A novel use of power laws

Dr. J. Sylvan Katz j.s.katz at SUSSEX.AC.UK
Fri Jul 2 10:28:22 EDT 1999

[Complete reference and abstract below]
The universality of the powerful concept of self-similarity (power law)
in nature, social and economic systems (see Katz JS, "The self-similar
science system", Research Policy, 28 (1999) 501-517) has hardly been
explored experimentally in any great detail.  Many cases of power law
distributions signalling self-similarity have been examined but almost
without exception with the use of low precision data.  The authors
recently published paper analysing world records established by men in
running events.  Sport data was used because it gave the most precise
and extensive set of values which could found on non-linear dynamic

In the article "Power Laws and Athletic Performance" the authors showed
that plotting log (time) versus log (distance) of world record runs by
men in any one year resulted in a straight line with R^2 = 0.999.  This
was the case for records spanning a period of 70 years (1925-1995).
However, because of the precision of the data it was noted that the data
points did not scatter randomly about the lines but formed a pattern of
scatter which has remained effectively constant (to within about 15%
average variation) over the 70 year period. They show that this scatter
is due to the variation of the total energy available to the athlete
(aerobic and anaerobic) as a function of exertion time.

The most interesting part of this analysis was the degree to which the
data defined a straight line when these variations in energy were
accounted for. The running records during any one year now fell along a
line with mathematical precision of R^2 = 0.99999.  All 11 athletes
(running distances of 100m, 200m, 400m, 800m, 1km, 1.5km, 1mile, 2km,
3km, 5km, 10km) holding record run times during any one year could not
be distinguished from each other by the relative times of their runs, in
effect they were part of a self-similar set.  This has some interesting
physiological implications.

TITLE: Power Laws and Athletic Performance
AUTHOR: Katz JS and Katz L
JOURNAL: Journal of Sport Sciences, 1999, 17, 467-476
Document type: Article          Language: English       Cited
References: 26

In a previous study the authors showed that the 1992 men's world record
running times in the 100 metre to 200 kilometre could be represented
accurately by the equation T = cD^n where T is the record time for
distance D and c and n are positive constants. In this paper the study
is extended to cover the years 1925 to 1965 at 10 year intervals and
1970 to 1995 in 5 year intervals for event distances of 100 metre to 10
kilometre. Values of n for all years lie along a straight line with a
small negative slope. A regression analysis yields an equation for
values of n covering the period 1925-1995. Values of c from 1925 to 1970
 lie along a steep negatively sloping straight line. However, after 1970
the slope changes suddenly, resulting in another straight line with a
greatly reduced slope. The analysis provides two more equations.
Collectively, the three equations define two best fit plane surfaces in
<log(T), log(D), date> space for all men's world record runs over the 70
year period for distances of 100 metre to 10 kilometre. Also, the
authors had demonstrated previously that the actual event times, t, do
not scatter randomly with respect to the values of T but form a
consistent pattern about the straight line in a log(T) versus log(D)
plot.  In this paper it is shown that the pattern of  (t-T)/t  as a
function of date has
remained constant for the past 70 years. Implications of the slope
change in 1970 and this consistent scatter pattern are explored and
tentative conclusions presented.

Dr. J. Sylvan Katz
Senior Research Fellow
Science and Technology Policy Research, University of Sussex
Brighton, E. Sussex, UK, BN1 9RF
Tel: (01273) 877152 Fax: (01273)685865

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