A novel use of power laws

Dr. J. Sylvan Katz j.s.katz at SUSSEX.AC.UK
Fri Jul 2 10:26:06 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 systems.

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

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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