2.4 Power Laws
17
slow
fast
Fig. 2.4 Left: Typical shapes of clusters resulting from fast and slow particle aggregation. Right:
An urban agglomeration (a satellite view of Buenos Aires)
ture thickens before it has enough time to develop. Compare it with urban sprawl
(Fig. 2.4, right). Looking from above, we can see branches protruding along incoming roads, a signature of the built-up area expanding in directions of easier commute
and/or absorbing satellite towns and villages, while the older central core is compacted.
Is there a connection with planet formation? In the early stages, dust particles
were kept together by chemical adhesion, though they aggregated in quite a chaotic
environment. Compacting also followed aggregation during later stages when gravity became the major force. This is the reason why smaller heavenly bodies commonly have irregular shapes, while large planets are almost perfect slightly oblate
spheroids; the radius of the Earth is about 750 times greater than the height of Mount
Everest.
2.4 Power Laws
In all the above examples we see that aggregation leads to extreme inequality. There
is no law establishing the average size of a star or a planet or any aggregate or
agglomeration. Unlike a normal (Gaussian) distribution where the frequency falls
exponentially with growing deviation from the average, the frequency distributions
of this kind follow a power law going as f (x) ∼ x n with different exponents n > 1.
Both kinds of distribution of the frequency of an arbitrary variable x are plotted
in Fig. 2.5 (left). It is really hard to show them together, they are so unlike. The
power distribution diverges as x approaches zero, so that it should be cut off (or
replaced by another distribution) as x goes down to zero. At the other extreme, it
has a long tail, and for better clarity it should be plotted in logarithmic coordinates,
as has been done in the inset of the figure on the right-hand side. We see in this plot
that the power tail attenuates far more slowly than the exponential tail of the normal
17
slow
fast
Fig. 2.4 Left: Typical shapes of clusters resulting from fast and slow particle aggregation. Right:
An urban agglomeration (a satellite view of Buenos Aires)
ture thickens before it has enough time to develop. Compare it with urban sprawl
(Fig. 2.4, right). Looking from above, we can see branches protruding along incoming roads, a signature of the built-up area expanding in directions of easier commute
and/or absorbing satellite towns and villages, while the older central core is compacted.
Is there a connection with planet formation? In the early stages, dust particles
were kept together by chemical adhesion, though they aggregated in quite a chaotic
environment. Compacting also followed aggregation during later stages when gravity became the major force. This is the reason why smaller heavenly bodies commonly have irregular shapes, while large planets are almost perfect slightly oblate
spheroids; the radius of the Earth is about 750 times greater than the height of Mount
Everest.
2.4 Power Laws
In all the above examples we see that aggregation leads to extreme inequality. There
is no law establishing the average size of a star or a planet or any aggregate or
agglomeration. Unlike a normal (Gaussian) distribution where the frequency falls
exponentially with growing deviation from the average, the frequency distributions
of this kind follow a power law going as f (x) ∼ x n with different exponents n > 1.
Both kinds of distribution of the frequency of an arbitrary variable x are plotted
in Fig. 2.5 (left). It is really hard to show them together, they are so unlike. The
power distribution diverges as x approaches zero, so that it should be cut off (or
replaced by another distribution) as x goes down to zero. At the other extreme, it
has a long tail, and for better clarity it should be plotted in logarithmic coordinates,
as has been done in the inset of the figure on the right-hand side. We see in this plot
that the power tail attenuates far more slowly than the exponential tail of the normal