Predicting Crashes in a Model of Evolving Networks

8/13/2019 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

24 C O M P L E X I T Y 2004 Wiley Periodicals, Inc., Vol. 9, No. 4


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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:

2004 Wiley Periodicals, Inc. C O M P L E X I T Y 25


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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.

26 C O M P L E X I T Y 2004 Wiley Periodicals, Inc.


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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


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