots use a linear
interpolation between data points that are located at the crossing points of the grid.
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crashes. However, it has to be noticed that the confidence
intervals are very wide.
On the basis of our simulations, we investigated the
fraction of observations representing crashes occurring in
the time periods [t T; t T 100], where t denotes the
current time period, given that t2
Ikand its growth ratetJ1. The intervals are determined as follows: the variance is
divided into 18 equally wide intervalsIkbetween 0 and 2.5
104 and a final interval for all values exceeding this upper
limit. The growth rate is divided into 17 equally wide inter-
vals Jl between 1 and 3.5 as well as one interval for all
values below1 and one for all those 3.5.
We choose to divide the variance and its growth rate into
19 discrete sections as a compromise between a sufficient
sample size in each section and the accuracy of the values.
We determine for each k, l the fraction of crashes for this
parameter constellation over all 4,000,000 time periods and
denote this by pkl,T
. This fraction, given the size of the
simulation, we use as an estimator for the probability with
which a crash occurs in this time period, conditional upon
the variance and its growth rate being in the appropriate
intervals. Theex anteprobability of a crash in any of the 100
time periods is 0.036. From Figure 2 we see that the pre-
dictability of crashes with our variables is increased signif-
icantly.
It is obvious that a large fraction of crashes are associ-
ated with a high growth rate and until a forecasting period
of about 150 time periods the variance reduces for the
maximum probability, as mentioned before, and after this
increases again.
From inspection of these plots is becomes apparent that
the dependence ofpkl,Ton the variance and its growth rate
is very complex. It is beyond the scope of this article to give
a complete characterization of this structure; however, it
should be noted that in many constellations the number of
observations is relatively small. As Figure 3 shows, most
observations are concentrated in the lower left corner of the
plot. Only future research can determine whether the small
number of observations has caused this result or there are
indeed more complex dependencies to be found.
It is also apparent that combining the growth rate and
the variance increases the precision of the forecasts as can
be seen from comparing with Figure 4 which only uses a
single of these variables to predict crashes.
The analysis shows that for certain variances and growth
rates the probability of a crash is increased. But we thus far
only investigated a single time period in our analysis. Addi-
tional gains in predicting a crash should arise from observ-
ing the development of these variables over time, especially
the forecast errors should decrease. We will therefore in the
next section propose a score function to predict the occur-
rence of crashes.
4. THE SCORE FUNCTION
We here propose to use a weighted average of the pkl,T in
distances of 50 time periods, whereT 500, 450, , 50. Let
Tt502
Ik and Tt502
Je, then we know from the
previous section that the probability of a crash in [t 50;t
150] is pkl,T. Let us now for notational simplicity define
this probability bytT.This allows to predict whether a crash
happens in the time periods [t 50; t 150], i.e., in an
interval of 100 time periods starting in 50 time periods. Let
T/50, we then can define a score function as
St 1
10
t50.
Five different weights
are investigated:
1. Equal weights: 0.1,
2. Linear weights: /
1
10 ,
3. Quadratic weights: 2/
1
10 2 ,
4. Logarithmic weights: ln( 1)/ln(11!),
5. Decreasing weights: 1/
1
10 1.
On the basis of these score functions, we determine the
fraction of crashes in the time period [t 50;t 150], which
is shown in Figure 5(a). Obviously there is a positive relation-
ship between the score and the fraction of crashes. However,
forSt 0.1 the results for the various score functions diverge
and the relationship becomes less obvious. This observation
can be attributed to the small number of events with a high
score (see Figure 6), giving rise to wide error bands.
FIGURE 3
Relative distribution of observations for the variance and its growth rate.
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We also estimated the probability of observing a crash in
the time period [t 50;t 150] using a logit transformation:
ProbtCrashe01St
1 e01St,
where the sign of1 should be positive. The results of these
estimations are shown in Table 1 and visualized in Figure 5(b).
It is apparent that the most recent observations should receive
higher weights as this increases the goodness of fit, but the
improvement from equal to quadratic weights does not make
the choice of equal weights unjustifiable as an approximation,especially in light of the few observation with a high score.
Using such score functions as developed above, we get a
relatively reliable predictor for crashes in our model, which
improves substantially the ex-ante prediction of a crash,
which is 0.036.
5. CONCLUSIONS
We have shown first that the variance of the relative popu-
lation and the eigenvalue of the adjacency matrix are close
substitutes; hence, macroscopic and microscopic properties
of the network are interchangeable. We then continued to
investigate the relationship between the variance and the
probability of a crash occurring in a given time interval and
developed a score function to predict these crashes.It is observed that a short period before a crash the
variance is minimal; hence, the homogeneity of species with
respect to their relative population is high. This finding is
FIGURE 4
Probability of a crash depending on a single explanatory variable for different times before a crash.
FIGURE 5
The score function and the probability of a crash.
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consistent with the observation that in financial markets
before large changes in the market most investors agree on
the optimal investment strategy, i.e., they show a great
degree of homogeneity in their opinions and are thus easily
disturbed by minor events causing a crash. Initially only a
few dissidents warn about the overvaluation of stocks and a
possible crash, but they are mostly ignored. Their presence,
however, increases the heterogeneity until the change is
triggered by a minor change in the opinion of a single or few
investors.
How these results fit into empirical data in the stock
market, besides anecdotal evidence, and other disciplines,
e.g., the extinction of species in the evolutionary process or
chemical processes, has to be shown in future research. It
would be of further interest to investigate the behavior of
higher moments of the distribution of the relative popula-
tion, e.g., skewness or kurtosis, to develop more precise
predictors of crashes.
A note of caution should be made at this place regarding
the application to social sciences. Knowledge of a crash or
any other uncertain future event and the subsequent ac-
tions based on this knowledge can easily change the out-
come. Hence with knowledge of a coming crash we may not
observe a crash at all; it may be postponed or occur earlier.
This result arises from the fact that people will rationally
exploit their knowledge and so change the course of events.
The prediction of crashes in social systems is thus much
more complex than the simple model provided in this arti-
cle and has to take into account these additional aspects.
ACKNOWLEDGMENTS
The author acknowledges the beneficial remarks from the
anonymous reviewers. All remaining errors are the sole
responsibility of the author.
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FIGURE 6
Distribution of the score for equal weights.
T A BL E 1
OLS Parameter Estimates of the Logit Model Determining the Proba-
bility of a Crash
Equal Linear Quadratic Log Decreasing
0 5.631 5.392 5.223 5.502 5.525
1 46.76 40.04 35.90 43.00 48.46
R2 0.141 0.149 0.151 0.146 0.109
30 C O M P L E X I T Y 2004 Wiley Periodicals, Inc.